Vector Mechanics for Engineers, Statics and Dynamics, 12th Edition Test Bank by Ace Test Banks

Vector Mechanics for Engineers, Statics and Dynamics, 12th Edition

By Ferdinand P. Beer , E. Russell Johnston , David Mazurek

Chapter 2 Computer Problems 2.C1 Write a computer program that can be used to determine the magnitude and direction of the resultant of n coplanar forces applied at a point A. Use this program to solve Probs. 2.32, 2.33, 2.35, and 2.38. Fi

θi θ1

θn

x A

Fig. P2.C1

2.C2 A load P is supported by two cables as shown. Write a computer pro-

gram that can be used to determine the tension in each cable for any given value of P and for values of θ ranging from θ1 = β − 90° to θ2 = 90° − α, using given increments ∆θ. Use this program to determine for the following three sets of numerical values (a) the tension in each cable for values of θ ranging from θ1 to θ2, (b) the value of θ for which the tension in the two cables is as small as possible, (c) the corresponding value of the tension:

(1) α = 35°, β = 75°, P = 400 lb, ∆θ = 5° (2) α = 50°, β = 30°, P = 600 lb, ∆θ = 10° (3) α = 40°, β = 60°, P = 250 lb, ∆θ = 5°

C θ

2.C3 An acrobat is walking on a tightrope of length L = 20.1 m attached to

supports A and B at a distance of 20.0 m from each other. The combined weight of the acrobat and his balancing pole is 800 N, and the friction between his shoes and the rope is large enough to prevent him from slipping. Neglecting the weight of the rope and any elastic deformation, write a computer program to calculate the deflection y and the tension in portions AC and BC of the rope for values of x from 0.5 m to 10.0 m using 0.5-m increments. From the data obtained, determine (a) the maximum deflection of the rope, (b) the maximum tension in the rope, (c) the smallest values of the tension in portions AC and BC of the rope. A

B α

Fig. P2.C2

y C

x 20.0 m

Fig. P2.C3

04_bee77102_Ch02_p001-002.indd 1

3/13/18 7:54 AM

2.C4 Write a computer program that can be used to determine the magni-

tude and direction of the resultant of n forces Fi, where i = 1, 2, . . . , n, that are applied at point A0 of coordinates x0, y0, and z0, knowing that the line of action of Fi passes through point Ai of coordinates xi, yi, and zi. Use this program to solve Probs. 2.93, 2.94, and 2.95.

A2(x2, y2, z2)

A1(x1, y1, z1)

F1 A0(x0, y0, z0) Fn

Fi z

An(xn, yn, zn)

Ai(xi, yi, zi)

Fig. P2.C4

2.C5 Three cables are attached at points A1, A2, and A3, respectively, and are

connected at point A0, to which a given load P is applied as shown. Write a computer program that can be used to determine the tension in each of the cables. Use this program to solve Probs. 2.102, 2.106, 2.107, and 2.115.

A2(x2, y2, z2)

A3(x3, y3, z3)

A1(x1, y1, z1) A0(x0, y0, z0) x

P AP(xP, yP, zP)

Fig. P2.C5

04_bee77102_Ch02_p001-002.indd 2

3/13/18 7:54 AM

I Z.Cz I

\ t ,t Fr,.,,,\ ar..- r'1

A^tD 0t,02,...0a FrND:

lrc-\I" |

n.,: tto ,#,, q A G N f r u D EA

D/Recrro^r oF ,fl

)

gY fono 3 svPPo(TFg

Twoch$LTsAJSffDuJN sEr o? FIND: FoREAcFr

iffissrowrl husro( "'-:-'-

-eV I

X \\

RtJUafC'Nrl'tlsrre |

\q.d\./L

Frverv;

l+ I

v,yrrfrll

D A r n 0 F P R ob 5 , | 7,32,2,33,1,,25,An/D | L,3D. I

vALu6-oF?ryoL?,=p-st -toori{o'-qrs,dr'ircern'

E N r SL O : (hl?AcHCASLE (4)TeNslot't (b) v*Lu€0r o @n uJ*{cH le{c

\{ .-"iV -o\p

Trt-{'roNfN'7t+e ckg(e

--l A=p- e - -t -- v- -5T6 ; 'h

= Z f ti c a 0 ; , L=:

'Y d = f , f i s- i n 0 & L--t

Lg.TeI

,Ar{r r ucH Trt+T

g^ . e=h6'3 -6-

rErrsroN (,)1,,. l ftfTb ) t ' b N D r H c r w _ z r , . - 1 < oP.=- ,q 4^of o 4T,a o 5 : Ao=5' l f( tr))X X:=3Sf ,f , Pl ?: 1=5715 1

.-.:,|fvnaisrs

Yftu)e "aanE9 av eo. (:J |i""ffi-q -qoo-< +10",

lF R^?o ArtD RSZo t

On-=O,

rc_fr2or+ilt fr,<o:

0o= sto"1fi ( f) I

tF en<ot

,-r-

Aul r.lulrgeRtrflRc'rs,rv,l'*' ENT:RPRosce^^iluMBER

Data

Ptob.

f trtv;2furY

lrc,utS,"^/

2'32

Nudberofforcee:N=3

-tu-

(+PA \P "t*''{

'-' o) ' (q+ i>Q\- cu._

0 t'\ I

l t n u t t rut ? i xt'".-v tF1 |

FKoetq|4 autRfi

,vc |

-TT--

h= rat+t*i+il

QuTtrttEaFPR06R.AM

rRrn^,612 FoRcE

or C

Trlatt, t'rr I i.^-

'Ac

_-'r*.

;m+D-

(ftLH f*< . T + srfnLl^ flE t/5 J fH ?ncc LckgLq rt)Lq

-.1

= 311.05 a""t.."

-l ll

Data of Prob . 2 .35 Numberofforcee:N=3

...

resulCant

ahd

x axi6

Data of prob. 2.38 NumberofforceE:N=3 Force and.angle (F, TH)? 50,20

F o r c ea n da n 6 t e ( r , r n l r a o , g s

F o r c e a n d a n-i r e i r , s r i i i i o l i R e s u l u a n t2oL,915

-a -i

/rvc(E4Evr56e

o, Tac,-ra5-, I ct+Ectr i-ug: ti+.at T*" *t'tt-rscfiA€Fq,lat h'rii T ^ t l AO- t t. | A - /( t . \ FDp' F'a<)' i

n'Nt4\}rt

I tQocearq ourerlr | I

Force and angle (F, Tn)? 200,215 Force and angle (F, TH)? 150,295 TH)? 100,325 tnt. Lve,r.J and angLe lForce orce anq alrgle t(F, t, = 266.55

AND +. . qU-X

| ffiJ#:H.-;#;'i::*#:16

ii{ iiiir eo,roo

54..931 ReaulEanE= Angle bervreen reFurranr and x axie

between

'TH U{t{Eru]14 =7.L.,-.., flT iS, vt *E,Ap-0: c{+O , g= F

=, riffi.:,";'?:i"3;,? " ;:::: :13llSi:[X;Tfi]l33;31"

Angle

'eoss|

fI\r

COH?UTE O,;(S-?O*

ffi;;;i;

Tgc

"*G:il'Efq+ol

oF ?(beRAM l:::::13:1313 ll: Tilili!;i9. | ou-r.LrNr iAngle : * i .betsween : I e F Ire8uLtant ; , j : i . * iand z r ix oaxie , u s= { |e4-69 f f i t deSrees{l voV&luElFa,P,qNo

;;;;

ts hg

5 t t { t tK l t? o s E r o L E

(a) f;,W t3:$" ft:2i;,,rr:2'!r't,: ltz) rsr ll'

ttrE rrl+vE R={ffi ,+r{r ^ * 8E T*+E

Y2-...1'r

| v

\\il

I | I

""'"""{l | | |

-Al iI |

Ansle between resurtant andxaxie! 33'23 aesreee<ll(b)

geENo.l

mgle ALPltlt = 35 ,lngle BETA = 75 Magnitude a!,srLuss vof r rload vaq 4O0 rP --

':::::'*"'_l:*=: THETA

-15.ooo _10.000 -5'000

TAB

TAE

_o.ooo 4oo.ooo 37.100 385.?89 73'9L7 36A'642

9 1 41s9..51818?:31296.. 90 9 s ..o9o9o9 1 8 3? 10.000 179.895 300.995

s> il,ili tii.ii| ii,i,ii| { 30,000 300.995 L79.896 35.000 40.000 45.000 50.000 55.000

326.083 348.689 368. 642 385.789 400.000

145.588 :.tO.L72 73 .9]-7 37.100 -0.000

2.C3continued

2.C2continued

fM, flNy l-,'

Set No.2 Angle ALPHA = 50 Angle BETA : 30 Magnit,ude of load P = 500 of THETA = L0 Increment TI{ETA

TAB

-0.000 105.795 2 0 8. 3 7 8 304.628 39L.622 -10.ooo 46G.7L7 0 . 0 0 0 527 .63L 1 0 . 0 0 0 572.5L3 20.000 600.000 3 0 . 0 0 0 609.256 40.000 600.000

-60.000 -50.000 -40.000 -30.000 -20.000

(b)@>

( b)b \ '-

TAB

-30.000 -0.000 22.L25 25.000 -20.000 44.o82 -15.000 55.703 -10.000 . 86.824 -5.000 Lo?.284 0.000 L26.928 5.000 L45.505 10. ooo t63.L76 .15.000 179.504 20.000 L94.465 25.000, 207.947 2 L 9. 8 4 6 30.000 230.o72 35.000 2 38.547 40.000 4.5.000 245.247 s0.000 250.000

Ac+cg = L

\fi:+b"- +t/14-rffi

TAC 500.000 6 0 9. 2 5 6 500.000 572.5L3 527.53L 466.7L7 39L .622 304 .628 208.378 L05.796 -0.000

(a-z)- + ? C at- 2,qr+,tY{'

2/ tm'

{

(c)

= I^t t z.qJL- &

4 t" ( z'+5') = (-{-'+ ?a7 -a")" {f-( - 4 L'x" 4L"1"=(/-o+2a2

( r)

?t='

eL ALso: ( : tac{'(#\

/3:h;;'ff)

\,d.'

' Set, No. 3 engle ALPHA = 40 Angle BETA = 50 Magnit,ude of load P = 25A Increment of THETA = 5 THETA

F . B , D r f t G " R A0rrt \ C l

i*v?'

70Rff TT td r-f. LE.'

(a+P) t-n,/ oF 5r vEs,' Tt, = Tac : u/

J-W ,frv

TAC 250.000 245.207 238 .547 230 .072 2L9.846 207 .947 1 , 9 4. 4 6 5 L79. 504

sfnf\

Jio{

-Ttlr s: Tr=r,v ;-f.,ffil Tor=ruffi;

ieg.iic 4C) 145.606 L26 .928 LO'l.284 86.824 65 .703 44 .082 22 . L25 -0..oqo

f /, i*l AgI:Lj ?_!f . PLo-6 ENTER Q= 2O n, /-='7A,I m, T0

FOA /=0,5 Co*1?u1e $ -t

C n tt P u ' rE C0l4 PuTc

(e)

lO,l

uitr.f6

Sin(4*

(s)

tV= BOO^/ O,5 sT E?5

FQortrfA.(r) FRoi.lFA5.(Z) A uof

Tsa A N D

Tsc trco^1 .51,ls.(3)

our PU-' BSoOnr-rnn

20.0 nr

-G r v E h l ;

A F V 6 Tt + O F R O P E= t - : 2 D , l n D I S I A N C FS e T W € P U A n u s I = Q , = ? O n t+T 0F ACRu$f e tlD 5A unrJct tJG eoLi'= W. BOOt'l Vr/Et6 loil f\ ssv t4E Nu strTn v a AND No E L*srrc DEF0prtlD-r 0F n,nPE F - INP:

XY 0.5 1.0 1.5 , 2.0 2.s 3 .0 3. s 4.0 4.5 5.0 s. s 6.0 6. s 7.0 7.5 8.0 8.5 9.0 9.5 10.0

TAC TBC 0.327 1426.03 LL93.97 0.446 1867.,33 L706.L4 0.534 2205 . 66 2.07I .69 2482 .42 0.605 2377 .4 8 27L7.45 2627.65 0.666 ' 0 .7r.8 29L9.79 2842.03 0.754 3096 . 14 302e .22 0 . 8 0 3 , 3250 .74 3191 . 13 0.838 3386.57 3334.19 0.859 3505.79 3459.85 0.895 3610.06 3559.95 0.919 3700.6s 3665.90 0..939 3778.51 3748.76 0.955 384.4.,45 39X9.40 0.970 3899.06 3878.49 0.981 3942,8L 3925.55 0. 990 3976.44 - 3953,.9s 0.995 3999.07 3991..05 1.000 40L2.00 4009.ot, 1.001 40L4.97 +O]-+.91

@,) lurAxtt|vt"lDe F L EcT,orN".Un = LO}l ftt

A L u zIsr(b) ^ N D T F { S I o x s}, r l ? vvALuEs I4AX|MOM4 " r E N s l o N DEFr€crreN6 ^ND-rE{sruxs 4 . 4 NANDTBc e T ^ , tuR | l ; M n X ' ^ 4 D rTENSI0N: -T -'T - b,O I K,V 0 F , ? R o t 40 , 5 m ' t o t 0 , 0 f i l , r l s r N i Q i - m - i * c t e N e r . r rIl 'Rc- 'g'c' F|P L:[g,0ra: FRuM DftrA og TA I r.rEg A LV Ff N' (e) tl44xlmur\A-DEFLEC-ir bN cr Ro eg ( b ) t t 4 4 x t r L t u pTt E N s l o N l N R O f F

C ) 5 ^ ' tA L L T S TV A L U E S o F T + . A r v D b

s v A L L E s Tv A L u E so F T n c P s / oU .

continued

f o n { = o , 5 n:

TAr=1,4?,6AN -T-

l8c-

l , F t +( t /

{

A2\x2,VZ, zZ)

\ \

2.C4continued

A1(r1,Vy zt)

PAOGRAr.4 0o-T?UT

;\ A,,( r n, !n,2,1)

Ai(r1,y;, ai)

FogcEs st+t)wNl / AITTNG oN Ao P Tl+F 'r"lftGfi/,TuOt rFQrttlr"/E To DZ, t T ( R o e R q N \ WR * v > D f R F c T r o N O F 7 r { 8 , R R F5 ULTANT A P ? L Y P Q O & €f i M 70 sc,LvE PQue'6' 2 ,q 3t A, 1 1 ,z . , q r ,* v D 2 , l 3 r ,

Data of Prob . 2.93 of A0 ? -4A,45,0 Qoordinates N = 2 for c es : N um ber of gi v en Magnitude of F (1) ? 425 of A (1) ? 0, 0, 50 Coordinates Magnitude of F(2) ? 510 Coordinates of A(2)? 50;0,60 R = 9I2 .92 48.2, THY = l-15.6, THZ = THX =

53 '4

Data of Prob . 2 .94 of A0 ? -40 ,45 ' 0 Coordinates N = 2 for c es : N um ber of gi v en Magnitude of F(1) ? 510 Coordinates of A(1)? 0,0,50 Maqnitude of F (21 ? 425 Co5rdinaLes of A(21? 60;0,50 R = 9L2.92 50.5, THY = LL7.6, THZ = THX =

5l-.8

E{inil Y '

ANftLY 5 rs P t q 5 T , F \ R € q c t + F O Q C E, F'; W e D e T E R t - I t u € T t + t D t5ih+c R eL; FRom Ao T; ^, ..

d'; = -Tt+E

.o t4por.lE f{Ts AF F, ff qt

?-& qW, (E);= G,I=ri ff , (5),= rr; f ,

(.2)

.r-tL

T{tE

cDr"l?ouEur5 oF Tf+E R7S'L-TArVT f7

ARE

=*,G)t = (t),, Rg K.=*, (-'l); , Ro, t, Tt+E {Y r+&N r70 DE 0 F Tt+E R E S U L T A r uT

R=/rc],r R i n R z h f ' , l bl r s

J(REcTroN

(3 )

Data of Prob . 2 .95 of A0 ? 480 ,0, 500 Coordinates N -= 2 for c es : N um ber of gi v en 385 Magnitude of r(1)? CoordinaLes of A(1)? 0,510,280 Magnit,ude of F(2) ? 385 Coordinates of A(2) ? 2IA ,4 00 , 0

l"* ]u1;3]r, rHY =

s 2 . s , r H Z= 1 2 , 8 . 0

{

(+)

cosrrvEi RPE

Ar= Rn/R, t= %/rc, A*= Rn/g T *E kN6 (FJ Tr+ft7 8

(s)

FoQtt4sv,ltTH Tt+€ AxE:5 ftQE

0*= co;'2* ,, 0J= coi'U , 0r= @s-'n, gE wI+ERI Vt+LvE5I eTwmt O ft,w l?Oosn>ut>

(s)

f LtcTED

Q U T L I N EO r P q o E R A I V I

Dat,a of Prdb . 2 .r35 Coordinates of A0 ? 0,0,0 Number of given forces: N - 2 Magnitude of F (1) ? l-0 Coordinates of A(1) ? -15.5885 ,:-5,12 Magnitude of F(2) ? 7.5 Coordinates of A(2) ? -15.5885,18. 6, -L5 R = 15.13 THX = 133 .4, TI{Y = 43 .6 , THZ = 85.6

Er\neR P(.oBr€M f.fuf"\BeR r NTER Lo6R-DrrJfl?f5 h, to ,?o 0F furxt Ao ENTF.R, Nt)MBte, 0F Fo (ctrJ 4v F?q rAa+ FDRcE f ;: ?NTEQ f4ft GN tTu D tr T; ENTsq CcloRprrVArFS z.; ,A t ,aL o F Po ru1 f,L c o t t - t P u f E d ; F R o m " d r n : , ( t, ) \

c o f 4 ? u r(tD ; , ( 1 ) r , ( i ) , F e o q E o s ,( z ) c6H?UrE n*, Rd q Faup{ r 05, , 3 ft) couPuTe ( FRot( €rq.(i c o q p u r y O . , 6 d ,D +f R o t l E a s . ( t * M D( d lF

Yoa IBTA1N A nt GRztve vALt)€ ?aF ANy aT

-rt+e trile LEs,A?D /?rf Ta rHnT vn LvE'

2.C5continued

2.C5

OUJt-tNe OF ?A.DGt\trY ENTYK' PAoetLeq dr-r'tt87P gYtvR CooRotNftr.-:s oF Porrrlr A, grurrR, i4AGvtT UDe oF u)Ao F ENT eR aoo &DtrvAzrS o F Ap , ht ,* z, frN6 As 1rSE2,Os,(r) *ruo Q) TD coMe:urE bouzoilEyTs oF ! use FQs.(z) ftN> $) To cot?vrE A*, At, AL flP f*c* CngLe t o t 4 P r t r t A ? R o Mf A . ( 6 ) A N o b , , L r , b , ._1 co rqPr) 16 F, , ?, , Tj FQDq fus. (7) *ND ?R.tr',1-

\ Ap{rp, p, zp) a

GttltN: REwIA.R<s : T f t q e E L f t h L e s t + T T A c t 4 6 gA r ? 6 , N T A o * N D F o R c E w t D T r L . l l 3 /

P + P P ( / e D f t T ' A D A S s + / o rrv. t. 7 o o E T E R . l 4 r v FT F ^ J s r o MF , ilUeite " e o * o o t i@pr6L:.,(t= t,zt3). P R o a ; . A M APP(Y l ! r?,, 2 T 0 5 6 L v EP R g 6 5 . , l O Z ) 2 .l O 6 , 2 , l O 42 , , 1 ( 3 f r t v D A MALY5 ,5 F\RST D eTE R

d- =/(., -xrf + (Ur-Jofr(?p- t),

oOretrr Arro Ap, P R o a r a R M

(t>

Tf+e a>M(ntrverurs oF P '+R E

=-frQ;Ur\, Pn= 1=f;{',-'s (z) 5f*r-rru), 7,

Nar 9€TIRnt,^/FF0e EncftcaBLqAoA, (i:t,2,3) D,sr+rucF dL=-,AoALffiD Tt+E P(RFcrtoN r*t cos rnl€5 (Vr)t, (Agi;, ( i-)i"': Tftq Drr.Ecr(oN

il/e-,UoWnrRfrr T*e.r&.utLtBRro^'l fOunTtorrj.fFoR ftor

P*=0, =e o,n,(E),*P* f,F),*E= f6sr1ft,* trr) Trr/ 6 (\)i FAorr{(S) nop

su _FoR_([f; , T4nr.rgFE RF.tN6 P*, p6 ,-?+ i

(v), Fr+(2r),F, * (?r).

, (F*)i

E =

- Pz

Tfi e

DeT€Rnr | ^/ANr

(2J, (xn). (a,'),

a= llBl?3.t?l; a

,!, f f f D T - * e D t T r R r g rt v ^N ? 5 A t , A . ,

Su8sft T.rrrN6 -Pl etrr{EN7s

oF a,

(d o B T * rt J e o O Y

-P,

Suc-cEsSfrrF[Y FoR,Tr+C ,-PJ, k{{9 Ttltqo Coluplil rf+E FIRSlSE*N.Dr 0F

u/E AgrftfY

2 . 1 02 D at6 of Pr ob of po nt A0 ? 0 , 5 . 5 , 0 Coordinates P= 800 of load Magnibude AP ? 0 , l _ 0 ,0 of po nt 'AlC oor di nates ? -4.2,0,0 of po nt C oor di nates o f p o i nt A'2 ? 2 . 4 , 0 , 4 . 2 Coordinates of poi nt A3 ? 0 , 0 , - 3 . 3 C oor di nates F3 = 415.5 F2=3 7L.7 F1 = 200.9

Data of Prob . 2 .L05 of poi nL A0 ? 0 , - 5 o , 0 C oor di nates P = 15 0 0 of l oad: M agni tude AP ? 0 , - 1 0 0 , 0 of poi nt C oor di nates A1 ? - 3 6 , o , - 2 7 of poi nt C oor di nates A2 ? 0 , 0 , 3 2 of poi nt C oor di nates poi nt A3 ? 4 0 , 0 , - 2 7 of C oor di naLee F3 = s27.s F2 = 829.8 FL = 570.9

{

Data of Prob . 2.:'07 A0 ? 950,240,A of point Coordinates P = 305 Magnitude of load: Coordinates of point AP ? 0,950, -22A A1 ? 1200,244,0 of point Coordinates Coordinates of point, A2 ? 0,0,380 A3 ? 0, 0, '320 of point Coordinates Fr. = e6o. o (pz = ++e .t) (tt = aar. z)

\. Data of Prob. 2 . 1L3 C oor di nates of poi nt, A0 ? 0 , 1 0 0 , 0 M agni tude P = 1 80 0 of l oad: C oor di nates of poi nL AP ? 0 , 2 o o , o C oor di nates of poi nt, Al ? - 2 0 , 0 , 2 5 C oor di nates of poi nt A2 ? 5 0 , 0 , l - 8 C oor di nates of poi nt A3 ? - 2 0 , 0 , - 7 4 . F1 = 973..6 F2 = 531.0 F3 = 532 .6

( b), q +(AirF"* (fu),E !=- pe ( 2 r )I ,+ ( ? * ) , f + ( t r h ; % / l r / 7 R oo d c f N G

f t N o ' 2 , 1 t 5 , c H o o s E F o RA ? ,

' k N Y ? a r N 7 D r Re c t u ' . A 6 o v E h ANY R:lrur DtRrcrrv rlrofP 4 t N PRo3. 2,,tOd,Ct+losl FlR Ap fN PR08-Z,/o7 hr{orDtrR P r+S Tt+E Tr1't1t1Nn N A FtcTtrr|US chSt-e "PftR*Lt?L To TttE 7( t+llS kNO 6t*?lsSg ruA A t p.N7 ?otN7 O tRtClLY '70 TkE R,GrtT oF A, 4 L5o c,>r\/sroffi 'rfrq TeNeronl rrttftb As fl+E Att)€N LZAD'

F,=+, rz=F E=f

Daca of Prob. 2 . l-Ls C oor di nates A0 ? 0 , 4 8 0 , 0 of poi nt M agni tude of l oad: P = 792 Coordinates o'f point AP ? 0 , 6 0 0 , 0 C oor di nates of poi nt, AL ? - 3 2 0 , 0 , 3 5 0 C oor di nates of poi nt A2 ? 4 5 0 , 0 , 3 5 0 of point A3 ? 2 5 A , 0 , - 3 5 0 Coordinat,es FL = 5L0.0 F3 = 535.3 FZ = 56.2

{

(z')

Chapter 3 Computer Problems 3.C1 A beam AB is subjected to several vertical forces as shown. Write a computer program that can be used to determine the magnitude of the resultant of the forces and the distance xC to point C, the point where the line of action of the resultant intersects AB. Use this program to solve (a) Sample Prob. 3.8c, (b) Prob. 3.106a. 3.C2 Write a computer program that can be used to determine the magnitude and the point of application of the resultant of the vertical forces P1, P2, . . . , Pn that act at points A1, A2, . . . , An that are located in the xz plane. Use this program to solve (a) Sample Prob. 3.11, (b) Prob. 3.127, (c) Prob. 3.129.

Fig. P3.C1

y P2 Pn

A2 z

Fig. P3.C2

3.C3 A friend asks for your help in designing flower planter boxes. The

boxes are to have 4, 5, 6, or 8 sides, which are to tilt outward at 10°, 20°, or 30°. Write a computer program that can be used to determine the bevel angle α for each of the twelve planter designs. (Hint: The bevel angle is equal to one-half of the angle formed by the inward normals of two adjacent sides.) 3.C4 The manufacturer of a spool for hoses wants to determine the moment of the force F about the axis AA′. The magnitude of the force, in newtons, is defined by the relation F = 300(1 − x/L), where x is the length of hose wound on the 0.6-m-diameter drum and L is the total length of the hose. Write a computer program that can be used to calculate the required moment for a hose 30 m long and 50 mm in diameter. Beginning with x = 0, compute the moment after every revolution of the drum until the hose is wound on the drum.

250 mm

Fig. P3.C3 1.0 m

Fig. P3.C4

125 mm

05_bee77102_Ch03_p003-004.indd 3

3/13/18 9:34 AM

3.C5 A body is acted upon by a system of n forces. Write a computer program that can be used to calculate the equivalent force-couple system at the origin of the coordinate axes and to determine, if the equivalent force and the equivalent couple are orthogonal, the magnitude and the point of application in the xz plane of the resultant of the original force system. Use this program to solve (a) Prob. 3.113, (b) Prob. 3.120, (c) Prob. 3.127.

y F1 r2

r1 O z

Fig. P3.C5

Fn x

3.C6 Two cylindrical ducts, AB and CD, enter a room through two parallel

walls. The centerlines of the ducts are parallel to each other but are not perpendicular to the walls. The ducts are to be connected by two flexible elbows and a straight center portion. Write a computer program that can be used to determine the lengths of AB and CD that minimize the distance between the axis of the straight portion and a thermometer mounted on the wall at E. Assume that the elbows are of negligible length and that AB and CD have centerlines defined by λAB = (7i − 4j + 4k)/9 and λCD = (−7i + 4j − 4k)/9 and can vary in length from 9 in. to 36 in. y 120 in. 90 in.

4 in. A

λAB E 96 in.

λ CD

52 in.

C 36 in.

100 in.

Fig. P3.C6

05_bee77102_Ch03_p003-004.indd 4

3/13/18 9:34 AM

Bemn

SroBresret>

5YsTET"t oF

VER\\RNL

GgU,

ggqX6(r {.S.,L". oR b

SRC€\

Re.rruTnt\fi

Frx*r,

FoR€.

BoKc.s

GwE*,

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:Ab€s

\xc\Cs oF rd, aSi

Ah.rb n+E b\tnNee Kc- \O trS PDIN\ bF APPLTCATT$N

gs"

oR Frs.rbr

Beveu

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eAC!\

trt-srN€.R. OutuNre

€ox

?RpcreAl4

lnfn-:.t

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LeNGTT\

OF Nut-lBeR gllS-t\ FsRcEF

lpr>uT Foc.

\t p..tvstt

\\s vrruuE ls.rrr oeonne. REsrr\5rA\f,\R: \T5

IH*S

R=R+F

PDS\T\\I

rJesnneLM^ .. l\ = LM* +, xF csvte.r1E xc-i Xc-= tM" / n ?ruslr

Peoqe**

vALues

o*;t"*

(?*oeuErnb: IP t, Bc- a. rot,q. )

ANb

p- trr:r Ar{c,LE

(a) R = - 600.0 R e su l ta n t 3.13 m Xc =

(b) ll = Resultant Xc := 39. 56 in.

-tL.?

N - N$MBCTR.

sr\ES Hxve-. \ r = srrll$J + cosFb

+=(TI-H.I- T T\

z1\

2-N

Grv€H:

Sv:rem

\/TRT\CAL

FSRC€\

NPPtr€b

rN Tl+€

X2Frsr\:

co,p*.!?- H)i t s,NF)* coEF''Sq- Hl U

P\AN€-

Re:.lwF\s.rT

P'oINT

\\S

AN\>

\PPI-\CNXION

= cj)ap srtrrR *spco:F \rNF R UB fi i + F )j i * cbsF.o: I \+\ N t\ BY \, ANb \z Trre Xtrx'ue soRvtEb .ANGLE 'TI-\E X T\T\CE BE\€L

E\oAL To

r'. cDb Zo( = \, ' \r. l,.,P.rr

UNfiS

lsrP.rf

NI\$nB,C.R e.Rs$

Foe

I*lp*rt

VNLI>E-

R.ESToUTHHT R:

OroPSE

THe. PsrhtT

FrSRcE

FoRCES tr

FoRgE \TA

l+.rPuf

ANb

LENG\A bF

=(o1csp5,Nt)+ (us$)(srruF)*(cssFXcsF*=R) = s,nlt$+ cu>zpc"sffi

Q = R+ F

cDoRbtNNTea oF'

(X.e)

rTS

NPR\CNTrsxl

+ xF tJeblse LM. : L['\a = l\ tJeuq1e LM*: LMx INtx + 1F gooRbrNr\rE5 (Xn".tn ) oF Tt+E ForuT Gmp.rre Otr

NPP\-\CAS\O\}

vNLvES

; = i. coS'( =*'p +.o='F.osH I

?e\h.rr

vALuES

t{f

xtrtb

oA,

CF R:

Xn = EMI/R P*rsrl

Foe. 'N s\bEt : N\ = 4. S, t, q c c Frcc. Ttur ANGLE: p: F = tO, Zs. js Foe. E\c\\ Cro1l1ts\$,.lArr\SNsf$ N rtsS p At\K>Le 0(: CowrPsRE. bEvEL -=

? R .= L t ( * / R .r(.R,tR AN\

NTIMBER OF SIDES

ANGLE

BEVEL N{GLE

10' 2t;^" 30"

4 e. l 4 " 4L.64' g z. 7 6 "

1 0"

3 s. 3 2 '

ro' zo"

zg.so" zg.oz"

30"

25.66"

' ( FgoBuems::P 3. tlr 3.lzl, i. \29) (a) -8O.0 kips Resultant R = z = At x = 3.500 ft

TILT

3 . O O Of t

3.048 m correspond to (c) Let the y coordinate 405.0 Ib Resultant R = , z = -2.938 ft At x = 12.599 ft

10"

22 .L4"

20" 30'

21.09" 19.3s"

3.C4 CONTINUED Denrm oF

DtN

O,\r-m

oF _ ncg*.xla OrJTurnrE Cr>nqpu-riE flfi$1$! r (S)

oF Erycg Ro\^l S: S= t, ?,3

c(s) : I o.iz€ *(s-r)(E*t){3l Row

?grr.n

1.0m

\ 5 T't+€ LEnCiTlt OF l{o:E hfouNb oN -nrE bRvrxt j drnue = SrJ tntrn Mlx, BCC$NNrr.lG fu1!,

N$lnRER

Rs\l

\: VALuE

Cos,apufe

otr

Cpvrn>-rg T\\E

LgN<jTt\ oF bRur4 I

K(r\ = Zft r(t) X (s\ = X (N-$ + Lrf r(r) v\clwrENlT

Mtxr) --c(r)

Fre:n NcrTE T\+NT Tr+e !./lNxw\\)rv\ nunvrBFR sF c$rL\ \b rN A RcNl GrvExt tsY = zis vnnn = 'J Cf*oEE ASS|.)MIE

bRuvl

Tr+Aff

T!,tE

mrn

!\stE

X tN)

x(o) = o

Auxuv:rs

Now

h\sse

NFrER eAcH

C.cnrrpote

vrrsr;t r\

srs TRE

st+bwN.

CSIL N: \r\:Q-.-r4

rr\

REvot-r-rn,$l bF nrE bRu4

€S

rcF E-\q\\

t" (rrr\ = t). oS (Z- s)

At.ts rt'\€!.t

AS X=o 125mm

Foe.

$"'

MCxr)

\Nsuts\

N= 2.,314

trcR. E-\c\\

-tr-1.(,#ioot\-Stl

ttz\ts) oF N NN\ VNLuE\ ?ernr z: F'c'e R'o\l C,ovtp'J'iE yN.Lr.rE Otr ee DF El\C\\ cS\L N\ 1 N = s . , - . - . ,B rn 4* (trt\ = O,os ( N\- t-.s)

*R\*ff* T\+E

oS fHE

Er.l\

rxr\rcNr€t>

\\s:F- x(t{) \rb$N\

CotL

T!+e

LF-NGTH

T\+E_ R.r\brUS

t+EL\c,\L-

T*g ts

SI+APE

T\+Nr

corLS

STNTTI

\SSUMS\ N Crs\L oF

LT\f,

GF- lrrE rN

ONE

\AJr{ERe

corL

(TF+$sr

Tt{E

crDrLS

raO\l

NN\ gta.1,\l

Tt+E

Nf=f,T

r?C\N

I+NVE

R.t\.N\

RAbn

TbtE

o\rRR

Rs\Ns,

L-ENGT$

HssE

txtoro11t\

r-HE\l__

r a€

E\\

TFIE

Rsg

CO}4PLETEUV T${Rb

tr o\t

rUor.lU\ \r

C.SVTFLETEJ>,

bRuVt

BESRE

Is - K(N\ CsvrR-yte

+ L\\ r(j)

> 30 \

Tr\Er{ v\ovrENT Mtu\

E\r\ FsR. gpg$

VKNI\=r(b\ $aruT

= $. ZSl."rn ) a'i-1 O.oSG)vn Row 3 1 Lj = S x Z1$). 3* - tz.9jt rn Idrr*s. T*enesoRq THE FtsrE (L= 3o yy$ \AnLL eE 'r*€

ctr r+sse-x.ru)\rsuu\ o\\

a.gl\ffi:* ,K(ts\ = X(\J-\

A GtvEtrl

*'i.?itl.* o+..qiFF)* 15t{

te(x) = o.oS (\\- s).

rCrUsieer$.,

E"**.

Row l: Lr = 3* LT\(o.3*\L1*

3oo'- *11

R.rxr vNLuES oF N NN\ M tu\ FoR. Rorru 3: csn*pore, vNLu€S str ?" oF gpCH cs\L N'. \\ :9, to, _._

TF+N\

r*€

X (N) = X(N-r) + zn f (Z) Qosnp.rc Mo$4ENT MtS) FoR. E-NSH q$rL'.

\'\tsl\ = c(L\

\\brTroxt,

q,b\L:

N = O, l,--- \EUoTqS

N-m:

It,.r

?r<sgRAst

.TbIE

vf\Luel

c\\L'..

\tosfr-S]l sF

Ah$b

\1tn\

o$TPoT REVOLUTION

UOMENT, l{. n

o 1 2 3 4

96.96 90.74 g4.23 77.48 70 .56

5 6 7 8

7L.65 53.34 54.82 46.L7

9 10 11 L2

40.87 30.43 19.83 9 .L7

Flxve-ANb

\1.1 8oo.6

lrret

uFbAl By Pers5 r " Agt

5YbTG.r{ otr N

tsR(E : EouvxLENT CD\)PLE S\STErrt filT oR\Grt\

\+e

Eoe1yr1.a*

S., Tt+E

A.r{b \T3

fesrs-T otr

coh4pcNEr*ns

Ngo\f; / THE- oR\GrN

MDMENT

C*o

(\)

..THE

ARE

; THEREToRE, REb$c€b Ts FoRCE. "

OR\SINAL

Tr-tNr Tt\E

S\NGI.E

ftswgrtex,

troRcE\

EQorvAt-El-lT

SY:fiEM

C5UPLE

Tr+E

ANb NCiT

THEy N SINGLE

CAN

E\UTVN.Ehf\ \s

PNRALLEL

ARE -FOR(E-, REI>\)CEb SORCE- B. Ts

Tl+E

PLANE... Enrb

(z\ On(s) = o oR. Sats\=rgS + Tue FoRcE: (r\ lEEd

NRE

NPr{rQb

oN T\AE

Y\r

OF VAUU€ ?srt..rt Er.rt> (4) ?nrtrr, " THE ErosrrvA\-ENT trsRcE nsth: -tr\e couR-E- NRE Er\)wA\.trfr oRTRcGDNNL; THUS, TI\EY c\N Be RSUNNLQTIT REb\X€h, TC A STXKTT-E-

FIRCE B. cooRbrp$irES (X-a) sF Tt\E Colneu-r€ Str hPPL\c-FSroXS OF B. P\)\hST ?nrur

r=-H {=H '( ot NNb Z \NLuES

)

for c e: (-L47.7 Ib) i + (-758.4 Ib)j + (0,0 lb)k Equi v al ent c oupl e: + ( -0.0 Ib.ft)j+ M = (0.0 lb.ft)i l-7232.5 Ib.ft)k i n the x y pl ane s o t h a t T he or i gi nal for c e s y s tem l i es to a force-couple system can be reduced the equivalent for c e R. egui v al ent s i ngl e T he l i ne the r i ght

Sb TSAfi \TS LINE Otr XT PLANE qgftsN tNliEJR.tECS -TI\FS \^l\LL NoT

FORCE K.

(Feoeue.nns ', 3, \\j, j. tzs, j.\z'l

C=o \Nb brtst = 9o" =) fue FoRcEs Ecsr ARE ppfitr1-1-€.1 To rr+E Pu\HE-, xe RruTt

EQ\)TVAL€NT \^/RERe

ctN{

s\NGLe

Be_ Reb\x$}5

Tv\e s\sT€!v\

Ess

Enrb Te

bo T\Nf

FoRcE-- cO$"Le

[= Tq

oF TF\E Yo :

F,oRqE

QSUPLE

onrrf"c'oNAL

x.*1 P\ANE

(a) Egui v al ent ' 'R =

ErbrorVALEhfT

cAt rucir BE ESurvNLE\n

THE

K \xrs

CNSgs:

EQUTVA\.ENfT

T\+e

F,cR(€. bYSTeh4 L\E\

oRrqrNAL

(osanri'e

'tT*\E

Pe.rUtl

?Utlr't€:

lHg

ETSOWN-EI.fT

to*ol'=iI,J:RT.-i.il_ B ANb so PssStF5LE

TrA6- X1

P<rst:

Mx = Mx * Fts)tt($ ccbs.ts) - A(\\ c,:sD.ts\l Iv\\ =M\+ Ft:tl(s) cosb*t:)- r((S)q-bsSrC\)J M.' 't = V\, + Fts\[ x(\) co\ts) - yt\) cos$*cs\l Rarsr , B ='Rx\+ Rr! * RaE So = \'4x).*M.r!*,M.8

(pu:rr>ge.

B. (K, e)

ExrS (3) D*ts) = goo \su Z($ = O $Tr+t FoR(EsECSI

Tr\E :CALAR,. clcMpoxr€tfis eF Tr{E ogsxiE ReSr,r\SNxlT soRCE R.: Rn= Rx * Fts\ cosD*ts) R..,fI Rt * Fcs) crcs$\ts\ Re = Rr t Fcs) crDsbrts) .OF T|F\E Cor4hNeNT: TL+E :fA\.AR fsru"r (s): X.(s\r R>>5rrcn.r VEctoR C \(s\r a tS) scAu\R

troRcE

cooRbrN,S€,S

L\E

T!\e

:Y:fiE}4

6 .stNGL€

sS R n.PPL\ Ct\i\rsxl ', M5 c=--I\ 'oF x ANS t YALrr€S

?irrrT

PIAUE

FsRgCuN\Tb oF LEN(fT\\ r\r\ Isv.r^r NuMtseR (S) oF FoRceS ItrRrT Foa EACH ForrcE F c:\ srlN(rt\t\Tub€ \\5 INi\'r Inrp.rl Tt{E \r.!GLeS S^ts), Dats\,, \(:)

OPuF$E

PodNn oF y\ == {€rt

I\FP\-\CNTISTT}\N TT+€

\ FsRge- c-sppLe

tGs>wNLeN\ Ccr.nPute

QrauVA\.E\rt pursrgr-E) trcfi<cE- (rs

FoRgG.b l\RE

Nx\5 \ T\F\usr T\r.e

\\e

Rqbr>c€b

CANI Be

STNGLE

Kt.

oF T\AE oR\qrNAL

PgsqAL\-EL To

troRC.eS

PAR.A.LLEL To TF\E 1 Aa\S, ( coxl-trNrreb)

of R i nter s ec ts of, ac ti on of the or i gi n.

the

x ax i ,s

9.54

(b) Eguivalent ft.= Egui v al ent M =

force,: (0.0 N)i + (-419..8N)j +,(-339.4 N)k c oupl e: ( 1 . 1 N ' m ) i + ( 1 6 3 . 9 N . m )j + ( - 1 0 9 . 9 N . m ) k are not force couple The eguivalent and the eguivalent to a single they cannot be reduced orthogonal; therefore, force. eguivalent (c)

Equivalent force: R = ( 0 . 0 N l i + ( - 1 0 3 5 . 0N )j + ( 0 . 0 N ) k Eguivalent couple: j ) + ( - 2 6 6 5 . 0 N . m )k M ? ( 3 1 5 5 . 0 N . m )i + ( 0 . 0 N . , , m forces are parallel Alt of the original to the y axis; force-couple system can be reduced thus, the equivalent to a single eguivalent force R. The point plane is x =

where the line

of action

2.57 m

z =

of R intersects

the xz

3.05 m.

GUEU, t>*.rsrr NB, Cb,

CONTINUED

3.C6

M\N\t\Auv\ r$€ " stsReD UesxrEvNu$€ crs d NNb THE eoRRESP$NS\\SG VA.L$€5 sF

NNb tsb I 9 rN,I LAs.Lcui3L r*.

\^s=er(ri-t)*,*E) \.o.$(jt1*4i - +E)

t\t

L f.g ANb L"u T's T1{AT T1+E b\siANCE FRSHTT:OrX\ E rO \OC-T Bb

t-A

NNt>

Pqruf

T\\e

vXUr.r€.

\us

n\E

uerrGTtl

(ra)

( rv\rsrv\uv\

\Dtr T:\xxs

v\L\r€)

NR tss)

N.$)

MtNlMufvt

For the

the minimum thermometerr

distance between the straight and CD = 9.00 4B = 36.00 in.

The minimum distance

58 .34

portion in.

in.

AHx\Ysrs

Orrurr.re os PtOcg.mrl Txtp.r-r THE CooRbrNNrES ( X, Y, e) A, (r \N\ E IsrAry

T\\E-

VEcTcR: 'rHE Is.rp*X otr

bosis

sS brN\\ .trlE

brRECTrc>N

CS:$NES

Axr\s \^* INCREMENT

\.o P Tr1q1 T\E- LQNGTI\S 'CI) ARe T'D Be. lNcRJEt\sEb

ANb

EAC}T TRINL bueT AB: dn* = 9 rN. T,o 3L

\r.JtT

AFTER Foe. oF

FoR. b\-rt

Cb':

rN,

rN.

tNgREtqEN\b

rs.

rs$REprEu\a

porNr

rN,

Cou?r.lrE

dgs = 9 rN. Tg) jL T*E

csoRbrNFirE\

ts:

KB = XA * nl (c,o:B*)fe

y s = y n * r u( c o s b ' ) r e e$ I LH+ ts ( cor Sr)ta

9srssit

ComPuTE TF\e Coof,\,1sq\TE: otr Xs = Kc-* v( css $x)qs \u = yc-+ vr(co: $"{)"o tu . tc. + s.\ ( cos Sa).u C-ovrp.vre TUE

LE\KrTb\ sF

9t beqT

d = L (xs-I*)a* (ru- \")\ Comp'rrE AttD

TldE fse

1rALAR

Porxlr b: nqs ab Bb:

tto- r"f l'l'

CDvlpoNE\fTs

\*o

$sJx = (\s-xsVdeu XgE=Xe-Xg -(\"o)1 = t\o-rg)/deu \ee YG - YB -te)l (\il. (tu = d"s LeE ' te- 2s Comp,,.rre rHe b\srAr-\KE.d BTT\^IEEN,ForttT E \sfS \lx-t Eb: d = \ [ (\eo)..r te. - (hsb). \.eJt *t(\eu).x.. - t\;.i* tolt r r r, nL (\eb)x\ee - (\e).,Xrc1r 1'tz \c.'u.rtrNu€b) Copyright © McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.

Chapter 4 Computer Problems 4.C1 The position of the L-shaped rod shown is controlled by a cable attached at B. Knowing that the rod supports a load of magnitude P = 50 lb, write a computer program that can be used to calculate the tension T in the cable for values of θ from 0 to 120° using 10° increments. Using appropriate smaller increments, calculate the maximum tension T and the corresponding value of θ. 8 in.

T 16 in. A

θ C

12 in.

4 in.

15 in.

Fig. P4.C1

B 1000 mm

4.C2 The position of the 10-kg rod AB is controlled by the block shown,

which is slowly moved to the left by the force P. Neglecting the effect of friction, write a computer program that can be used to calculate the magnitude P of the force for values of x decreasing from 750 mm to 0 using 50-mm increments. Using appropriate smaller increments, determine the maximum value of P and the corresponding value of x.

4.C3 and 4.C4 The constant of spring AB is k, and the spring is unstretched

Fig. P4.C2

y θ

a B

R A

R a

Fig. P4.C3

400 mm

when θ = 0. Knowing that R = 10 in., a = 20 in., and k = 5 lb/in., write a computer program that can be used to calculate the weight W corresponding to equilibrium for values of θ from 0 to 90° using 10° increments. Using appropriate smaller increments, determine the value of θ corresponding to equilibrium when W = 5 lb.

x W

Fig. P4.C4

06_bee77102_Ch04_p005-006.indd 5

3/13/18 7:58 AM

4.C5 A 200 × 250-mm panel of mass 20 kg is supported by hinges along

edge AB. Cable CDE is attached to the panel at C, passes over a small pulley at D, and supports a cylinder of mass m. Neglecting the effect of friction, write a computer program that can be used to calculate the mass of the cylinder corresponding to equilibrium for values of θ from 0 to 90° using 10° increments. Using appropriate smaller increments, determine the value of θ corresponding to equilibrium when m = 10 kg.

y 0.2 m

D 0.1 m

A z

0.2 m

E 0.125 m

0.125 m

Fig. P4.C5

4.C6 The derrick shown supports a 2000-kg crate. It is held by a ball-and-

socket joint at A and by two cables attached at D and E. Knowing that the derrick stands in a vertical plane forming an angle ϕ with the xy plane, write a computer program that can be used to calculate the tension in each cable for values of ϕ from 0 to 60° using 5° increments. Using appropriate smaller increments, determine the value of ϕ for which the tension in cable BE is maximum. y

E m 5 . 1 m 2m

C 2000 kg x

Fig. P4.C6

06_bee77102_Ch04_p005-006.indd 6

3/13/18 7:58 AM

?a=/o$ @, E .r/avt:!2 Btq 7T 7VE zEtr €buLY Fl,up: lvntuaTooe o7 P JT 400mm Fot? yt?tttg or x n?o,y

BO-*

To O tt9'yt

,Jo zro

/NC44agr76',.

(4uo Ftre P-., (orz?BPtuYg/rt,tl,iloE * 7

Frzee pooy AcD

€eo

tS = FOz

-t R9.sin6 -Er

ME7..Y

nBcoso +ftr

Mn' F(t^u, ac iot)+ cD ror")

(*stn)si's

Qx=-Tsrz p AA= P - rcasp

1"o c6e

Fr?et BooY 8oo AB

A = (o:* ^;)"' 7/-t ,<tg

P/Za gftAr'a

ENTFE.' pnzn : P&adafurr,

P =,{o

fiB , /?hv //v .s€64u?,vcq

/} r 4F= 6 A y EF, g rn, Co , 1/ir, BC= /€/U,/ 7,ae r?B,auE 6/./

VAt ui or O , EvnLuA ,fua EtW PrartrZ W)t ox o F T O^, flr, rh,o A.

8, + A

Ec, utl rtuvJ.

coSG

beta deg.

T 1b

A Ib

Ax 1b

Ay tb

0 10 20 30 40 s0 60 70 80 90 100 110 t20

-15.9s -LZ.OL -8.13 -4.33 -0.65 2.88 6.2t 9.26 tL.92 14.04 1s.34 15.40 13.45

60.668 95.937 114.835 L z s. 3 2 4 130.601 t32.22\ 131.071 L 2 7. 7 0 5 L22.54s LL5.962 108.367 100.338 92.943

18.534 48.L67 65.719 7 5 .s 6 1 8 0. 6 0 6 8 2. 3 2 6 81.544 78.768 74 . 3 4 4 58.537 6 1 .s 8 6 s3.797 4 5. 8 1 7

-16.667 -L9.957 -t6.237 -9 .470 -1 . 485 6 . 541 1 4 .1 6 9 20.sAL 2 5. 3 t 3 2 8. L z s 28.670 26.643 2L.62s

-9.333 -43.838 -63.681 -74.96s -80.592 -82 . 0s8 -80.303 -76.O42 - 6 9. 9 0 2 -62. s00 -s4. s06 -46.737 -40 . 393

5 0. 4 2 50.44 50.46 50.48

.7t/,{tott

= /32

WHrtt

2 U 6 = Jfo,V{"

= **-^g S LLc ac asse ,

'o oF

QoTztne

(z)

P/2o62ttPl

,{rvZPn D/} 7fr :

fu =,/o&g,

/, = /oaz>rat

= ? tt-n,/g" /7 = #ao

?7a

frcrtB,o-, BQ) 7,qgv 4:6 k). tlUtaert 4ma Prznrr %/ 8, '+no P ftl" ' VfrLDE Otr 7 tr,4q+, ?,f> 4rez 7e O AT 6O

in cable-T tb

L 3 2. 2 2 7 L 4 2 L32.227Ls8 L 3 2. 2 2 7 L 5 8 L 3 2. 2 2 7 L 4 2

/44fWu/v/

8 +

tttul

theta deg.

:----MarLmum tension theta deg.

05'

- hm

/tYcEe;Pt6;'YzS

x, mm

theta deg.

4s0 400 350 300 250 200 1s0 100 50 0

4L.634 4s.000 48.914 53.130 s7.995 63.435 6 9. 4 4 4 75.964 82.975 90.000

P N

x nm

theta deg -

282.9s 282.9A 2 8 2. 8 5

54.725 54.730 54.735

40.453 43.3s4 45.734 47. 088 46.735 4 3 . 8 72 3 7. 74 8 2 7. 9 9 L L 4 . 9 76 0.000 P N

47.198383 47.L98387 47.198383

Grurx.t F=/oh,; 4 =lo u,

G r veat Q = lO i2.1 a = Zo f'' Sfata: fi =Jf tJ/irt, utttftzFralEowrtT 6=0

,SPQtNg,*-€lby'r, UNsTeETIHE, llttc, g=o F/fto: b W

(o)

6=D

Veue op9

Ft'vOz (a) W cqeeisWztrYg To O=O TOfuo nr too / NCr?€lnrnT. (/) Vruoe

ta Qy=6/6,

o t= 6

W coeBf*Po*auut: -o fu" r7r /6"

/ace6'6aEr't7g^. (b) fbrz

tClsrtt9Rturn

/?o2

EOUTL/8 e/olv

l fl = 6/6,

RdnO

At, -- a--Ecasa F,f.,= R s'i' o

fre=(auir^B;)"^

T=$lrn -(a--"r] T s = T# $E/,4c=oi Oaaanr

p=O

a-W

PeaeAn-,

Dl?7n'

AB, = a-

/78=(aafrae] * as;)"'

(r)

T= -& (^ u -a-)

(s)

F\BS = 2. (/ - cos 6)

ar rnaaznra

fiyftA

(r) (z) (s)

frBr= I si,

E2 = ,at tn -i a = ?o ,'o.; -2 =6 U/A.

Ee u0Vo*e 77)fZnoudd /6) (a\ ;vruD/t7E v,qluts oF TqFa lzv pa e FrzDN O To ?a" >2iirz/ poTngpg.rTSrvzs (l) 8f frnt vqLu? CE A /:aa tHH/c# fal=t/l et?refu*,r.r /H €f,-a;ueart, 4ND pr?ur7

RBx

(6)

C = - T 'hs theta deg.

T lb

lv 1b

0.0 10.o 20.0 30.0 4 0 .o 50.0 6 0 .0 70.o 8 0 .o 90.0

o.000 1.497 5.705 11..966 I 9. 567 27 .924 36.603 45 .288 53 .747 61.803

o.000 o.505 3.503 9.655 18 . 0 8 0 27.451 36.603 44.66r 51.O19 55.279

+)r/42= o:

?o ta,i 4 = {/A/D. /:N7/-TZ O/+T/): R= /Otn .i A--/A/ Eaunrto*s -Q/-auw.Ft Przoaofl*, ft) fuzauat+ ( z) (a) Aunt unzf rtna PErn,7 7 apn lN pu? Vntues oF 6 trf?a+-t O ro ?ao ugtul too /nalnMe6 Vdtup at= A twt G) BY T74A/ ggfEn44tuE tc,* /b. € Vttt l// (stable)

lAl so: For W = 5 lb

( unstabl e )

= 175.7 deg

l,y=^6

6F Fza6rznlv1

O u2l/vE

For W = 5 I b determi ne theta 2 2. 9 0 2.491 4. 983 22.91 2.494 4. 988 2 2. 9 2 2 .497 4. 994 2 2. 9 3 2.500 4.999 22:94 2.502 5.005 22.95 2.505 5 . 0 10 For W = 5 l b theta = 22,93 deg theta

+C.8 +wf? <o i

theta deg.

TW tb

o.o 10.0 2 0 .o 3 0 .o 40.o 50.o 60.o 7 0 .o 8 0 .o 90.0

0. oo0 0.379 1. 4 9 6 3.295 5,687 8. 564 11 . 8 0 3 15.282 18.877 22.474

0. ooo 0.033 0.252 0.797 1. 7 2 9 3.O21 4 . 5 71 6.228 7.819 9.175

For W = 5 1b 62.59 6 2 .6 0 62.61 6 2. 6 2

determi ne theta, 4 . 9 9 7 13 12.686 4. 99878 2.690 5 . OO044 2.693 5 . OO 209 2.697

For W = 5 lb lAlso:

For W = 5 lb

theta

= 159.6 deg

62.6 deg

(stable) (unstable)

Gtuen,

Ftpe Tagau tp Ceete BO $Ho g ftQ. rnLa6 aF 6 Fl?dq o T> ld Ar so/W17qs775. ALe, Ppp y4UE

?o-&g ?nret o; aF

Plass Ftrot CYtrno*-@

o/, Q roe- whal

To EQAL/glzlur+r Foe

Vhues

l:/?ap

O 7b ?oo

/^t

FTA{ lttt:t't

Te,vSraq

CABTE

BE,

/oo /na,ErvavEi,

fl lSat l;rnP aF 6.FoR

n+UE

AAgC rs tautATmAL

m=lOQg

/. =go, gE = (?"re'rr,S1'/z (?u6ne)0,1t n/6) 17y=

,' /$'

(t)

(z)

7L 8e

-sJ +t,su) ?-'(6/i(-z! '/,g-h) + =(rs/zX-zf- si lbtn=G-)i

\ lw %t-tr,

( "g, =(ul ,os:ri'ars + i, +Gil sineo'i +(.?n') eos3ooErh 4 XY=o tAn = e: r6,6/f*+_hn*Go*9,

CPz=,d - a- sino

CAS: ,C f a cos 6

(2)

cQe=%

(e)

cD=(.rf * cof + .o")"t

(*)

COx

4=ffT

2M4i 7

(c+L!,- (cAr'<

eoEFF=

d/e,

ryll-P* (s)

For2.

ntZou++(z),

theta deg.

T N

P@!ftfr?1

Pt?o€Enyr,

tN #auexct)

7=aS,

ril=i

EvltLLt,|7Z

itNO

?'ra.vZ

FQol-t

O 7V too

[V,4Lu/77€

.'?FA Pru,-7

m kg

sToT

TtqM.rr.

0 .o o

o .o o

0 .o o 0

28.78 5't.41 69 . 58 84.45 96.95 108.07 119.26 133.31 15 7 . 0 4

2.934 5 .241 7.093 8.609 9 . 883 11.016 12.157 13.589 16 . 0 0 8

For m = 10 kg determi ne theta 9 7. 9 9 9.9886 5 0 .9 0 10.0003 98.10 51.00 10.0120 98.22 51.10 For.m = 10 kg, theta = 5l .O deg

10.oo 15.O0 2 0 .o o 25 .OO 30. oo

3 5 .O 0 +d.oo Determi nat i on of 36 .84 36,85 36. 86 30 .87 36.88

hQuo7ryS

Q ) naaup

(+)

(tr!

Cable; kN

.</lCtc

Angle pht deg.

(+)

'1^/O E v4Lu6 Foa uJrpc €o txczEt*+rzv"3 ! Buf puF ra A CttEL€ ?LLllRlB

?Eaatac

BEcor..t/^/6

(s)

*l *)

TE=o;s(3wlcass")( #*

5.OO

o.o0 1 0 .o o 2 0 .o 0 3 0 .o 0 40.oo 5 0 .o o 6 0 .o 0 70.00 80.00 90.oo

= o

/?'vO G :

ou7/./x6

Ou74/u€ OF PrZodain: ?cz4 (to neou€). Przua,a4 /tt :ffiDptcc f/vep (t)

cos,goo cosd

) eos 3a-as

( s wl/e

b-- o,s(zwzcosE')(- #r

T(co"r) - w ta sro a=o = l/vErQfl-roF CyLlA/oEa-3

eas SooSra{

= - (ewl/r,c)co-s?o"srod

7D -fr

fQunTtops

t 3w ?,i G/z - *tC/t tol + ( 7€/L - g w I

TD t Q = {aue

/*o# Tavyrav

CaEfror g: coee or &t

4= co */F =6 J -Q,e=f

/WzoF T&staftT: Feanat

COg

-/' c4) GIS t(7oh)Fz!=tj +t,dD + Gilr?tt') (4! - s{ g + 3 srrzzo'i + 3 cos?o?c,r+4),'( w1) =o + b cus-s-ncos( =a ?G,t*rs.y)+"f (14-r,ti) swaf 3aZ"s{&*svvas so"cfil-L

16 . 5 8 8 14 . 5 9 8

7<a) Cabl e; EB KN 16 . 5 8 8 18 . 4 5 3

2 0. 1 7 7 2 1 . 74 8 1O : 2 9 9 2 3 . 15 3 8. O23 24.382 5. 687 25.425 3. 307 2 6 . 2 75 o. 902 cabte BD has become slack. Derrick col lapses 12 . 4 9 6

TE = max o.ol4 0.010 o.oo5 -0. ooo -O.005

26. sg7 26. 539 26.540 # 26 . 541 26 . E13

Tee

lvnrlkluM 47 cot/afsE

Chapter 5 Computer Problems 5.C1 A beam is to carry a series of uniform and uniformly varying distributed loads as shown in part a of the figure. Divide the area under each portion of the load curve into two triangles (see Sample Prob. 5.9), and then write a computer program that can be used to calculate the reactions at A and B. Use this program to calculate the reactions at the supports for the beams shown in parts b and c of the figure. w0

wn+ 1

300 lb/ft

B L 01

L 12

400 lb/ft

240 lb/ft

3 ft (a)

4 .5 ft

5 ft

2 ft

420 lb/ft

150 lb/ft

4 ft

3 ft

(b)

3.5 ft

(c)

Fig. P5.C1

5.C2 The three-dimensional structure shown is fabricated from five thin steel

rods of equal diameter. Write a computer program that can be used to calculate the coordinates of the center of gravity of the structure. Use this program to locate the center of gravity when (a) h = 12 m, R = 5 m, α = 90°; (b) h = 570 mm, R = 760 mm, α = 30°; (c) h = 21 m, R = 20 m, α = 135°. y

R α z

Fig. P5.C2 B

5.C3 An open tank is to be slowly filled with water. (The density of water 3

60°

is 10 kg/m .) Write a computer program that can be used to determine the resultant of the pressure forces exerted by the water on a 1-m-wide section of side ABC of the tank. Determine the resultant of the pressure forces for values of d from 0 to 3 m using 0.25-m increments.

2.1 m

Fig. P5.C3

07_bee77102_Ch05_p007-008.indd 7

3/13/18 7:59 AM

5.C4 Approximate the curve shown using 10 straight-line segments, and

a y = h(1 – x )

then write a computer program that can be used to determine the location of the centroid of the curve. Use this program to determine the location of the centroid when (a) a = 1 in., L = 11 in., h = 2 in.; (b) a = 2 in., L = 17 in., h = 4 in.; (c) a = 5 in., L = 12 in., h = 1 in.

L–a 10 h

5.C5 Approximate the general spandrel shown using a series of n rectangles,

a L

Fig. P5.C4

each of width Δa and of the form bcc′b′, and then write a computer program that can be used to calculate the coordinates of the centroid of the area. Use this program to locate the centroid when (a) m = 2, a = 80 mm, h = 80 mm; (b) m = 2, a = 80 mm, h = 500 mm; (c) m = 5, a = 80 mm, h = 80 mm; (d) m = 5, a = 80 mm, h = 500 mm. In each case, compare the answers obtained to the exact values of x and y computed from the formulas given in Fig. 5.8A and determine the percentage error. y

Δa 2 y = kx m

d c

d' c'

b a

b' Δa

Fig. P5.C5

5.C6 Solve Prob. 5.C5, using rectangles of the form bdd′b′. *5.C7 A farmer asks a group of engineering students to determine the volume

Cord

of water in a small pond. Using cord, the students first establish a 2 × 2-ft grid across the pond and then record the depth of the water, in feet, at each intersection point of the grid (see the accompanying table). Write a computer program that can be used to determine (a) the volume of water in the pond, (b) the location of the center of gravity of the water. Approximate the depth of each 2 × 2-ft element of water using the average of the water depths at the four corners of the element.

1 2 3 4 5 6 7 8 9 10

Cord 5

... ... ... 0 0 0 0 0 0 ...

... ... 0 0 1 1 3 3 0 ...

... 0 0 1 3 3 4 3 0 0

... 0 1 3 6 6 6 3 1 0

0 0 3 6 8 8 6 3 1 0

0 1 3 6 8 7 6 3 0 0

0 0 3 6 6 7 4 1 0 ...

... 0 1 3 3 3 1 0 0 ...

... 0 0 1 1 0 0 0 ... ...

... ... 0 0 0 0 ... ... ... ...

07_bee77102_Ch05_p007-008.indd 8

3/13/18 7:59 AM

,l IOn+ |

Grvesl r N BExvr AB C'\RR\ING

:ERre\

g\E*'.

Sr**><xrRE- :+lswN, 1q5\rcA \\

otr

Fr?csrt

trNE

Rous

hrxr\ i

FABR\<AS€.b 'Iil+lN \TEEL

tg$NL

DtAt'vtETtR. [,ntS.,

[-ctAin51$

str

CTNTEA.

Otr' GIANV\TY :

Tb\E

Arrru-y:r: Tl+A$ Tt-tE Reb: ARe 1as:,osnG. 'rt+Ei CENT€R. OF (,RAVIT.( Otr

frga:;T 5D

TI+NT

:tiRr-X5$RE

cORRESbN\ttr\G

\+L

orfius.lE

l':tPl-fi

T\+E

Fcn

E-NCq

1r..1tLL COrucr\€

br5Rr\3gl..'€r\;

b\SrT\BuT€\

oF r-SN\

t-s\\\

(S= 1.,Z, - -.., N)

str rHE uExSirH \X(S) BEAM \+e s\ER rT 61$re\\ AqrS rr+C. VNL\JE\ IxPuN OF Tb\€. b\:STR\B\'iE\ tTs uoN\ r{T LEfi' LW.rsfJ r\N\

LoN\s

XE.L = L*.t-

R.tr)

T.r\E

TT}E qtNnRS'\

a o

zR/n

hla

elz

hlz. o S.orot h l z E s,*t

rffi

qo:cr {R-.hz

I YIL - -I qL

= |u'GT ?u =LeL c,

LON\ E,srxu: UeUNT€ 'nrE . TCsTAL XppLr(\ Fr.r, xL = Qon,*u+ [Ru($ + R.* tsrl r\tsost

K o

.[ffi.

R.( S)= | [ uxrsil \^-\^tL(si \t:) =ittxrsr]twe.(s)l

Oi'r.rme

L ZRnR.

3 .fRqiF -R/a

Ir.sP*f

R\G\fi Ett.r\S Lwo(s)l Rr\urvi\LE$fi Cor..^vn-rrq n+GR* tsl: \slu

lt+€

LINE,

?RSGRX\-'\ N$|^B€R

W\Til

Hovre(;€Nletl.)S

Sr>!.4 str T1AE XrsVrel.NS MX su?PoRT \:

NIx = N'l^ + L L*.+ | Ax(s)llR,_tslJ

* 1L* + l,rxtsrJLRc(\\l

Opunrt

n+G- b\:rNNcE

L. = Lu + AX(s\ T\tE.

VQF'T\CAL

\$PR>RsS

COr.NP.'rC. N\

B\=\,

\s.r\

L. '-r'c rHE NRx\ \^lLLs

REtrRS\SN\S B,

HSr\

\ R.ES?EC.INE.L\:

f = 3 [F;[3

R.r CouesrE

a€br\ER'

A\ = Ron*u - \

THe thu.rt VAL\rES r**rs\ sF \Nb

Tr\E

\N\

,\Xts\, B\

WLt\),

Tr\E

qsoRb\NN3q:

Dtr

(Q\Y11'11

right dist. load 300 tn/fE 300 Lb/fE 400 l.'b/fE

length, ft left dist. load 4. s0 24O \blfr 1s0 Lb/ft 3. oo 4.00 1s0 lblfr' 3 . s0 420 Ib/fr at A = 1264 .33 Ib at B = 1600 . 62 lb

right,

dists. Ioad 0 lso Lb/ft 420 lb/fr 0

X1 Y,

AND L

oF Tf+E

E =(Z*t ies\No([E!F')/L_

?e.,rn $gr6*ru"n

length, ft, load Ie.ft dist. 3.00 0 2.00 300 lb/fr 4oo Lb/tt 5.00 at A = 1 2 2 0 ,0. 0 l b at B = 1 8 3 0 . 0 0 I b

Ro\Sl

V' it- Gffi/t-

\NN

THe VALUES

or= X, Y, \Nb a

DuTP.;T

(O-)

R =5m h = Lz m XBAR = -0.0000 m = 3.6L62 m YBAR t.27SO m ZBAR =

(b)

R = 760 h = slo srn XBAR = 46.2642 IIEN = LzO .L978 tlm YBAR Z B A R = L g 7 . 6 5 9 6 nun

(C)

R = 20 m h = 2L m XBAR = :1.0802 m = lll: 4.8L22 Y-BAR ZBAR = 5.2945 m

lc) regio:r 1 2 3 4 'fhe force Ttre force

+ ZR.+rrR_

1 = Z R e o s KF * f btr

(b) region 1 2 3 The force The force

ur:UcinA oF Jsrs.Yr n+E r;rsr-r! cF R' H., r\Nb K f.xtP.rr fl+e VNLu€l .'rr{E C-onzrPurE T\\e tsTr\L LE-b,KtH L ctr

c=90o

d=30o

d, = L35o

G,nqsri Txun oF l- w\ wtbtt\ B€rslC, SLorr.JUYtrrLL€b NrN *t/*' \^JNreR.'-D=lot FrsrD, Ret*rxrol oF n+e pRE:huRe

Wbhct\

B€

A"FRox.\$44$Eb

LrN€-

ite,\Klvr-

SEcrldEhf\s

FoRcEs R troR. d=o,,D.asm, 9.3o., - _., 3. SO fn oN 5rbE

st\o\lN,

Grr.Ve

GrV€X,:

F..NbY \r.t\\Eu

trr*rsii

(G) O= \ rN., L= \\ rr{,.,

ABC

h:. L rN. O= L 1N. r f=\^l rr.l.,

tb)

\=drtt.

Ar.rxuy5t: FoR di a-\ rn

tC)

A "5 rN., L= l?,r.rr \:

.. S€.GI.AE.NT

\ rN.

S..

(Xn.\c)

( K'-.y.) -t-o<

d > Z.l vn

l-hve-. P = l A +

: itthAxd)(?qd)

=ig3d'

o. (s_t)a^ ax

\il = c a.Y

= ( 3 [ l * ^ -(d-z.r)+d - Z - - . f f i jzr rY\r

\NI)

- *"1

H x v e - -\ e - \ u = h ( r - * * . \ - h ( \ - t " ) = a h ( i

=(iP3\ 4 . L ( d - t . o S )

Frr.,lNutx, R --

'lr{E

vAL\reS otr &' L., Axtb h '. . Tt\E 161rt);'nA /\X Otr EACts\ :EGMC,I fT

JrsPsr

TAN \) = =P-

* VJ'

Cosnp.riE

t\X= is t L- o) Foq

€.r\c,U yRFT\A d I d = O, S.LSVrnr- --,,?..ODyvr;Z.lDm Qome.rrg FotcE . (: tr{E

sF d, R., AxN b VALuqS Foe Elqc\r btpT\\ d r d = Z,aS y$, a.Sovl\)- -. . s.Sbm (P) UoeraorWsL hNb yEF$rQ\l (W) CovtP\rre q-ovtF'oNeNIT\ btr R:

4,L(d- \.sS) TRE

brREClnoN

h/lAGNrrube bF f,:

R--(ffi-

Pr<,*rr THE Peoc'nsn depth d, 0.00 0 .2s 0.50 0.75 1.00 L.25 L.50 L.75 2 .00 2.LO 2.25

2.s0 2.75 3.00

vNLuE:

force, kltl 0.0000 0 .3540 1.4150 3 .18s9 5 .6538 8.8497 L2 .7 436 L7.3454 22 .6s52 2 4 . 9 7 74 28 .64L2 35 .L744 42 .2470 49. 8669

Tr= Xu* i. Xx

qi u,ir- +)

L€rfGirt

tr+E

Otr

T$E

SEG$^Eh{T:

Ls=f,-o^1.*[ah(i- *")T

Xxr\

r1\E

b--TN\i'+

btr d' R, Nxr\ S

o.5aeuT m

* [':il-^tXe')

LEFT

EN\5:

XR = Xu* AX Covrp..rre Tr+E cOoRb\Nt\SE\ (l.",C{r) os T\\E c€NTROT\ str rrrE tEGtqENn:

CorqP.rr€

T\NGs. towreurg

t*''*jll

b=-3o

R=ig3fr"," \

P*.rrst

S=l.,Zr---r\S Foc E\CA SEGM€\n Si TrtE X goor<btNg1E5 otr \Ts Con^gsre

e, degrees -30.00 - 3 0. 0 0 -30.00 -30.00 -30.00 -30.o0 -30.00 -30.00 -30.00 -30.00 -29 . gg -29.36 -29.59 -27 .72

UPDN.IE

'f\\E

rcsrAL

Lts;t*L=

Leh\Cf$\ Lr.t,xL* Ls

L-",-r.u:

+ LLt 1I'

UroprreL*Lsi Lxf,s = ltLt UPbAsre LqL5'. lqL=--It\'* Cow,prne T\+E CooRbrt-g;reS (X,Y) ()tr T$E gENT\Ro1b Cr.rtVe I

I = Llt-=/ L-orou ?e,tw

'T\+E vNLr->E-S otr

Reoge.nm

cF nAE

V = lq!=/ A.L'H.

Lrsrsu

X, ANl\r Y

o$TR)T

a XBAR E 5.8006 Y - B A R= 1 . 4 9 1 9

h = 2 in. in. in.

L, = L7 in. in. a=2 XBAR = 9 . 1 1 1 2 i n . YBAR = 2 . ? ' 1 6 2 i n .

h = 4 in.

Ln. L=LZ a'5 XBAR ' I . 4 9 2 2 i . n . Y-BAR, 0.3745 in.

h = 1 in.

in.

5.C5 continued GrvExt: Geser.nu :t h:tUN

sPrlr.tbRe

| \,..1*\Ct{

Tr)

N=PRoxsnsnE\ TERIE:

?e.ctexn'\ (a) a=

RESfAT-ILE5

ouTF.,rT

tlt=2 10 n= EOmn h= 80mn = L-404* terror XBAR = 60.842 mn = t L 0 . 3 3 3 ? ?error 2L.520 mn YBAR =

oF r$\e trcRvr bcc b' Frx.r\, i ,\t{\, Y, An.r\ ('l-enec*)* AN\ ('?. entcn)q wr{€N

h=80mn a=80tmt XBAR = 60 .24L r un 23.395 tnm YBj AR =

(G) W\eZ., Q=EOYnYnrh=8O w\nn

(b) a=

tU) rn = (.' 0.r BOvnrrn, h= Soo v\rn (c) hA: 5.,0.=8t) wxn, h= BOvrnm (d\ m=s, q:B$ss, hrS))

n= 40 = 0 .4013 ?er r or = - 2.52Lt ter r or

m=2

m=2 n=10 5OOlnm h= SOmrn 1.404t ?errOr = XBAR = 60 .842 mn = t 1 0 . 3 3 3 t t e r r O r YB,AR= L34.500 rrn

n=40 5O0mn h= a=80nun = 0.401* *error XBAR = 60 .24L mn = - 2.52Lt ?er r Or YBAR = L46.2L9 r tr n

m=2

Ausu-rsrs (c) m= n=10 80ffin h= a =80tttrn terror = 0.307? XBAR = 68.782 m n = * 2 5 . 4 3 2 * = t e r r o r fltrn LG .269 YBAR

L- (s-r)ao.--J ao.l*h=kax oR k=il" Fc^< ReCTNIICTLE\" -- l, = Zt t\ (K:-,)* J

f=G.,\=ht

h=80ntrn a=80tttrn XBAR = 68 .693 run , YBAR = 20 .444 m n

n=40m=5 *error = 0 .L772 *error = -6.2962

(d) 'h=5O0ntrn a=80mm XBAR = 68 .782 ntrn YBAR = 101.683 ntrn

m=5 n=10 0.307t ?errOr = = Z-25.4322 terror

n=40 h=5O0fiun a=80fltrn = 0.1778 terror XBAR = 68.693 mrn = - 6.296* = ter r or m n L27 .778 YBAR

i(*"")tt.-,)aal* Ir.fP..tf

T[+E VALU€S

lsrP*rr

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ENch\ RECTANGLE-'.

a,o=ft

t= \,?.,- --. t { FoR ENct{ REcjrr\NsE \: csvrPurE T\\e ccDRNNAsES (1.r,, i.) CEsstR.oru otr Tl\E REgiNr..lGLEl

THE

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RE(INN\GLE

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reproduction or distribution without the prior written consent of McGraw-Hill Education.

5.c6 continued

QrJ'rUrn-lt n\€ lne.tn IxrPuT TH€

PRoGBA\A

Otr a AN\ \f.f\ oF r\{€ :qt\hlbREL Otr REC.T,\N6LE\ 11+E 5TOMBER. N }*pur ( XlExFsf, 1 C-.:xru.xe Ars\ (\)e*rcr YAUll€\ olbER

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Po,tlfiS

2345678910

AOl Dtr EASA RRCI.NI{qLE I T\-\C rnsnrr AO.= t,S S= 1.,? Foe. E N(n ReCTNNgLe \: )N (l\ir) cosR\16qr€-S T\+€Sg T}\E ts,\npuTE otr Tt\E QExrTR.or\ ReC.i\NgLE: n\E

E:S>. (o)

f;*

R==(5- o.s) Ac\

trS

9q11111'(

(q)Exocr = "

FeEs)

"sb'\\ GR$

Z-$t

I 2 3 4

Colr\p$Te

Connp.rre

(rs{

beprus t\

otr

I-<E(TNN6LE:

As = (lq)qz!.,) rr-+E -r$TNL NREA Atcsrruu', At-rxu = A-or*u* \s gpspsr- ZxA,. }.=Ns LX A = El\+

0 0 I I 3 3 0

E5 66

7 8 I 10

0 0 3 6 6 7 4 I 0

00 01 33 66 88 87 66 33 l0 00

; I 3 3 3 ,1 0 0

0 0 I I 0 0 0

Uputre

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THE csssr;5pite! Otr THe \RE\: CENrr<srb

Csvrr.rte.

( i.,Y

)

otr

Xx*r

Y =lqA/A'n--o.

1= Ll\/\-or*u

)

Hx\bs1. etr G. h, Nxrb N\\ At,.lD t-l- excs*.\q

ax=Nr-= \A,H€R. zIt

\L.*".o* :,itilS:***.i$

(a)ExNcr

Pe'',srr -r\\€- VAuse\ Y i t'1- E*qoe)i.

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l.?-eeesa\ =

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-rHE

6iqgH

qJ (x.a)

DEp\r\ v ER]\ cAL

ELEr.llE *rr

E\c-\\ oF

?OINT

\^lAfiER

(x*

-- 1 Zr- -, ,9 ;

Z n l = t . 2 . ,- - - . , 9 ) n=10 m=2 *error = -0.251? 0 . 5 8 3 ? terror =

h=80ttun a=80ntrn XBAR = 59.991 tnmYBAR = 23 .99L nun

n = 40 m ; 2 *error = -0.01-6? 0 . 0 3 6 ? = teror

(b) h=500ntrn n= L0 a=80nun m=2 = -0.2s1* X B A R = 5 9 .8 5 0 tnm ter r or 0 . 5 8 3 t = = YBAR L49 .L25 mn *error lrun h=500mn a=80 X B A R !r 5 9 .9 9 1 mn = YBAR L49.945 ilun

-n+E

Covrp.rre

(a) h=80fltrn a=80nun XBAR = 59.850 run 2 3 .8 60 nm YBAR =

n= 40 m=2 = -0.0r.6t ter r or 0 . 0 3 5 t = terror

AyeRNC€

bEpTF\ dxve l

= t ta ((*,t*)* d(xu*,,t*\+ i(* * 3**,) dx.rE

- \x+do...E)

\hbsilE Ll.v, r;$:H'vtl.r)t-

OesrsELq\ : fqv = f qV * ( ido.,=)t+dove) \)esxrEZZY: f eV = fiT * (ZaN-\X4d^u.) Ui"r>nsE

TcrrNL

1r\t

voLuMe

]tlrcn,u-l

\T-.rr*. i \l'-t .- + ( !dl"d) (r>r,.e*r'in rr+E- coDRNuns€S (X,Y, a) cesrre.R

otr

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rr\E

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rr\€

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oF Tt+E

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VA\$E\

-L=I{At,f,rAL

Y ,\

Asb Z

(c) a=80ntrn h = 80 nun n=10m=5 X B A R = 6 8 .2 2 8 IItrn ?error = -0.500& YBAR = 2 L . L 0 2 mrn terror = -3 .283t h=80 mm n = 40 a=80nun X B A R = 6 8 .5 5 0 m m ?error = -0.0318 = YBAR 2L.773 run terr:or = '0.208t

m = 5

V = 628.00 ft^3 XBAR = 8.650 ft

YBAR = '2.264

ft,

ZBAR = 9.274

ft'

(d) a=80fltrn h= 500 mm n=10 m=5 X B A R = 6 8 .2 2 8 nun terror = -0.500? Y - B A R = 1 3 1 , . 8 8 6 mn terror = -3 .283? !!=80fltrn h= s00 m n n=40 m=5 X B A R = 6 8 . 5 5 0 rmt terror = -0.03Lt Y B A R = 1 3 6 . 0 8 0 mn terror = -0.208t

Chapter 6 Computer Problems B D F 6.C1 A Pratt steel truss is to be designed to support three 10-kip loads as shown. The length of the truss is to be 40 ft. The height of the truss and thus the angle θ, as well as the cross-sectional areas of the various members, are to be selected to obtain the most economical design. Specifically, the cross- θ sectional area of each member is to be chosen so that the stress (force divided by area) in that member is equal to 20 kips/in2, the allowable stress for the F C E G steel used; the total weight of the steel, and thus its cost, must be as small as 10 kips 10 kips 10 kips possible. (a) Knowing that the specific weight of the steel used is 0.284 lb/in3, write a computer program that can be used to calculate the weight of the truss 10 ft 10 ft 10 ft 10 ft and the cross-sectional area of each load-bearing member located to the left of DE for values of θ from 20° to 80° using 5° increments. (b) Using appropriate Fig. P6.C1 smaller increments, determine the optimum value of θ and the corresponding values of the weight of the truss and of the cross-sectional areas of the various members. Ignore the weight of any zero-force member in your computations.

6.C2 The floor of a bridge will rest on stringers that will be simply supported

by transverse floor beams, as in Fig. 6.3. The ends of the beams will be connected to the upper joints of two trusses, one of which is shown in Fig. P6.C2. As part of the design of the bridge, it is desired to simulate the effect on this truss of driving a 12-kN truck over the bridge. Knowing that the distance between the truck’s axles is b = 2.25 m and assuming that the weight of the truck is equally distributed over its four wheels, write a computer program that can be used to calculate the forces created by the truck in members BH and GH for values of x from 0 to 17.25 m using 0.75-m increments. From the results obtained, determine (a) the maximum tensile force in BH, (b) the maximum compressive force in BH, (c) the maximum tensile force in GH. Indicate in each case the corresponding value of x. (Note: The increments have been selected so that the desired values are among those that will be tabulated.)

x–b

x b

3 kN A

3 kN C

E 5m

3.75 m

I 3.75 m

3.75 m

Fig. P6.C2

08_bee77102_Ch06_p009-010.indd 9

3/13/18 8:01 AM

6.C3 In the mechanism shown the position of boom AC is controlled by arm

8 ft

A B 3 ft

800 lb

5 ft

Fig. P6.C3

BD. For the loading shown, write a computer program and use it to determine the couple M required to hold the system in equilibrium for values of θ from −30° to 90° using 10° increments. Also, for the same values of θ, determine the reaction at A. As a part of the design process of the mechanism, use appropriate smaller increments and determine (a) the value of θ for which M is maximum and the corresponding value of M, (b) the value of θ for which the reaction at A is maximum and the corresponding magnitude of this reaction.

6.C4 The design of a robotic system calls for the two-rod mechanism shown.

Rods AC and BD are connected by a slider block D as shown. Neglecting the effect of friction, write a computer program and use it to determine the couple MA required to hold the rods in equilibrium for values of θ from 0 to 120° using 10° increments. For the same values of θ, determine the magnitude of the force F exerted by rod AC on the slider block.

250 mm

2.5 N·m

θ A

B 150 mm

Fig. P6.C4

6.C5 The compound-lever pruning shears shown can be adjusted by placing

pin A at various ratchet positions on blade ACE. Knowing that the length AB is 0.85 in., write a computer program and use it to determine the magnitude of the vertical forces applied to the small branch for values of d from 0.4 in. to 0.6 in. using 0.025-in. increments. As a part of the design of the shears, use appropriate smaller increments and determine the smallest allowable value of d if the force in link AB is not to exceed 500 lb. 30 lb

1.6 in.

3.5 in.

1.05 in.

θ B C

80 mm

100 mm

0.25 in.

150 N

D 30 lb

0.75 in.

Fig. P6.C5

6.C6 Rod CD is attached to collar D and passes through a collar welded to M

Fig. P6.C6

end B of lever AB. As an initial step in the design of lever AB, write a computer program and use it to calculate the magnitude M of the couple required to hold the system in equilibrium for values of θ from 15° to 90° using 5° increments. Using appropriate smaller increments, determine the value of θ for which M is minimum and the corresponding value of M.

08_bee77102_Ch06_p009-010.indd 10

3/13/18 8:01 AM

6.C1

6.C1continued ,,ecT ro NftL Pta.ql5 t rcREk-tt MEH6EF.' %tt=if

tA' =* ( " vl At&,ft+rtL, r %rr=*4't''a'7 %t

qf t'j!4Bir, LzuEreS-!uq;-,J"offl"r=blqna, L Lts=Lr{L

WEteHT_Gr -W{-amt.t.t !R,v55

riULTtPLY rT e Tfl r TcsTftL VcrLur'tE Arvo (: 8Y TftE 3?(CrFiC ru€,6ni-

l0 kips

=z/ ( \"1 ^, t fi^rLnr*Ar{inr*r} noh&= !J (rO t11/

6+veP; TRgrr ;TEEL TR\)s1 t3 To 3e Dr-s I6NEr> r-r: 5uPP0 nf Tt+e LOADs Jil+on/tVtN MosT Earuo,llrcAl ToIAL wEt&HT W*y (t4ar 19, wtTH Tfl€ SMSt-r-eS7 5leee OF Sfeet-) KNoyrrrrVG jS=D, TftAT Fon ALLoHheue5TRess = Coll = 7"Okips /lin' wet Q*t

5?ecfftL

= N = 0, ?DLt tb/ir1

FtNp;

tTtSNnTlr't Ct1Dfu) ($Ncr DE tSA #Raronct AengER, 1uTLtNE 0r TRog+af"l 20f;p/ii, t= 0,?B[iblns furcR, , Gall= r AO : fo| 0 = Or TO 0z *'rl lN1netrfru -bE, CnHNTE Fae, fh,c,FsrrFrr, % FRonFff. (i,n@ CtrplftJ rE AAg, frfir, h',c,Acu, AEe , &o Tq,NE o (g) f'lqcnr Et93'F) tittcTtls W FROM EO (IO) COI-IRJTT

( a ) w P : G H T o t = T R 0 5 s A ^ J D c < o g l - s € c - r r o r v r lA r -( E A )oplattI-

OF eAc* l-lAD-gEARrr./G r-\trvttsi-(r() LtrFTor DE flRERs Fp1UT' 0, Wrn.,1g"*LL .@sS-$ECTro.JAL FoR lftLuEs oF 0 FRoH zd To Bo" uslN6 ,o ,NCREMFNTs ) usf 0l = zo: Oz=sa", a 0 = 5o (6) u5E Succtss tvEi-Y CbseR VAr-uEsOf 0, fttiD 0e (b) |PTtMoN {ftuve oF 0 ftND coRqts?oNarNG -SECTTOU4LAND SHALI-ER /MCREf'lEN's bo V KLUES AF UlCI&fT OF TPU55 hND C€OSS *AERs OF V^R.,OIJEHFI*S€R S, , t a 1

v a

' t r v r r v -

9 ,

PR06R,At*t otrrFuf (e) ASALY515 -W ffi5F. Vore e{+cH oF T{-tE 3 Lo^os 8V I FNs> d e gT rHe e lb F ME,!,tBE R 8Y b, TTTCT FN GTI-\ O E ftCT+Th6QIZ, 20.000 692.55L (f*rS, P=lD t,p 2 5 . 0 0 0 5 5 9. 2 6 9 huo b =/Dff) l=Ror',tFB -DlAGQen oF Eruv RE 7Bvss, \^/E ftNl T++AT a j--A W e N o n JD E T = - e H t t t E T u € T O R C T I N E f t C H f4rfigr-,p, T0 ft*E LEFT oF O E 8 v T H e l4ETf+oDoF J,tN'T3 I -AB

TH degr ee

!n,

w 1b

s6.7s0 3L2.349 56 .750 3t2.349 55.77A 3L2.348 55.780 312.348 56.790 3t2.348 55.800 3L2.348 s5.810 3L2.348 s5.820 3L2.349 55.830 3t2.349 56.840 3L2.349 56.8s0 3L2.349

7f,cl =o Zt=o i

JorNT

2.L93 L.775 1. soo r-.308 L.L67 l-.0510.979 0.915 0.855 0.828 0.798 0.776 0.762

A(AC,CE) A(BC) in^2 in^2

A(BD) Ln^2

A(BE) in^2

0.s00 0.500 0. s00 0.500 0.500 0.500 q.500 0.500 0.500 0.500 0.500 0.500 0.500

2.747 2.145 t.ii t.428 L.]92 1.000 0.839 0.700 0.577 0.466 0.364 0.258 0.175

0.731 0.592 o. ioo 0.435 0.389 0.354 0.326 0.305 0.289 0.276 0.266 0.259 0.254

2.O6t ' 1.608 L.zss 1.0710.894 0.750 0.629 0.525 0.433 0.350 A.273 0.2010.132

(6) wrrH IN CREP\ENTs AO= 0.01"

A {+\o

30.ooo 472.22G 35.000 4t2 .288 40.000 370,094 4s.000 340.800 s0.000 322.020 s s . 0 0 0 3 1 3. 0 5 s 6 0 . 0 0 0 3 1 4. 8 1 8 6s.000 330.497 7 0 . 0 0 0 3 6 7. 7 3 L 75.000 445.487 80.000 52L.897

A(AB) in^2

A (AB) i n^.?

A (AC,CE) A (BC) in^2 in^2

A (BD) in^2

A (BE) in^2

o .897 0.897 0.897 0.897 0.895 0.895 0.896 0.895 0.895 0.895 0.896

o .492 o .492 0.491 0.491 0 . (191 0.491 0 .49r. 0 .490 0.490 0.490 0.490

0.655 0.55s 0.6ss 0.655 0.655 0.654 0.654 0.654 0.654 0.553 0.653

0 .299 0.299 0 .299 0.299 0 .299 0.299 0.299 0.299 A.299 0.299 0.299

0.500 0.500 0.500 0.500 0.500 0.500 0. 500 0.500 0.500 0.500 0.500

{

E 0

[r/

:'153 sinE

=Q(n alr'-oi6e

=ffi.,#,rfi

continued

6.C2 -

B" p $ zts=oi rn^(}") - P(t-: fu)a =o , o,: b c T-ffiio.' T

t r l Y - l v . ' r - , \ - - - 2 .

la L

lor -a*fs

fu= t,.f)

Enr

Fogc€1 cREftrto

lfil ,mE H1ERS 0F 0^4-'rRU55

uli: f i'Hil T:, {

o.urLr^,e-pr r(06 RAY1 '4= 4"t, *, b = 2,?5m, P= 3 kN, Ax= Q'75m FNTFR

GtVer.t .i. ,eY 7wo Ftmg Af BRtoez 15 sJ?Qo4tel As srloh/^l TR)sStSt

Ot(E OF Wt+rc++tl 5 H,oWx( , TRULK 6F Wet&rlT

b = Q,2Tn 1 A / =l Z A r u N r c # f t r ' - * f t r D t s r f r . N C E r R f r v E L | ' ,o N 8 R r - D 6 t W r T H t T Su / e ' 6 # 7 ' * Q u A L L y 'D tSTRtSu"EJr ory 173 F0urE WHEELS. R lrE F(o eRtl:l - To c A Ltt) l*'rr: Fol< cet t t't B +"J T o l 7 , 7 T r r r , , v t T +a' l, 7 | - m hND'GH FOR i,:o E,t{eArTi , 'NCR

Ftr'tD ! (a) /&Jdtx. ?r^/sfLE PgRcg tN" BH a# CoflPqFSS c0 r\,l ,b i F 5 5 tVC tvf tt t, ',, t ) ) Af ,, t, b) hAY, l T <, 4 TEx,gtr-€' ,VAX. ftR<E IN GH tt )

lA Lu€atrz tt

,, ,.

'fRv55

x (m)

P = * W = 3 k,V Q = 3,71n

=oi'RF(+a).-P(+a-4=o

R F =P ( t - h ) PRzU A To B o ( n { a ) , +)ZM; = o; A

F to

tr{1r9

l:Z fOp x-= D T0 4A+b WtTt+IN.-RE14TN-|S (e) l) ( AilD Fos. r/se tf tc { aI F a<L <2 L uSE FQs' ('t) ftNp(tt ) cl.-qE f,ls, (5) AilD (6) lt= uaL <x s 4a tF 1.>4a- ulRffe FaH=\r*=0 (< Prncs n $t' ,2.-b hND FFtrE4rASIE STER A D . D I A / 6 "xI<Fo nreil 6,r:Fgr=o", fto p Tt+e vBr u ?S O|mr N i:D'/?) FPFy'rot/s|$LUES RESTaR,E oRl&tNftt CooRlt{ft7Ex 6Y,*OornGb PRtNt X.t =e,+, ftw Fb,t (rOTo rVfx7 :TE" PRD('qAM ouTPdT

AN*LYs t5 WE ?RST CoNSlDfR 7{+€ toftD P fxERrre tv ONE FRorrrT WHEELpru !N=: AF Tu-e TRuss€S: fn/7r RE

6 sursr trvr'''t

,*}f'a*; rg-z)- P(t-H& =o'

(r) ?rr:#b-fr-e+4 E;frfix-

+ 1 X F=* oi ? ( t - f f i - ? g T : r = o

Frr=-ftlx (al

0.000 0.?50 1".500 2.250 3.000 3J50 4 .sOo 5.'250 6.000 5.750 ?.s00 8.250 9.000 g. zso r0. soo 11.250 12.000 L2.7s0 13.500 L4.250 15.000 15.?50 16.500 r7 .250

F or c e

in (kN)

0.000 -0.188 -0.375 -0.563 -0.938 -;1-,313 -0.938 0.553 -0.188 0.938 2'.063 2.438 2.8L3 g. rqe2 .813 2.438 2.063 1 .688 1.313 0.938 0.563 0.375 0.188 0 .000

Force

in GH (kN)

0.000' 0.338 0.675 1.01-3 1.688 2-363 2.588 2.8L3 ?.038 2.8L3 2.588 2.363 2.138 l-.913 1.588 1.463 1.238 1.013 0.788 0.563 0.338 A.225 0.113 0.000

(o) UAx.TFdslLE ruRCEr^/I H =.3'J9-kNruR,t =9,75n

+ffir4;,

'J'

f,*=#Et4a-r)

(s)

:# t,u-*") rgr+

(4)

Br+P(r-fu) rxF,:o;F(t-#o)RE ir^+= o frn' ifP(t-f,: Bu]= +r (t-L-_r:t\

ilf F4Rce^ (t) n*y,colutpRrssr rrJs H : LJ13A,\,FDRd: J?Fn dn*X,ffirtSttf FOR(eN Gtl : 3,04 (N troe7= iraCIm {

_Ftoa, CouPta fl Foe fuutlBtztura FoA oF

VAtutrg - goo ro

Frzr-*t

?oo

ustrrrg

/oo t *cturua*}s 6 lzrz

3ft

l=tng

b7/2 Anox

. ft t-sq y'|nrt* ftnp

-\g

/N AnBOt lrtw ecc)</,rs

c --ll"*J'- ?bdcos(?oo-ril''t

( t)

cosf= c"sd f

(z)

=,(6+f)= isnhs-e); reEF Saor Eetu1 nc

6.Cg continued theta deg

M lb.ft

beta deg

B ]b

A lb

gamma deg

-30. o -20.0 -10.0 o. o 10 . 0 20.o 30.o 40. o 60.0 60.0 70. o 80.0 90.0

2625.8 3076.9 3643.2 4035 . 3 4568.6 5 16 5 . 6 6855.8 6668.7 7697.3 8 4 6 1. 5 8623.8 6042.6 0. o

-51 .79 -45.07 -38.15 -30 . 96 -23 .41 -15.35 -6.59 3.20 14.48 28.O2 44.80 65.71 90.o0

566.6 679.4 903.7 941 .2 10 9 4 . 6 12 6 6 . 7 14 5 8 . 6 16 6 5 . 7 1866.8 1996.2 1884.O 1247.1 0. o

632.6 677.g 624.2 484.3 480.6 638.6 670.2 868. 1 111 0 . 4 1342.6 1432.A 1172.4 900. o

-46.4 -33.7 -19.7 0.9 26.2 51.5 75.5 -83. 9 -65. 1 -46.7 -22.O 14.2 go. o

Determi ne theta for maxM 66.90 8684.315,37.46 1975.3 14 2 6 . 8 65.91 8684.316:i 37.47 1975.2 14 2 6 . 9 65.92 8684.3141 37.49 1975.0 1426.0 maxM = 8684.3 lb at 65.9 deg.

-32.6 -32.6 -32.6

Determi ne theta f or maxA 68.40 8628.4 41.85 1929.0 1436.061 68. 50 8623 . 5 4 2 . O 4 19 2 6 . 6 14 3 6 . 0 5 9 68.60 8618.5 42.22 1924.1 1436.031 maxA = 1436.1 lb at 68.47 deg.

-26.3 -26. 1 -25.8

FoRCF

AC ON

VALabT oF e 2'5 N'm

p-F#

G c-fa.*f=o

+)E\=o:

*t&= o: n z- B sio'P=o +f t7 =o: AJ - P +easf =cs

^ul&:

R; /3 abg

(r)

OA= P- e'as?

fi=(nfuafS't.

fu = /ai'

ap=

0l -,

cos7ves:

Aew

\\ fi

-e[t

/N AAEo'

fr?Ef tuoy ftela Bp

/vrle*'("

F eon ct ro / %t

cas7, &)

( 1" +a'-

ta-

?l a.c os o)

/ nvv cstr 9te&S

si4

flno

4--nP

(e)

s;ta?

t -- ? s r o Eu

t./

.os+='(t-tu'f)" FPEX WT,

0U Ttrtlg flvzen

O t"

/s RelZ,o,ut

DnTn:&= 8fr,

b=€#., J= 3(T

' 'p=

F = Saol PEo6ErtM/, r,v ,SEdu&cEt

E@uAT/atS

Q) Zzeotrn{

(8)

tvntuarr nNo p/t/*r 44, p, q A, ano ( t=a/? VAtun o F 6 preoru -lgoo To ?ao gst*i /Oo

POO BPI

trvcEarpae,t4<,

ouruNf oF PEL6,EM| fttn /=o,?Sa,6t: o./,fa\, $=?.5^1'1', FEUPnvt,/N ffiacnc$ Eq Un7laNs2frbuta(e). EVfitunE nND PfUr^r

flao trla fuR

Q=o

Ta /zoo ast*4

/o6/,rrc77giua7t2t

oyty ylttteroc -say a*? cNgL lzxaurzeq /voT Q rzsar) thetA deg.

(co,v-zzrvrQ

r(4 o)

AD mm

F N

MA N.m

10 0 . o 0 106.64 120.61 141.69 16 6 . 9 7 19 1 . 8 1 217.94 243.62 268.28 291.66 3 13 . 0 9 332.64 360. OO

10 . 0 0 0 10.319 11 . 0 5 1 11.790 12.286 12 . 4 8 9 12.464 12 . 2 6 1 11.979 11.662 11 . 3 4 2 11.O40 10 . 7 6 9

1.OOO 1.O89 1. 3 3 2 1. 6 6 9 2.O39 2.396 2.714 2.987 3.214 3.400 3.661 3.672 3. 769

EauuB.R1eH ry, Vfrttt* f*ont Usilr{

oF O o /f ro ?oo ,fottcn*qqzg

Also, llnr* Ahg. co4ryfa*orwd It4tus or Q.

AA= O,ge;a,

fuZ: F//VD!

,*lfttrytra\€

oF Vm-r/c,f L f-aAcFS ftPPtlED 7o Bntfute 'ir P Fa? v+t ucf trtzavt o,rt irt, To e,6 in usirya Oc d O, ?Cd,

? (r) (z)

/HC Etl."rttu7J,.

(") o.?sirr.

a tosO

P= 3o tb

P(z,.rr>) + F* (o,zsrn)-G G,?9h ) Two - funcf

A. 9/a e

Faa: goof : Roa cD

AB = Q,&dla.

tvctwt BHz

lv a

F ' g- -A&q d

+dd 7F =o D sn€ - fas$ ^-P u-

,(0

frBy=(oe"- ,/')"^

' 3 ' =' YFAL B y ' 'P

(e)

3s P to,?,rFy- FY& )o,zt

tan g

ff?cf

BoAf , ftv7rT

7=(4"tr;f"

(+)

rrrlEa+pzvr-f *r

+)Et/r=o

(z) fr+ro,afur,

M-Dd-P/z=6

(+)

y'4= J2J *Ph

it4

/,G,€/h - J

lJ?r,t =o:--Q(r.d,b)- 4Qas-l)-

Q(A&r+o'zsla

e= #lK,,N-r)40q,p'&f

("t)

OuTztts ar F?o6enn = 20n r P= 3A U, 48 o,Vs)". fRoennnitN straunNcq ee unvom5 0) weaan (€). (Uore, ?A.G) *flhas @roa€ fo,(z) rn ftBovGANAtysu ) *'u4Lu+/E Oe d in.

0.4000 0.4250 0.4500 0.4750 0. 5000 0.5250 0.5500 0.5750 0.6000

trb^4

F apo Q

O,f/>1, To 0,61)1, 4r

,rcrz VALueJ o.o?S-

Ert7T.

ABx in.

Fx 1b

Fy 1b

F 1b

0.750 0.736 0.721 0.705 0.697 0.669 0.649 0.626 0.602

700.0 573.7 4 8 1. 6 4 11 . 1 355.3 309.7 2 7 1. 7 2 3 9. 2 2 1 1. 1

3 ' 7 3 3. 3 3 1. 2 300.5 2 7 7. O 2 5 8, 4 243.2 230.6 219.7 2 1 0. 4

793.3 662.s 5 6 7. 7 4 9 5. g 4 3 9 .3 393.9 356.3 324.9 2 g g. 0

634.4 531.9 456.9 3 9 9 .7 354.3 317.3 2 8 6. 4 260.0 2 3 7. 1

F 1b

q 1b

Find d for d 1n.

ttD

'PUrvf

ABx in.

Force i n A B = F

= 500 lb

Fx 1b

a .4732 0. 706 4 15 . 6 0.4733 0.706 4 1 5. 4 0.4734 0.706 4 1 5 .1 For force in AB = 500 lb,

Fy 1b

theta deg.

d mm

D N

M N.m

15 . 0 20.0 2s.0 30.0 35.0 40.0 45.0 50.0 55.0 6 0. 0 65.0 70.0 75.0 80.0 85.0 90.0

6 4 . 10 9 40.843 2 5 . 15 5 13 . 2 0 5 3.339 -5.283 -13.137 -20.523 - 2 7. 6 4 1 -34.641 - 4 1. 6 3 9 -48.737 -56.O27 -63.601 -71.557 -80.000

s59.808 412.122 3 2 1. 6 76 259.808 2 14 . 2 2 2 178.763 1s 0 . 0 0 0 12 5 . 8 6 5 105.031 86.603 69.946 54.596 4 0. 1 9 2 2 6. 4 4 9 1 3. 1 2 3 0.000

0.89 I .83 3.09 8.43 5.72 4.06 3.03 2.42 2.10 2.00 2.09 2.34 2.75 3.32 4.06 5.00

Deternine

278.5 500.4 278.5 5 0 0 .1 2 7 8. 4 4 9 9. 8 d =rights 0 . 4 7reserv 33 in.

Ptzoaan /:N7Tv Derat a =o,o)ta; fi=O./+t, f=/'So A P?oder*,1 tu -*aud*, E6 uftvoyt (r) zzaoua+ (e). fVnt urtzE 4"vr2 PCroT 4 D, 4ra .k/

Oufrprc'oF

403.3 403.1 4 0 2 .9

59.9 60.0 6 0 .1

Mmin & corresponding

-34.501 -34.641 -34.791

t{inimum M = 12.00

value

86.952 86.603 86.254 N.m when theta

theta

12.00004 12 . 0 0 0 0 0 12.00004 60.0

deg.

ed. No reproduction or distribution without the prior written consent of McGraw-Hill Education.

Chapter 7 Computer Problems 7.C1 An overhanging beam is to be designed to support several concentrated loads. One of the first steps in the design of the beam is to determine the values of the bending moment that can be expected at the supports A and B and under each of the concentrated loads. Write a computer program that can be used to calculate those values for the arbitrary beam and loading shown. Use this program for the beam and loading of (a) Prob. 7.36, (b) Prob. 7.37, (c) Prob. 7.38. 7.C2 Several concentrated loads and a uniformly distributed load are to be applied to a simply supported beam AB. As a first step in the design of the beam, write a computer program that can be used to calculate the shear and bending moment in the beam for the arbitrary loading shown using given increments ∆x. Use this program for the beam of (a) Prob. 7.39, with ∆x = 0.25 m; (b) Prob. 7.41, with ∆x = 0.5 ft; (c) Prob. 7.42, with ∆x = 0.5 ft.

ci P1

A a

Pn B

Fig. P7.C1

b a

B P1

Fig. P7.C2

7.C3 A beam AB hinged at B and supported by a roller at D is to be designed to carry a load uniformly distributed from its end A to its midpoint C with maximum efficiency. As part of the design process, write a computer program that can be used to determine the distance a from end A to the point D where the roller should be placed to minimize the absolute value of the bending moment M in the beam. (Note: A short preliminary analysis will show that the roller should be placed under the load and that the largest negative value of M will occur at D, while its largest positive value will occur somewhere between D and C. Also see the hint for Prob. 7.55.) 5m 20 kN/m

D a

Fig. P7.C3

09_bee77102_Ch07_p011-012.indd 11

3/13/18 8:13 AM

7.C4 The floor of a bridge will consist of narrow planks resting on two simply supported beams, one of which is shown in the figure. As part of the design of the bridge, it is desired to simulate the effect that driving a 3000-lb truck over the bridge will have on this beam. The distance between the truck’s axles is 6 ft, and it is assumed that the weight of the truck is equally distributed over its four wheels. (a) Write a computer program that can be used to calculate the magnitude and location of the maximum bending moment in the beam for values of x from −3 ft to 10 ft using 0.5-ft increments. (b) Using smaller increments if necessary, determine the largest value of the bending moment that occurs in the beam as the truck is driven over the bridge and determine the corresponding value of x. x 750 lb 3 ft

750 lb 3 ft B

20 ft

Fig. P7.C4

*7.C5 Write a computer program that can be used to plot the shear and bending-moment diagrams for the beam of Prob. 7.C1. Using this program and a plotting increment ∆x ≤ L/100, plot the V and M diagrams for the beam and loading of (a) Prob. 7.36, (b) Prob. 7.37, (c) Prob. 7.38.

dn dk d2 d1

hk A0 A1

Fig. P7.C7

An – 1 Pn – 1

A2 P1

Ak + 1

Pk + 1

*7.C6 Write a computer program that can be used to plot the shear and bending-moment diagrams for the beam of Prob. 7.C2. Using this program and a plotting increment ∆x ≤ L/100, plot the V and M diagrams for the beam and loading of (a) Prob. 7.39, (b) Prob. 7.41, (c) Prob. 7.42. 7.C7 Write a computer program that can be used in the design of cable supports to calculate the horizontal and vertical components of the reaction at the support An from values of the loads P1, P2, . . ., Pn −1, the horizontal distances d1, d2, . . ., dn, and the two vertical distances h0 and hk. Use this program to solve Probs. 7.95b, 7.96b, and 7.97b. 7.C8 A typical transmission-line installation consists of a cable of length sAB and weight w per unit length suspended as shown between two points at the same elevation. Write a computer program and use it to develop a table that can be used in the design of future installations. The table should present the dimensionless quantities h/L, sAB/L, T0 /wL, and Tmax /wL for values of c/L from 0.2 to 0.5 using 0.025 increments and from 0.5 to 4 using 0.5 increments. L A

B h

Fig. P7.C8

7.C9 Write a computer program and use it to solve Prob. 7.132 for values of P from 0 to 50 N using 5-N increments. Copyright © McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.

09_bee77102_Ch07_p011-012.indd 12

3/13/18 8:13 AM

7.C1continued E^/?€R PRobtert VuMsrR 3 e t e c l 4 P p R , o P R r l \ TueN t 7 9 O F L E N G T t lR ( Y DF o ( C t eNlTr? V*t-u€s oF Lf &, AND b Zt.ttt Q tYur.l.g[R 4v tr \- }eP s Gtv*N', ,rL , FDRL= 1 ro Tv ) ENTe( .P,, nilo oL CoMLeN FEr{u SoppoRrlN6 m*N6tr!6 coqpvrE. R',8 FRort4Fo ' ( t) : TRA'leD LoftD1. 70CALCvL*rt F(opt Fo. G) Cott purre R; il/Rrr€ Cot{ Atvo P ? r u^,DFR B rF{ VftcuEg hUp A oF ff6 ANP FaE EnCH SegpfNG Mot\4ENr4f AD. Lo C o f i p u r r F ' A p F k o n a r 4 " 1 J ) , d s r w a o r r / L yL o A D g Co^rcE^ITRATED rJ g & o *Dr F Lo ^4 R}_gq'tl4 Afva wt+tctt E+t f c To FoA h?PLY 7 . C o t \ ? o r l N I A F R o r l F A ( 4 ) , U S t r v G6 N L f L o A D s b,7,36 , (b) PRoa ,1,32, (.) PRoB,1,sg (a) PRo fOA W+tcFr C; I5 -ANALr'g WE PRtrurrYR AND MB F r R 5 7 b E " E R M , N F T f ( E R E f \ L - f t o h It n r .TO @PI?ITE BENyING AoI4ENT N; tI^'D€(. LaW Pd 4t TftE >vPP1RTS, -cr) =e 'lf tt<&;USE E&,.(S)

rtZMA=o', %L- F,P;Gi f+ RA- = l U L

P;k;,-o)

+qz5= o : R S *R * - 4 , 7 ; = Q 4

t't2

(t)

i-l

K =/

i=,

03EEQ.(6)

IF C;

Tt - R,

'(a) [nr ore' TrrAr r t/ Eo 5, (5), ((), kNfr (7), T*e 5 uI\43oN LY

BErtD,N6, n4o i,rEN\T A T A :

&lBi n -fl^

lF a<ci< L+a:

f P:.t sooY,'.t".T1_nflG^l o

-lJ f^nuuoe Tr+x LoA;Ds Pk FoR Nt+tctl k I't, T A 6 U L B T E R E S u L T S p B T f t r r . f1D F o R L , = 1 , 2 , , , r r T1' t ' ( z A r NN TG f L) ci, pL,kN'o M,,

BobLAM Ott-reur (0-)

(;)

/\ -h' )

fF? fli=-

Bendi ng

m om enEs under

0 kN.m 0 kN.m l oads

the

c(i) m

P(i) KN

M(i) kN.m

0.50 1.50 3.00

40.0 32.O L5.0

30.30 39.75 7.50

Problem 7 .37 ReacEions: RA =

Mi=

(c)

- (Lro)] * R^(oi-ilscrl% , (7)

(cDNrl NUED)

kips,

5.50

Bending Bending

moment at A = moment ats B =

Bending

moments

IF C;

-R D

A = B =

m om ent at moment. at

37.50 kN

.t_t

A;=-E,Pt@,-')

R -F

RB =

50.50

Bendi ng Bending

1 2 3

B E T D l N G / - 4 o n . t E r uM7 ' L , N D E R 'L D A D -Pl r .

{F C;(d,i

Problem 7 .36 React.ions: RA =

under

RP =

15 . 00 kiPs

0 kip. f t, -e kip. ft Lhe loads

c(i) fr

P(i) kips

M(i) kip. ft

2.00 4.00 8.00

6.0 L2.0 4 .5

L3.00 14.00 0.00

Problem 7.38 R eac t i ons ': R A = 2 4 0 . 0 0 l b ,

R B = 3 0 0 . 0 0 1b

1200 lb. in. 1800 Ib. in.

atA= atB=

Bending Bending

moment moment

Bending

moments

c(i) rn.

p(i) Ib

1 2 3

0.00 35.00 70.00

i z o .o

o .o o

300.0 r.20.0

1800.00 0.00

under

the

loads M(i) lb. in.

Gtle rl r WsuP?oRr€D

BEnnl

7 CZcontinued

Cft RRyrnlGr'tn/ dl Ct I 7t7ft7XD }tor Ll NE O F ? A O G Q . * r , 4 N untBER totos p-ittD4 uwtFaRr4LY Erur< e PRo6cEt4 '}E?STR (0 t5vreD t&q D tntE unjrTi otr LE$GrttanrDF0RcF s?Leq A?? PR

vl,.RrrQ*o v?n=e ?eut(ftA f rJrtsR

-I r0 &tcu LA E r*E s#fAQ tN ftNo A? No k M ot l?w-r g r+,gRftMS /t/st 16 GNEI\) lrucnet"ENTsa Z-

F$:f sR

rcR t ':=l TA {L eil"rR P; A^lD c; l) AH u-Tc R, PeootE 0 . ( O Ro FR.onE:(i,( Z) Ccn^P AZ t r oR L = O r 0 L u S t N 6 t N C R E r q F h ' T S (r) M F(opl ( te)' VC?t) FR'owl( rr) AND b|I?uTe etsFt've-e wttEqe V;(4 kND At(r) ftce

(a) PRW,7,3q NrT+l*tz'* 0,1;!n (b) Pfil.a,T4l wnH L 7 : 0,5 ft c ) P R 0 8 , 7 , 1 f 2w r T H A v =

AND A{|ALVE S oF L,, &, b ,d, Nt-tpfgeR. n, 0? CoNCE*t T Rn TeD LOAD 5

NG_ uFoN&, ^ (s) on ros,(0 I F-F'ty?r 8v Fc{S,

O,Tft

ArD Wrteee Y,, (r) n vo n* @) ARE D FFr MEo 8Y E@5,(4), E'rls,(7) n*D (8), oQ, 6re5,Cil ht'/Djl D E r e r , J D r6N U P o n tT r t e y ' R u v e 0 F r C . ' FR rrur vR trres oF tC) V (a), AN I n (x)

UlE rrRSTeeTeRplrMg7+F Rr* cttonlS r+T T#F s!?PoRTS;

PAu eG.A+4 o\JT PuT 2"--{ N(b-a) +1 EMA=*o; ffi;t B' %r:t,Er;_il(b-r)i@+g=o t} ?gitElr'J'nt'(b-'oxu*61

|(a+l)

(Crl

Pr ob . 7 - 3e xVM m

+t L,g- *

R n* R u3 41_

Ro= VPt JI{EA(

(b -a)= o E ,Ot'4 $ + ei(t-o)-P,s A

4 v> E. tt, _burE T 0 & AT e

Vs(x) = K^

p6r4:r,tc2 L F*M

Mp (r) = ftl*

(3)

TD tofrnP;; J I - I E R Cf t r { D t , A 4 . } / C tF x. S ci : V;(.) - o ) tvt;(r) =Q

tF tL

lr)c-<tu: Vn)(*)-'o, M*(x)=fl lF Q<z3b:

,"6 --Q

- ur(x-cr) Y"(^) = $*r-_.i(x-a,) L

0.00 0.25 0.50 0.7s 1.00 1.25 r_.50 L.75 2.O0 2.25 2.50 2.75 3.00 3.25 3. s0 3 .75 4 .00 4 .25 4.50 4.75 5.00

lvl ' t

'filAv

lMl*l' =12.0k''{

xVM ft

V , ,( * ) t - a ' ( b - )

il*(r)=- N(b-c,)[z.'ik$| ,

A I>TKL ir+taa AnlD.6Zrtp rN6 H oF\Er.fT I c o MB I N l N 6 ' ( k E E / P R r 5 S r o N sA i } T t r r u E 0f t " & > v T ,

V ( r ) = R Ar Z V ; ( r ) * V - ( x ) = Roic+ Z Mi(a)'rMr(z) FA(a)

(rr) (ta

'wreRE RA

ts 6,vl=Nr^l(z) ANPdrtFRe V; (ar),Mt(| M^r(il frReorrrr./ED by E6is.(,r) ftND % C"),

-f

lo), rr*f C.t*o/ca pF rr+E ftlPRo eiatAT e Ha.ov 6 H (--bE eeND lt'{i 0v T t+E VA '-JE OF /' rorV 5Q'JAT

LJED) (contTn-t

0.00 11.50 23.00 34.50 45.00 57.50 59.00 80.50 92.00 88.50 85.00 81.50 78-00 73-72 57.88 50.47 5r.50 40 . 917 28.88 t5.22 0.00

= btt-.o kN

( b )e ' " u . 7 . 4 t

tF z)b:

kN.m

kN 45.00 45.00 45.00 46.00 45.00 45.00 45.00 45.00 -14.00 -14.00 -14.00 -14.00 -14.00 -20.25 -26.s0 -32.75 -39.00 -45 .25 -51.50 -57.75 -64.00

kips

{b) kiP. ft

18.00 0.00 18.00 0.50 18.00 1.00 l-8.00 1.50 10.00 2.,O0 8.00 2.50 5.00 3.;00 4.00 3.s0 2.00 4.00 0.00 4.50 2 .00 5.00 4 .00 5.50 -6.00 6.00 -8.00 6.50 -18.00 ?.00 : L 8.00 ?.50 -18.00 8.00 -L8.00 8.50 -18.00 P.00

0.00 9.00 18.00 27.00 35.00 40.50 44.00 45.50 48.00 48.50 48.00 46.50 44.00 40.s0 35.00 27.0O 18.00 9.00 0.00

' | 8,DOkips lVl. t ' 'h(X I

IHl*., = 48,,k;P'fL-

Prob . 7.42 x fr

VM kips

0:,00 15.30 r-4.05 0.50 r-2.80 1.00 1.50 1r-.55 10.30 2.O0 2 .50 9.05 ?.80 3.00 3.s0 6.55 5.30 4 .00 4.50 4.05 s.00 2.80 i..55 5.50 -1r-.70 6.00 '-r-L.70 5.s0 - 11.,70 7 . 00 -11.70 7.50 -r.1.70 8.00 -1r_.70 8.50 r-r..70 9.00 -1r-.70 9.50 - r - r .. 7 0 r - 0. 0 0

kip. ft 0.00 7 .34 l-4.05 20.L4 25.50 30.44 34-55 38.24 4L.20 43.54 45.25 46.34 45.80 40.95 35: 10 29.25 23.40 17.55 11.70 5.85 0. 00

I Vln*r= 1530lr,ft I Ml.,or'- tf6,0k,?,ft

GtvrN:

7.Cgcontinued

B t A r { A g H r N G e sA r j

AI{ntY s,5kND f u?m:q-rro Ay R.Ott@ l 4 P r r r + ptv E I 1 4 l o F T H 6 Tt> tIt tt t M tz-E TkE AA,y>cvTE '1 frT D CARStt5 ()N tTrRULY -T /N Tt+ E Et API ) ulE n4usr F++Yr D lgfR I e,uTAD Lo rrD S+tott)ll 13Er.l D t ^{cr HaHE^l 'i0 H tt,A ^,1t7? 8{NJ, tNG hl ttt'r 1-rvT rn.t

,rl

gE n f't

'lMol = M€ \M*,r\=Mnrr/ oR I ' R o A ( l ) tA D = i , , , r t

vhg-

lFr\' ^|a(?)= + N-A. t7

r* = b

;

,-\

(?) FRoAo

T;'t*b t-'wE D1-TtpryrilF Tr+t V=O.' D rcr-au'ce\7,) ZtNr E'r/H?R?

V =*!-.ttv'+= ()

F R"ot-te?/s SFnNCIN C

FoR ?ALtl vhLaE oF t

i"':'--rr >,' l* Lrd

r t ''J t,',J tcl tL

Rrh;a) = R^h;a) A: =

X ljtoPE)a

LER, IS PLACzD 0 MQe? Tr+tr [o"lD.'

l,(rrx,valrr {r

lvl

? e g t L M I n a R v A f t A , L YS l S

4rr0+

7 - E f f t E O T S T A N C El = R t A A T o e .

fr+VStDENOT(NG 8f

F4.(t): Recnn rNr6

I Mro,\= lr",oI =

k=,#

(s)

1=fu

-u+ZN E MF FQOt-EO.(4) ANJ rt+tr CoNPU-7 wr 'ro( rtE rlnuut oF A' trY ME; Qvr+Nr lMDl l M * * l >a I b ( b - t ) = ! r,^rh' SINCF a7 b:

Tt+uJ

el*c.eg

RoU E R. 5 r+oULD €E

RtAcTro^J Fo Ar RocuFR:

TkE lr/otV OETrR&1tfVE [i'lE7 MotV r

,+b JWb 3t

uN.Dg R r + + E L D A D

E3 l Be*u, AB

)> Mr=,o' o

(r) |

F " ( z b - a ) ur b( ? -

(z)t lte = Q

g;t*6t?*4

f4D a+-

;rr'4r

rctF-tMo ,l

lv't.a I a,xi, lMpl , fvtE-W,Tt{ -sHTILLEP A7 |IICRE/TIMIS Pgace6r,''Af Rfne -i'l DfSl QTo

A (t r_]R"Ac y

lg O81 n truED

flRoG Rn P\ ori T i:'tJT

(3) |

T*;

lYtrl, * *b" P\ nor et E

ul A'

-^$d

"h:nl*LYstt TFtff7 l7 F tLEf+R Flaopr Trl-E t(ro:ve T l r e ' L r e 6 r s 7 ' N E G * - r r v , E ' r h L V eO t A a c ( u R s f r 7 - b , f r t { g 7 f 1 f t 7 r r u l t L LB E f ' tt { t ' q ' t z v o F o R A VAuvE oF a- 'mft-LLf4. THAM a,. A ' t - S c r7 l { E L+RGEjT fosrTtVE VALvE oF M wtLL o((v13 AT Et BtTu/€trt D hND C' (6'oNt

Puti ?r-* PRTNT

\=-*'n'ar 11. = C) IF &

ro !l)

r?6P,Eo. (f ), f"!t>u

-^'i'^':t rM,feoyEo(4)

["t]J L

fRot't nQens u v l e €

| z

coPrPurrMD

con?vrE xF

(d |

lF a.=

O ?rtMur{,(VhLuF. Tv'l,tt NE OP iRoerl+rl I t U r e 4 f i = S-rrho, t ' t / = T o k t J / " 1 2 , r r v n + 4/ d c e F ' l - f l v T s D r L = 0 , 1 I fuo a=e orYtJa.f ,l Co\?urE R, trcz I

*L(|b) -

fi"

r) f / ! = o l

Wt+tcH T#tS cpur+,vTrTf tS Z7R'O' t$ ft{E DES| RFD

A m 0.0 0.L 0.2 0.3 0.4 0..5 0.6 0.7 0.8 0.9 1.0 L.lL.2 1.3 L.4 1.5 1.6 L.7

| 1''8

1.9 2.0

2.r 2.2 2.3 . 2. 4 2.5

XE m .75 .79 .83 .87 .91_ .95 .99 4 .03 4.08 4.L2 4.L7 4 .2L 4 .26 4 .31 4.35 4 .4L 4 .46 4 .52 4 .57 4 .63 4 .69 4.75 4.81 4 .87 4 .93, 5.00

ME kN. m

ur-luol

t40 .6 135.9 1 3 L .L L26.3 LzL .3 1 1 6. 3 1 L 1 .3 106.L r.00.9 95.6 90.3 84.8 79.3 73.7 58.0 62.3 56.4 50.5 44.5 38.4 32.2 26.0 19.5 L3.2 5.6 0.0

140.53 135.80 r30.72 t25 .36 LLg.74 r . 1 3. 8 4 107.58 L O L. 2 4 94.53 87.54 80.28 72.74 64.92 56.82 48 .44 39.78 30.84 2L.62 12.10

kN. m

i.t+

-18.14 '28.80 = 3 9. 7 4 -50.98 -62.50

AXE mm ' 1L .. 99 10 00

kN. m 4.53 38.4 4.64 37.8 4 .G4 37.2 4.G5 3G.G 4 .65 35.9 4.66 35.3 4 .66 34..7 4.57 34.1 '33 4 .58 .5 4.69 32.8 4 .69 32.2 J

93 20 0 '. 1L .. e 1.940 1.9s0 L. 950 L.970 l _. 9 8 0 1.990 2 .000

ir m

' t -920 4.'64 L .927 '4.64 . L . 9 2 2 ' 4. 6 4 t.923 4 .64 1 ,. 9 2 4 4 . 6 4 t .925 4.64 L.926 4.64 t .927 4-65 ' . 'Lr . 9 2 8 4. Gs It.gzg 4 .65

ME kN.m

m n -l u o f kN.m

2.3r t_.31_ o.32 /l -Q.69\l -1.59 -7.7O -3 .Zr -4.72 -5 -73 -6.i,5 -7 .7'7 4

ur- luol kN. m

37.2 o.32 37 . L o.22 37. l0.L2 37.0 35 . g -0.18 36.9 -9.29 36.8 0.39 36-7 0.49 i ' ze.z -0. s9 36.6

3:33{

,N I/ED)

continued

-x 75ollr

o u r L tN{. Of PROGqq${ A,lrer. v *tuFJ of P) b, L , Anrg bxTo b Wtrt ix6;gE64gNr1A?c-; Fo? p-b

: o .( t )

l^tr;[il,,,

20 ft

GLVEN: uks f tW Of 6R tffi, ConlStiTSO F N A RRord t"Ln Eft r45P?R u B TED PLY N & P 5 OF.tTh/0 .gr'r't REgTt Drstnrvcf S]TWrEN4XLEg 0F 3OO0-lA TRU6F T R A V E T - ( N 6o ^ l A R I D 6 € t 5 6 { t R N t "{?l&HiL)1" ?:RUC,K lS g.O rJn r-tY O /grR t g vTE.J) OVER t Ts Foug Wtt E PtS ,

M,r,o, -4 I NcREA E tv f5

A7-"

,LR isou (+) FtroN F0, (,) elrub F coHeuTER; FRIH ?a 6) (t) corlpurt Mp Tipou N CZIf *, 14RFeon G) DEreRu rNE Hm^, FRor^'t flR ) H, MK'>+({D Mnnx\ PRlrVr ArxF, FIND: / N r a e n e r l TA s z F{e PaRT(h),, r(0^/ o tr N Ax tfuiJr"l5',ero pl & U S e g M A L T E R ffi1,ag6 ^t t-ti)De 6 rr0 LocA OSING P,ftA grer'r UT OUrtr ITft, ruR ^:--g frlol.{thfr tr.l 0^te{t PROE %t4PurE ^

"-0 0,i- {t tnlC REP{En'lTg (D L*R&e57 Vft,-tJT0F BEND,/V6Hourilt A E r^lquci( ls 'R,vtsN 0v Ee e F |DG"E *ND co{Re5 (4t'tltNG

XXFXR fc fr

V n w r o FL . ATVAwstS

L = 20

P:75o rb, b-3{t,

\= TLb

(r)

F^t*-=- L

TtfuS: ftno+=MF = fte*r

0) C'

P? Ltr

x c =Lfb

ftuo ir

= z-4(A)

+) ZMB=O'n

Pft.4h:[:,t)

Et,

K o =(Ptt)(zL-zr-xn)p) -T l+t/5; Mo = R e L F f ( 2 6 ) l,l R = R o Z - X

(e) (z)

T H e r . / 'tVI'rhry- M F tF MF)tln, 0 TftrcUlrS-E-/ V *n, = M p

(s)

toA

/n vul

MF kip. ft

MR kip. ft

Mmax kiP. ft

-3.00 -2.50 ' -2.O0 -1.50 -1.00 -0.50 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3. s0 4.00 4.50 5.00 5.50 6.00 5.50 7.00 7.50 8.00 I.50 9.00 9.50' 10.00

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.50 6.00 5.50 7.00 7.50 8.00 8.50 9.00 9.50 10.00 10.50 11.00 L1.50 i.2.00 12.50 13.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.50 1.00 L.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.50 6.00 6.50 7.00

0.0000 0.3655 0 .7L25 1.0405 1.3500 1.6405 L.9L25 2.L6s6 2.4000 2.6t56 2.8L25 2.9906 3.1500 3 .5437 3.9000 4.2L88 4.5000 4.7437 4.9500 5.l-188 s.2500 5.3438 5.4000 5.4188 5.4000 5.3438 5.2500

0.0000 0.0000 0.0000 0.3655 0 .7L25 0.0000 0.0000 L.0405 0.0000 . 1.3500 0.0000 1.6405 L.9L25 0.0000 0.0000 2.L5s5 0.0000 2.4000 0.0000 2.6L56 2.8t25 0.0000 o. oooo z.ggod 0.0000 3.1-500 0.6188 3 .5437 1-.2000 3 .9000 L.7438 4.2L88 2 .2500 4 .5000 2.71,88 4.7437 3.1500 4.9500 3.5438 5.lL88 3.9000 5.2500 4.2L88 5.3438 4 .5p00 s.4000 4.7437 5.4L88 4.9500 5.4000 5.1188 5.3438 5.2500 5.2500

x fE

xF fr

xR fr

MF, kip.fE

MR kip.ft,

Mmax kip.ft

8.00 8.L0 8.20 8.30 8.40 8.50 8.60 8.?0 8.80 8.90

L1.oo 1L.l-0 11.20 11.30 11.40 l-1.50 11.60 11.70 11.80 1L.90

5. o0 5.1-0 5.20 5.30 5.40 5.50 5.60 5.70 5.80 5.90

5.4000 4 .5000 5.4067 4.55L8 5.4L20 4.5020 5,.41-584 .5508 5.4180 4.6980 5.4188 4.7438 5.4L80 4 .7880 5.4L58 4 .8308 5.4120 4.8720 5.4067 4.9118

5.4000 5-4067 5-4L20 5 .4158 5 -4 1 8 0 5-4L88 5.4180 5.4L58 5.4L20 5-4067

MF kip. ft

MR kip. ft

MMAX kip. ft

5.4185 5.41-85 5.4L87 5 :4L87 5.4L87

L z 1x

B B c * u : r o F i y r y . t t E T R y , J *uu€S O B r4rru E) 1=uR x c kr/ kLso BE d sFo F o R L - x ,

XXF fcfr,

XR ft

8.45 8.46 8.47 8.48 8.49

L1.45 LL.46 LL.47 L1.48 11-.49

5.45 5.46 5.47 5.48 5.49

5.4L86 5.4.185 5.4187 5 .4t87 5.4187

4.72LL 4.7256 4.7302 4 .7347 4.7392

8.5L 8.52 8.53 8. s4

11.s1 LL.52 l-1.53 rX.54

5.51 5.52 5.53 5.54

5.4187 5.4187 5.4L87 s.4186

4.7482 4.7527 4.7572 4.76L6

8 . s o 1 r . . s 0 s . s 0 s . 4 . 1 9 q4, . 2 . 4 ? 9: . + + 9 9

(conl l NrJrp)

5.4187 5.4187 5.4L87 5.4185

{

I Prl

7 .C5continued I

B& +.'l

r./r

7'36 (a) Problern7.36

(kN)

/ -i

M ( k N .' m ) 1

., 1

: 0

0.5

kips) v ((kiPs)

2.5

1.5

3.5

'37 (b) Problem7 (b) Problem7.37

(ft)

(in. )

M (kip'ft)

7.38 (c) (c) .Problem .Problem 7.38

v ( l1bb))

M (1b'in. )

(in. )

* 7.C6

G t v eN i BEAM Ah/DIOADIN6 0 F P R0 8 , 7 , Ce (A/KrcE*'uhwr.r TRoeg,nu-r0 PL7TV

*'7.C0 continued

nv' A DlnCYQhlt4|, rJ5lt'lC lNcRfMENre Az = L/loo , APPLYTIfl1 PRyGR+H

v'(kips) 20

6) ?nofr,7,It,G) mQ!),l,+? A N A L / S t 5 . 5 e f - S o t ^ r l T t Dt t 0 F ? R 0 9 . , 7 , C Z 0-F PRoc>e.nM , llrltn{E g gf EMTER PRo6t ,?M N'LUU FNTE( VAuUEJ oF , &, b, AiNDt^, E d T E R N U Mg E R O F C D N / C E N 7 rR: -4D L O AD 5 PoR,; = t To qv EM rC( P; kvD A; (Z) 0F ? 7,CZ Cou p urE & o kMD Rt+trRou Fos. ( r) AruD s * v a r c D T A G R s / v ll D n + U r A n /p v A r F s * N O P R r X r L A B EL - u / tS lo F 7 R 7 - = 0 T o L , t / - 5 r N G f. n / C R 6 - A , l E v /t T Ctst4pv.rE-Ua b\ FR orq Ee, G) oF p 7,Cz Fon i=t To & i c o H P u t E V ; ( z ) F R o M . F Q6 . ( L D , ( 5 ) , 0n ( t ) COHPUTE,/ruU) l=Q.omEo,(7) oR (?)

(D)Problem7.41

0 -1-0

-20

( kip'

ft )

50 40 30 20 10 x

(ft)

Cor4?uTEV (x)' Lt4?ou ro( l) lc ?LoT potNt , v (zll '. Bevar N e -p{oM g ru; ot A &R ttn RTprat aBlut ?R,I:EDUPtwtrH A /dsrEnD1Fv rz;tlpurrN/G U (z) 1Q-oH ro. (,2) AwD Ptp-rrrN L

Porl,rTIx, 14&)1 , P R Dc n s n t o u r p u r

(a) Problem7.39

v (kN) 60

(kips)

40 10

20 0

(n)

x (fr)

-20 -10

-40 -60

-20 -80

M (kN.m) 100

M (kip'ft)

50 80

30 20

tof.J

20 0

0 x

x (ft)

(n)

Gtua' Cneq' aNo G:avtrzn , loftDtnz

#/oa>'+/.

vvr?E P/aannn 7Z usF /N cllLc ucA 7/44 '\

@")r^- (A,)g 7aF/y

USE

Plbgrcnn

Pk*t

5*s

7.C7 continued Printout for Prob , 7 .96 Span of cable = 32 Elevation of An above Ao: ho = 6 Number of loads: n = 3 = = Load 1: P 300 d I Load 2r. P = ZOO d = 16 - Load 3: P = 300 d = 24 Value of k for selected load: k = 2 Vertical distance to point A sub k: hk Horizontal Vertical

Component at Component at

An = An =

2666.67 900.00

@zt,n,

7.?,t1,7.?64 nro 7,77h Printout for Prob. 7.97 S p a n o f c a b l e = 10 Elevation of An above Ao: ho = -4

(e")y

@r),

Number of loads: m = 3 Load 1: P = 5 Load 2: P = 5 Load 3: 10 .P = Value of Vertical

,l; jPo

d = d = d =

k for selected load.: k = z distance to point A sub k: hk

Horizontal Vertical

Component at Component at

An = An =

21.4Zg 1.429

(t)

Gturot cnstE oF l.avrs 8" fiya F FEaT /unr

'Wer?E Qz6rann e

Dtvetep

(z)

+)fvn*=o' (A-),U,-Jd -F)rt o (&,?)- (ea,Nhn)

OUTZIPF

(n,)r=ItuL J, ho

C/Z

-H: ?ir;

4t-tO FQet

T/.t,

7o O,S

USr'e6

a( Ag'yC

O,g

Lo/-t O,ofS

/NavAEHETvT=.

/*7: ,b =

$e=

7v7nl CrtsLElfllfi/t

"-2 h' 9e- rc

(fr)x

oF Pb4fZnr't

Printout for Prob . 7 .gs Span of cable = 32 Elevation of An above Ao: ho = Nunber of loads: m = 3 Load 1: P = 300 Load2: P= 200 = Load 3: P 300

d = d= d =

I 16 24 g. O

An = 800.000 An = 550.000

+ €P | f4

(C o*Vxuea)

Trrry= -3 A lzvrca

VetuE (ae rytrU)

t+rz

Euatmzt FtU6n,tu, w ,feourx€, Eoutzra*s (r) ruautt(g) ADrwrlnEr H f?E:@u67ED SprctAeO Vfrttct oe d ANO Pra*r

k for selected load: k = 2 distance to point A sub l(: hk = Component at Component at

"-") 4A= 75s

(ta 7,/q 3e= E esh I = {c(et+ Z= ura

Horizontal Vertical

I zo

Tr/E FotJa,/P/46

h/t , Sn"/t A'?

FrZUn

(o (ea,?,ts) Sa= c, srhlt b = *-(.i-

p; - r/*, ho, 7.o6os farpn', ( ano ( Przumn .Fas s) +)

Value of Vertical

TlTttr

(cl

o P- Fr2Ol;fZ,qpt:

PA,yT

(nu'E

&u*rv

?-/

.AuALmTF pluo

Fon

/naZgrtAzS

- ( hyr) ,1',rfrGr;-ro) E,orut Y/tt6

Evsrevl?sr

PACaEzvzg

T,*T

ttJ?

yrrtx:

h-l

E*l)

pln

'7/tel?

f Lfw67//

a Co/wt PAzezZ '#o u-?F / 7 7a

g@ /h

c/L

s/L

h/L

Tolwl

Tmax/wl

0.200 0.225 0.250 0.27s 0.300 0.325 0.350 0.375 0.400 0.425 0.450 0.475 0.500 1.000 I .500 2.000 2.500 3.000 3.500 4.000

2.420 2.O52 1.813 I .650 1. 5 3 2 1.444 1.377 1.324 1.282 1.247 1. 2 1 9 1. 1 9 5 1.175 1.042 1.019 1.010 1.007 1.005 1.003 1.003

1.0265 0.8253 0.5905 0.5944 o .5225 0.4667 o.4222 0.3857 0.3554 0.3296 0.3076 0 . 2884 o.2715 o . 1 2 76 0 .0841 0 .0628 0.0502 0 . 0 4 1I 0 .0359 0 . 0 3 13

0.200 o .225 0.250 o.275 0.300 0.325 0.350 0.375 0.400 0.425 0.450 0.475 0.500 1.000 1. 5 0 0 2.000 2.500 3.000 3.500 4.000

1,226 I .050 0.941 0.869 0.823 0.792 0.772 0.761 0. 755 0.755 0.758 0.753 0.772 1 . 12 8 1.584 2.063 2.550 3.O42 3.536 4.031

Gtv&zv:

Wrea oc

/r*rmt-f*Zootr Massfar I Tn ' o, ?-Eg/e,r SPau /. AHD &€ O 70 -ro fv

/=t*p Forz

US/il6

-U-N

/NcEEprFNB,

20 m P m

Tmax N/m

ybc mm

0 5 10 15

1 9. 6 2 0 20.247 22.021 24.697 28,017 3 1. 7 8 0 35.846 40.124 44.553 4 9. 0 9 1 53,712

10 . 0 0 0 10 . 3 2 0 11.224 12.599 14.29A 16 . 1 g g 19.270 20.451 22.709 25.021 27.376

z0 25 30 35 40 45 50

ArzqL

COrttft'v&7

er = 1.962 N/m Lh mm

0.000 2.549 5.097 7,645 10.194 12.742 15.291 17.939 20.397 22.936 25.494

0.000 10.582 14.541 16.565 17.694 19 . 3 6 9 19.794 19 . 0 7 9 19.274 ' 11 99 .. 54 11 95

10 . 0 0 0 7.771 6,127 4,942 4.096 3.455 2.990 2.612 2,320 2,095 1. g g 2

(t)

sa= * s VfQfrent

ke/n

= -t

oF cnate

lWaH

m=0.200

Shncrp*

eT g

(z)

Bg= @tJ-sn

(s) ,Fa 78 l:azt?:

(+)

rvrfl ar= 413 !

(s)

j;-*;

t/)

la=ae-p

tq,2/7,'

f" = .sral b

fa?./S: A=nsrLl f ;

aa.

2 = ,srtr|-'3

t= +

/er

(7)

gah-'I

TG= '

/) cbt(or+r1utJ

l= Zve

gi= ?oaz r ? = o, Z*g/or, X=23/ tu ,9eo u tuc, ?G u4 7 rc*f fzaazft*t, (r) Tt/r?ou6n (g)

4%L

7B= rc .uq t

EYrsZ

/tVaturlrli

?ND P2/N7

A eno /.

kazc:

PaunT?o^t I Fe hezes oF /NT€cR/fLf

,Aanrzav*T/cAL

49 S>wfa1 ,rfftruo&oop ly .*tc6&

{c4A,t*t

,{CrZtsS,

*?e €Ana,t-HtlL

(c*

uEo)

oo7lJ/vE

-ga: *rltant.cftn

Chapter 8 Computer Problems 8.C1 The position of the 10-kg rod AB is controlled by the 2-kg block shown, which is slowly moved to the left by the force P. Knowing that the coefficient of kinetic friction between all surfaces of contact is 0.25, write a computer program and use it to calculate the magnitude P of the force for values of x from 900 to 100 mm, using 50-mm decrements. Using appropriate smaller decrements, determine the maximum value of P and the corresponding value of x.

B 1000 mm

8.C2 Blocks A and B are supported by an incline that is held in the position

shown. Knowing that block A weighs 20 lb and that the coefficient of static friction between all surfaces of contact is 0.15, write a computer program and use it to calculate the value of θ for which motion is impending for weights of block B from 0 to 100 lb, using 10-lb increments.

400 mm

Fig. P8.C1

A B

Fig. P8.C2

8.C3 A 300-g cylinder C rests on cylinder D as shown. Knowing that the coefficient of static friction µs is the same at A and B, write a computer program and use it to determine, for values of µs from 0 to 0.40 and using 0.05 increments, the largest counterclockwise couple M that can be applied to cylinder D if it is not to rotate. 8.C4 Two rods are connected by a slider block D and are held in equilibrium

by the couple MA as shown. Knowing that the coefficient of static friction between rod AC and the slider block is 0.40, write a computer program and use it to determine, for values of θ from 0 to 120° and using 10° increments, the range of values of MA for which equilibrium is maintained. C

75 mm A

150 mm

B D

150 mm

Fig. P8.C3

250 mm MA

2.5 N⋅m

150 mm

Fig. P8.C4

10_bee77102_Ch08_p013-014.indd 13

3/13/18 8:13 AM

B θ

Fig. P8.C5

8.C5 The 10-lb block A is slowly moved up the circular cylindrical surface

by a cable that passes over a small fixed cylindrical drum at B. The coefficient of kinetic friction is known to be 0.30 between the block and the surface and between the cable and the drum. Write a computer program and use it to calculate the force P required to maintain the motion for values of θ from 0 to 90°, using 10° increments. For the same values of θ calculate the magnitude of the reaction between the block and the surface. [Note that the angle of contact between the cable and the fixed drum is β = π − (θ/2).]

8.C6 A flat belt is used to transmit a couple from drum A to drum B. The

radius of each drum is 80 mm, and the system is fitted with an idler wheel C that is used to increase the contact between the belt and the drums. The allowable belt tension is 200 N, and the coefficient of static friction between the belt and the drums is 0.30. Write a computer program and use it to calculate the largest couple that can be transmitted for values of θ from 0 to 30°, using 5° increments. Q θ

θ C

Fig. P8.C6

B 10 in.

10 in.

Fig. P8.C7

8.C7 Two collars A and B that slide on vertical rods with negligible friction

are connected by a 30-in. cord that passes over a fixed shaft at C. The coefficient of static friction between the cord and the fixed shaft is 0.30. Knowing that the weight of collar B is 8 lb, write a computer program and use it to determine, for values of θ from 0 to 60° and using 10° increments, the largest and smallest weight of collar A for which equilibrium is maintained.

8.C8 The end B of a uniform beam of length L is being pulled by a stationary

C B' θ θ

A' x

Fig. P8.C8

A L

crane. Initially the beam lies on the ground with end A directly below pulley C. As the cable is slowly pulled in, the beam first slides to the left with θ = 0 until it has moved through a distance x0. In a second phase, end B is raised, while end A keeps sliding to the left until x reaches its maximum value xm and θ the corresponding value θ1. The beam then rotates about A′ while θ keeps increasing. As θ reaches the value θ2, end A starts sliding to the right and keeps L sliding in an irregular manner until B reaches C. Knowing that the coefficients of friction between the beam and the ground are µs = 0.50 and µk = 0.40, B (a) write a program to compute x for any value of θ while the beam is sliding to the left and use this program to determine x0, xm, and θ1, (b) modify the program to compute for any θ the value of x for which sliding would be impending to the right and use this new program to determine the value θ2 of θ corresponding to x = xm.

10_bee77102_Ch08_p013-014.indd 14

3/13/18 8:14 AM

+o = /an-'<*

theta deg.

900 850 800 7s0 700 650 600 5s0 500 450 400 350 300 250 200 15 0 100

23.962 25-.201 26.565 28.072 2 9. 7 4 5 31.608 33.690 36.027 38.660 41.634 4s.000 4 8 . 8 14 s 3 . 13 0 s7.995 63.435 69.444 75.964

P N

Find x for

Pmax

354.95 354.90 3 5 4. 8 5 3 s 4. 8 0 3 5 4. 7 5

48.415 4 8 . 4 19 4 8. 4 2 3 4 8. 4 2 7 4 8. 4 3 1

43.027 4 5. 1 3 7 4 7. 4 0 6 49.832 52.406 5 5 . 10 5 s7.882 60.655 6 3. 2 9 2 65.584 6 7. 2 2 7 6 7. 7 8 9 66.708 63.324 57.003 4 7 . 3 72 .646 ,34 67.796V52930 67.796752930 67.796752930 6 7. 79 6 7 6 0 5 5 9 67.7967s2930

P m a x= 6 7 . 8 0 N ; o c c u r s a t x = 3 5 4 . 8 m m I theta = 48.4 deg. l

(t) (z)

e = /an-'(?")

x mm

g.c2

I We= 70 lb , as: o. /.9 n.?

(s)

I s

/r'af2:1v2/pd

4 Uslrv€ /6 -/6 ttuceeptenzf

Frzer

(+)

6?eY

Szae

A ssa/et W€>Wa /fn=oz rlr= Waas6 I\ tF=o: T-lNns)n6*4sNt 7= INn €ta O f 4sW^

-5=1t

tr, *l f

jr '

ftur

N - 2g -eurn.s=o

lr=o: /V= trg* h** S

?rtal ,

4,+B= /o'htt

fa,0-fa@t

UCt-un *f6-?7?4

h = ?or:&^,

*grot*=

ouTlttrE or fro6nn^4

x= ?,fl%'

EO(4,

?Ag

(t)naouth)

tN S1OaENG)

eQunT/trt

$uo

71/ o, ,qu7 P FA

PEtrtT :

?oO +>rd

DEcEFnfwTT.

(s)

taos= utffif ustu6

Faq,ry., furatuzrt:

-ftl -Wure6

(z) TWc s,IrO -4s 6ns6 (zug+w) Wasia6-4s W1s ctrQ - Wgflie t-% asa(Zw*tue) (wo-tvu) sin a = 4s cos 6 (g*o +wd

P2oeQA*l /ooo

(t)

-- Wcsin e 4 4s4 +1sil2 = o 7 W6sin 6 *4< tu^cas6-4s(wrrtkke

Fr= z75Nz

(-tl

casO

6=,sil, ,t;?::"+wccose

Fu=ct

P = R*+-ar.,l

EfiZ*re-

/Zf =oi

Eoo|'t Bzocr 6

(gl

o 4r+'1/kN- P=

C)uT/ //vE oF

FoZ

/oo4l

l4rg=O

- (*ruil)Aaas*

R, h *zsv

/nt oTrary

tui/clf

$) +)T4n=oi

l/tuue aF a

Foa

/ao4q+'

U.S/,vj:

lfln-Tttttt, 4( = o,E Featztta VUB= o 7o toa/b EvALqqTe 4ilr2 Pa*v7 O 4o to-r6

o"r*t*rr*, wB [ lb] 0 10 20 30 40 50 60 70 80 90 10 0

Ftztn

lNcr/€tat-vtzt.

oe kh)rs theta [dee. ] 24.228 4 6. 3 9 7 53.471 36.870 2 8. 8 1 1 24.228 2 1. 3 0 6 19.290 1 7 . 8 19 I 6.699

NoT6;

Foe

lNelwn

7vE

Wn-wc. Bl oc k A Bl oek A For any Bl oc k A Bl oc k A Bl oc k A Bl oc k A Bl oc k A Bl oc k A Bl oc k A Bl oc k A

m ov es m ov es theta, m ov es m ov es m ov es m ov es m ov es m ov es m ov es m ov es

dow nw ar d dow nw ar d blocks do not upw ar d upw ar d upw i r d upw ar d upw ar d upw ar d upw ar d upw ar d

move

GtvE/v: CrtlNp€e C, T, -3'uJ 4=

SVq Tlc trR/cTloN nTftttDB /]/?6es7 couNTaz"/ Cl&eeu:se Cou Plc tl Vtar

F/np;

/\/nr

tan

R.6ffi

t/y-7?r" mnt y =6.5 &3 t J = V6/ 46-, at-?O./.f,n (il. feaAvaxs (r)'ryraalt/ ftro;cmq, lN seou€Nce, 'Vqlu6 FoR f Vntuaze ftpo Przt*7 A4 exo 4r*-o 6/= As Faorr.t O 7o O,?o ushv{ A,Of, tnCeevr'2/w7s

rfi;3ttf3o.'ilu'lrtol

rT,,

tS NoT

P=+= I

g=/",i'!

0.037 0.o58 0. O82 0.109 0. 139 0.174 0.214

18 . 1 7 8 2 9. 7 1 5 43.526 60.359 8 1. 3 2 7 10 8 . 16 6 143.745

0 . 10 0 0 . 15 0 0.200 0.250 0. 300 0.350 0.400

co= $o

Cfo*'te7PY:

OotTure

BE aP4lP

CYI/,wAft 2 7o por4ze

8.Cg continued

CoEFFtcttHT

$ruen; 4s= o'fo Ftun: Pftatr op l/ftLt/ESoF fr74 ryZ //utPFNOtrt6 PrO TloN 1

flss una 5l tfrrr,tl //"1PENDC n T n ,

cnuofte c t ELI.= o

\ )

2.b N,rn /S lvt alN fAWb-D

tr=fi=FB

/tr

l/r+r* fauuquan

(z)

W=h S FAo= FAo

[.-t***!

/oo

/^t 4EytF/23

C YtrtvOtrz D: fMo=o

E=qtno

(i (z)

1'l=Fa

+ lI8 =ot NB*tp -rv-F- Firp

//'/-?PFt/zs ar

SltPPtut

Para

!/ruurs oP O rbkl O 70 /Zoo ustng

A [nof

(t)

frSSunt

!o= F

(+)

l-CgJ

/'tctnur

6r: cottAn

D /YtPtvaS

76trup12s C op K,

Fe ee Goov or r2oo frc

p'+

|.o=; -Fasp*^/Bd"p AB

/'a/2

SuBsvTuTc

,G -*tF

roEF =

- (triop)t..p)

Uu-rT

ruQ^/

nS.

,= =

oF cau2tl:

CuFcta

77/'nT

lrvP Nottt

(s)

(+)

"€

@FF'

n/Zta

/s,

Vl1Lu€

A F

p-+

ED= L

tCas

wtP-(!tsi'g) .os?

(/)

+) f ,u/B=o i'

For? t F?auazo

FRrc BeoY lZaP E-D

/v, = F a

F/fta

NB ' = cllrcx

= tu /anp

cosf -0 * snp)tap

7,,/f,nFa/Z€,

lt/tlcrrzuE

tr*F - (,* *r(r+st:gry =^

/ FT'

--o

;

Co*pct?tu7 -l/

lfGa

V4zut

|tf&C7/d^,

ft7

-+ -r't /,.1G= fV cos( g

lsi

(tt

L *s(p -+ - e)

N6t

futtno

tt4Lr?

ltr laorror+tor cottnt D /,vtPnflot fotu612p A ott ACs 7/E 4r&e + /s tuPtx6sP 8Y - # , 7/a 86uft7/oNs-

(ca-v,vreo)

Baoc,,

70 Ge

PrZOdn4peo

4n.6

€Ufl/wrfir/t?ED

(Carvn,vuro)

8.C4 continuedI

EilaEA On?n: /Vo= ?,i N,a,, //V S€GrvaxcE,

4S = o,fu &= o./,f+a

A=o,?Cnt , Feo9po*\

Gltupy: Saott --g)?,,,,, .45 = o,3O

otr Feaaaapt

Ou7/./nE

77/a ft4a4rt*li

/*r,.v= Zoo x

/ nr?6Ps7 @: 7or?a ue /,/t 74nf

€apa77a6

-a-

AO*= 2 cos 6

C*p BE T&^gkl77GD

4Pg= L Crbe'

F oa

/]D = (ror" *^r;)"' p = b*,,-' (nos/rtor) o6 O //vPEvrS

f laoZo*

/owr+aD C,

ttSE SuAScrz/pT

frrkt-t

f ?oo

ilh,

1rr3tH6

6a? /oc

/ao

vnLues

g.c5

/ar)o

!:r:,3";7!;-

?vzzft'

W%ue c>F f gPwffiO

= o,3o,

f = o,o8+z

fv4LutvE

hrrr*

rtvil> P/?J*7

Fcsz O= C) rv goo ftr 4'o/ncre.*tart

theta deg.

T min N

M max N.m

0 5 10 15 20 25 30

77.9322 7s.9184 73.9567 72.0456 7 0 . 18 4 0 68.3704 66.6037

9.7654 9.9265 10 . 0 8 3 5 10 . 2 3 6 3 10 . 3 8 s 3 10 . 5 3 0 4 1 0. 6 71 7

Canoncg --3o it

fu:

lNe=8/b = e: gETvtGEH

4 C>LL ftl'zg trrD

4rtnacsT,teo

F1o!

-{nrAttEs

o 7

lNfl/ crtf lS

€Eape77dt

p/A/H74t

frvZfA,

Paal regs Equ1votw; (r), (z), npa (e) f-oA

VAL(B theta des.

atr

90 80 70 60 50 40 30 20 10 0

3.264 4.892 6.295 7.478 8.446 9.200 9.736 10 . 0 5 2 1 0. 1 4 2 10 . 0 0 0

6 T lb

Faowr

ftND EyaLunzp

tra 2

Acg=L

G) AC = L-CB

lt/= /O /6, 4r

tuE

(r)

?=TrI4f€4.0-7) Atr p@6/?A/..1

tUn

EG)O lL /A Pr utvt

= lbo-

OuVtne

/Zo6

o, 3o : EFrroreo* 4S; CotzD rtN Ft xen €/*1F7 47 C

/46-//*A

4= bn'ax

-4

O), k), ,wo(g)

Otr Btoc? vnLues o tr Forz Frzo*n ?oo To o AT /o'rm@ufus Fanze 7z1pa6aP'

Torri+= zoo 4

(s)

Pr?ryaan Eount/es

€ut?ffiee5;

lqfil p ffr/H

lTuu-o,

aelawrf

- t'F

T*,; e"y

/o-t/ 8toc1< 4 @t = O, ?o qT ftLL a*

T,+ = e 4 ' F

T-r,

Range of values of MA for equilibrium Iexpressed in N.m] (= MA <= 1.000 1.000 <= MA <= 0.988 1.213 <= MA (= 1.12f 1.641 (= MA (= 1.336 2.225 <= MA (= 1.586 2.854 ( = ( = 1.844 MA 3.419 <= MA <= 2.093 3.861 (= MA (= 2.327 4.170 <= MA <= 2.543 4.366 ( = <= MA 2.742 4.474 2 .92s <= MA (= 3.094 4.518 <= MA <= 3.250 4.487

theta d e g. 0.0 10.0 20.0 3 0. 0 40.0 50.0 6 0 :0 70.0 80.0 90.0 100.0 11 0 . 0 1 2 0. O

EQ, A/4;

T-o,

[,,tol:':,9;;]:or*n*o n, usE susf,cR,r ?l Dz= tao/ (t cos(B-s+il) 'lor' ( o" os 6) n o fuo, *,

tryAeC,e&ZJ

p=re\ 7=r.F (,)

Uu/(/-cas(p-*-4))

iFuALUhTE ,?t-,n Fp/^/7

Ta Soo

@ =o

l/S/,y6 €"

aOrJO Tnpo

-l

\=coS

(z) (s)

?oo P lb

6.617 10.182 1 3. , 4 4 9 16 . 4 0 2 1 9 . 0 17 2 1: 2 6 2 23.099 , 2 4. 4 8 1 zs.3s6 2 5. 6 6 3

e=O+V

(+)

@orn*ueo)

8.C7 continued

s o AT eoct*z-r

Fa/c7/o^t

Cat24," B:

8.CB continued @) FOe BEap ,s,ttohq!

fa LF\T C

=4r, t/ a

i',.

FoR

aF BE/rtn € L

/ run

\a d,(-wo= o

fi.shY

)T,ur=o, (ztn)t-Nv +w(v- t*sa)=o SHnryl

For? Mozop

uyl peyDtNl:

6. = e4s? Ta. l\

(cnsF z)

uptxAr4

Zr=6..4s€

L = 3Ou,,

lM sEaumtcE,

EuatuaTg:

tftvo

t--lorn',

Motion of block B is--> gamma deg.

EauntT

WA=8/4 (s = o,3o (,) Tllrzaudt (g),

giatupFatzI:

A,vo (w")z

u9a6

WA l,rU 2 lb

infinite * 2 7. 6 2 13.25 8.52 6.15 4.55 0,00

(a) OUrUsr

infinite * 5 7. 3 3 4 30.272 2 1. 2 2 7 16.423 12.469 0.000

Pas/rtav

ltt

lS pultno

::,::*884M,f

n,,:.ffi:,, F/-o ts

BEfrvt

fl/Svttrct

€ltDtv'

rya

4No

CoengP6,vslu

(Ol

frxo

FAn

6l/OtN6 /:l*a

baut o Oz

nrtf eE

carzrz?s

.tur?

ftNY

tu?LuF

7zE

/EF7

,lrtD

O t

Vfrtba

FaE

hv/ PC|L/D //y6

7#e

2t A f

polvDrry|

: y

77rt

valDe

INI/ctt /tlvD

wCN

US:/^//

pKn (v/D*oA,

nN,

slanzz

Vrl/tE = Q.4 ]

,Find theta for max (x/L) 5.00 5 . 10 5,20

f ,- s

0.603577' 0.603578 0.603575

(il

r, C

ftcEPT

w l .I:

Faa

Fe4t'N

f=4iv fl6

lZcE

BaoY

:ratw ns./k rlar 4* Pflnr a, *ve-fnzurt -4s

€

f=A, N at[

7/Na eo)

e/= €

0.6000 0 . 6 0 13 0.6023 0.6030 0.6034 0.6036 0.6035

hv Ea (e)

,a;A 4*

OuV./.vc:- 6t PAzt'taAP, ftUt'nap

Vt{Z)*6

&(d

fu*a-:4t=Ctf, ftrw .fv'4ttfr7? T/z uArvt-

o,to3f,

ANo FlNt2 lsrzszsSP*urvwl$

FOR MOTION IMPENDING T0 THE RIGHT

( too

/lvcn€/v/tETavs

/ia2ra$fuv7rni

F O R M O T I O NT O T H E L E F T I m u k i n e t i c x/L theta 0:00 1. 0 0 2.00 3.00 4.00 5.00 6.00

= O,+

V/z- uNztL e 4xo V/47 r',/ftX/l,v1uftt V4LuE ot: V/z //r15

lslrvz

F/.vt

Oup pq*Z

BPAU

u//t1.5

4/^sa

Evffi

(A)

/7 /g Avr?aryT

AB frr,lg gy crz4yF

c48L€, nFrETz

oF, Preet an4tvt

P&oazri-pt fa

AS=O,5D 4X=O,1O /s /t//7cLLY Bran

{o)

(s)

=o,/:

I Y E E : t u a6 = o , T = l - / * = / - o , ?

For theta = 0, cord BC is horizontal Thus, Block B is improperly constrained.

;,,

-4* !=CAIO-Arsr4,O L * a n 6 ? a * /-

/O" //WEEMF*vTf

DOWN:-------

I l.._'J

(z)

-= =totu z [cos 6 -Ak sia a)

tu&vr-Hato i4FMBcr.zs oe 0) 4m(z) y _tcaso caso ,vr- , - , =w. = | -4aL Z(C2SA -4* 6la A)

v,qLuT

Fon

We subl Ib

60.0 5 9. 7 58.9 5 7. 2 53.8 4 6. 2 0.0

0 10.0 20.0 3 0. 0 40.0 5 0 .0 60.0

(*r)

Pzt+z

(w)r= 4.d" Y Eouhztcws

6 7o /oo

6 Fpp*t

theta deg.

t w(t*e)=o

(Nr = t F

Photma,

(r)

)z/r'tu,o1t'ln)ts)aa -/vlaso

fnohc,

Eil7TAt

Y-*usb a-4^L

N - _W

theta

x/L

10.0 20.0 3 0. 0 40.0 50.0 60.0

1. 3 3 6 1 1 . 18 0 9 1;0245 0.8632 0 ,6962 o .5245

Find theta for 55.41 5 5. 4 2 55.43

[mu static

= 0.5]

max (x/L) = 0.6036 0.60377 0.60360 0. 60343

Chapter 9 Computer Problems 9.C1 Write a computer program that, for an area with known moments and product of inertia Ix, Iy, and Ixy, can be used to calculate the moments and product of inertia Ix′, Iy′, and Ix′y′ of the area with respect to axes x′ and y′ obtained by rotating the original axes counterclockwise through an angle θ. Use this program to compute Ix′, Iy′, and Ix′y′ for the section of Sample Prob. 9.7 for values of θ from 0 to 90° using 5° increments.

Write a computer program that, for an area with known moments and product of inertia Ix, Iy, and Ixy, can be used to calculate the orientation of the principal axes of the area and the corresponding values of the principal moments of inertia. Use this program to solve (a) Prob. 9.89, (b) Sample Prob. 9.7. 9.C2

9.C3 Many cross sections can be approximated by a series of rectangles as

shown. Write a computer program that can be used to calculate the moments of inertia and the radii of gyration of cross sections of this type with respect to horizontal and vertical centroidal axes. Apply this program to the cross sections shown in (a) Figs. P9.31 and P9.33, (b) Figs. P9.32 and P9.34, (c) Fig. P9.43, (d) Fig. P9.44. wn dn

C3 y x

C C2

w2 C1

Fig. P9.C3 and P9.C4

9.C4 Many cross sections can be approximated by a series of rectangles

as shown. Write a computer program that can be used to calculate the products of inertia of cross sections of this type with respect to horizontal and vertical centroidal axes. Use this program to solve (a) Prob. 9.71, (b) Prob. 9.75, (c) Prob. 9.77.

y = kx n c

9.C5 The area shown is revolved about the x axis to form a homogeneous

solid of mass m. Approximate the area using a series of 400 rectangles of the form bcc′b′, each of width Δl, and then write a computer program that can be used to determine the mass moment of inertia of the solid with respect to the x axis. Use this program to solve part a of (a) Sample Prob. 9.11, (b) Prob. 9.121, assuming that in these problems m = 2 kg, a = 100 mm, and h = 400 mm.

Δl

Fig. P9.C5

11_bee77102_Ch09_p015-016.indd 15

3/13/18 8:14 AM

9.C6 A homogeneous wire with a weight per unit length of 0.04 lb/ft is used to form the figure shown. Approximate the figure using 10 straight line segments, and then write a computer program that can be used to determine the mass moment of inertia Ix of the wire with respect to the x axis. Use this program to determine Ix when (a) a = 1 in., L = 11 in., h = 4 in., (b) a = 2 in., L = 17 in., h = 10 in., (c) a = 5 in., L = 25 in., h = 6 in. y = h(1 – ax )

L–a 10

a L

Fig. P9.C6

*9.C7 Write a computer program that, for a body with known mass moments

and products of inertia Ix, Iy, Iz, Ixy, Iyz, and Izx, can be used to calculate the principal mass moments of inertia K1, K2, and K3 of the body at the origin. Use this program to solve part a of (a) Prob. 9.180, (b) Prob. 9.181, (c) Prob. 9.184. *9.C8 Extend the computer program of Prob. 9.C7 to include the computation of the angles that the principal axes of inertia at the origin form with the coordinate axes. Use this program to solve (a) Prob. 9.180, (b) Prob. 9.181, (c) Prob. 9.184.

11_bee77102_Ch09_p015-016.indd 16

3/13/18 8:14 AM

g.c1| ^

9.C2 continued

Gf}EN'

11.,T..1\ AND Jry oF ,rN AR€N) Axsue otr RsjT\T\oN E sF t*e

Frr.rs:, ]';t':il: qd oF oF

s: o.So,--.,

= l:cy = -0.380 in^4 1.090 in^4 Imin = 0.225 in L.257 in^4 Ima>c =

AREA oF Snvrn-(

?nseuer..r9.1 O-JTuNe

o$TP$T

(a) Prob. 9.89 0.392 in^4 Ix = 0m = z-23 .720

i['FoR

ttrE

?RoGRNN\

(b) Samp'Ie Prob . Ix = 10.380 in^4 0m = 37 .720

Pr<oqraNv\

TH€ r.)Nr-I\ OF Tt\,E MOvt€NT: l.fsa;tf AN\ Otr TNEKTTA. PRFbogT rt\E sF THe Mor.4Er{-r5 NN\ lsrqor v,\Lu€s sF \NEKTrtr'. PRsbsgi T*, frn, 1*t. otr S: $= s, E ^:., qo Foe. E-Ncrr vALuE Cornprrr€ 1* r\Ni) l.y, :

9 .7 ry = 6.970 in^4 Im ax = 1 5 . 4 S 3 i n ^ 4

Dcy = -6.560 in^4 L. g9? in^4 Imin =

g.c3 Gr.vEN,

b>,

L* : i$*:1:) * i(lx -\)qu:zs - l..r s.'uzs Iq = (I*"f..r)- T1

?sY \

FoRv\eb

\nl

NReN

REGNSLC.S

:ERle:

Ix , Tr, H* AN\ h\ For<-nr€ NRE\ ( (c) F'rcS.P 9.3\ NNb ? 9.ij (b) f'rcrs.Pg.Se ANs ? g.j+ (C) FrG.? g.Aj (d ) FrG.? 9.+4

Co*rPuTE .1*t.1. t + I{\ = i (f*- 1.n) :rs.rZb * }x\c-sz-B pgrsn .-n\E yr\uug! ctr b, l* ,1y,, Nxrblx,t o\)TPsT

ft.s6Rmn An gl e

deg

Ixt ,

in^4

10.380 1L.493 L2.52L L3.432 L4 . L98 L4.796 r-5.209 Ls.423 15.431 15.235 14 .839 L4.256 13.504 L2 .604 11 .586 10.478 9.31-6 8:135 6.970

0 5 10 ls 20 25 30 35 40 45 50 55 60 55 70 75 80 85 90

Iy' ,

in^4

r x' yf

i n^4

-6 .560 -6.164 -s.581 -4.929 -3 .929 -2.9LL -1.803 -0.64L 0.540 1.705 2 .8r.8 3.846 4.757 5 . 523 5.121 6 .534 6,748 6.755 6. s60

6 .970 5.857 4.829 3.918 3 .1s2 2.554 2.L4t L.927 1.919 2.11s 2 . slr3.094 3 .846 4 .746 5.764 6 .872 8.034 9.215 r.0.380

OF UeSGSt T.nrpl.n Tr-t€. \rNn\ \+E t\ N$*.tsER **P*n Otr RESINi\K>LQS Fop. ENC)I RECIT\$>.L( b: S=1.1r---)N 'r\\E Ixrp.n 61r\$A wts) \Nb rHE *E\Gt{\ h(s) qtsll etr r\\€. lspt:rr \Ae titsr LcsRbrqpffib cEl'.rgot\> $esH-rs.

*+E ToTNL AREN $rnrnnu: NrcrTld i \-",-rr'r- + o-(s)*t' w c\)J t \' tsrJ \N\\ERE G(S\= \ oR O(\) =-l \F Tr'fE a,CeA

ARE.\ T\AE

SNupue. ?tr.sg. 9,1 OrTurue

F\REN

os' Peg6etvr

Tt+E TJNTTS oF THE MOMEr.fiS }.lp.rt otr tNERSt\ T\+E vALu€: fsnrr oF rl+E MovlE$\S PRbbusT sF INERTT\: lx, ln, T.*f

F\N\

?RSb$sT

r\bb€b sR. 5sQirF-ACr€\, \s Ts g€ R€sPtg:fwEL\ \>tr Tr+E C_SSRNTNNTE ,rXgS bbE\ T1AE OR\grN fs (.€snRstb' p.1crt CorNCr\€ Str \AI\TI\ T]*E Tl*E- I\REN: R.ECTNIC.LE S i $= \, Z, --- N Fcn E-\cr\

UpsxrE Er \: ZxN= L-xX*ait)tits\tth,ts)ltht Nr Lq \= lq \+o.tr)Lisr)i[wrs$th(l drcncre Ec1

\nrrb

(X,Y )

Gsnnp*rre a1\€ qsspsnslrSe\

covrpurE TF\E AN6Le s,"1 Tr+\t Tr{E PRINCIPAL AXES treR,sq w,rTr\ tr\E esoRb\NAsE Axesr

CET.ITRSI\

TT+E

sF n'\E

XR€A.i

s^q= l**-'(-#tt I*oo

ANb

\no,

t,(r**J.,). {[{u.-il _I*r (\*I1)

CcnnnnE

'FHE

IM\x Dtr T*,,T.1 , Arss 1*I

VAL\rES

\ b* ,,Iwrrrx r

AstU I*r,.,

GsnnrrE L*^g.n{s Ert h'* = m..srrrL ?nrr$

(coNrrxrll€b)

f\F.\bt\ TrArG-vALuES s€ L* Ahsblf i L* n without the prior written consent of McGraw-Hill E

(conrrrxr..r€.\),

ducation.

9.C4 continued

9.C3 continuedI (a) Ix = 37L426.8 Kx = 2L.33495

(a) Ir.y

=-1760000

mn^4

(b) i:.y

mn^4

=-4.254

Iy = 64256 ttun^4 Ky = I ' 8?3843 tIEtt

mtt^4 mn

(b) Ix = 40 .44445 in^4 K:( = 1.49897L in

Iy = 46 .5 in^4 Kif = 1.607275 in

(c), Ix = 191 .3 in^4 KfG = 2.525206 in

Ty = 75 .2 in^4 Ky = 1.583246 in

(d) Ix = 18.12817 in^4 IU( = 1.90411 in

Iy = 4 -507946 in^4 Ky = .9495205 in

47lo4o

in^4

fu|E.tll:

E!t$

GrvEsr, Nu ,F

l-txt\i

NRer. tror\M€\

By A sRRrEs

R€gT\F.TGLE.\

Dtr

Lv.,

trSR

Tkl'E

NREA

Otr

(o-) PR.oe.9,-l \ tb) ?e.oe. 9.15 (c) ?eoe. g.'l-1

hREA St\o\^Ixt\ \Nlr\Acil \s r\> N\3o$T T\+€ X \?(\5 B€- REVouvEb TO FDR.tv\ A SOUTi> Str I.4F\SI t!\= Z Eg O-= \oO rnrf\ AND I Jx \^IHEN h = 4Os w\Yv\ FoR (a) Snr.nHe ?eoe 9. \\ (q) (b\ Pe.oe. 9. \a\ (o) Puxue.

AEPqOX\$44fiG. Tr{,E AREA \I\.T\T A

:ERreS

str 4oo

FoE.

Recms\Kt€s bccH

SEGMENT.

s: = ?V* - p ('i .t.t $) q>tr f.nsVEtfT

IvtAS\

lN€FStA.:

Gr). = i *..11 =i(tn sl)ft oF LE\sGiT'\\ Uxrrr5 rSor.nB€R. N otr \aESINNcr\-e\ FoR. EACA Re,cf\Nc'LE- !5', S=\.?, --.r N }{n:n Tt\E \rr\iT\\ \^l(S) NNb rr+E $E\6$T \ts\ oF TF\E [ Xt:f, q \\il Jxre.ry T\*€ c-ooRbrsrxrE\

ltsnn lNa;rf

Foe. "rH€ lsLrb:

rr*E rU€

,\Nb

=):(lx)s= t roil [.qt =i fr L'tt

CENTRSIb

Tl+E SR\Grxr sF Tr+E CsOR.,brxrRfi-E \xe: eorxn-rb€ w\T\\ Tl+E G€NTRSTI\ otr rrlE Nfr Foe tr\crt REcttlNGLE S I \= \r?, - --) N OpspirE rr+E tcsagL NREA. \-csnq I

Arqrv- = N-**.+

\OE\ r\r?EA'.

ct(\\Lwcslllhtsl]

\^l I+ERE O(\) = I oR O(:) " - 1 \F -r*E SR. rS rO tse Abbeb AREA . S\)STRASTC-b, RESPEST\VELY

PeOenN.l or OrrrurNg oF Q,,. Q.,, N r A,trrb m T*E VAuu€S Lsp.l.f COvrp*r-rE Tr|E VALuE- OS TF\E c$6qSt1*1q5H:

Te tz

v(No\^lN'.I a \7qI

t<No\AlN"

IHE

LENG'TI\

COSqP,-rE

H = \thf A\

€FCH \EGSII€N\:

a\ = bl!' 4oo

UpsxsEL*, A,'fi N I f.* f\ +o(slxtsr'l\tw(sit\(s!1 Oeuxie tq A, Lq N.Lq N*_ot:rtqt$t*,tst']lhttt j Fog a\ct{

Connp..rrET*e crccrlRbrNAres(X,Y ) c€sreorb str -rr\€ xRE\ I

X=LrNl N.*^.

Foe EAcu \$Q-TNNGLE S :

oF rr\E

Y 'IqN/ A-.,'^,-

S= \r2., --.. \l

ororxie T*r, ]xI = T^r +G(\\[lt r - X]tit=l ;Y I * \ twts'\lt h tsrl I r*., hr*n r*E vAL\r€, otr (csurrxrrlQb)

RECIA}{GLES'. S= 1.,?)---.4OO C-owrP\trE Kc: \c- = Q,* t:- f) n l tosapu-i€ \c-: yc-= h.xl

rJesrrretrl r I1.1 = L"rl * yl 0pu,ve L.1"*' f rl =f"tt* \J -Ix'

C-omp,,-r-r€ ?e.,nrr rrr€

I* = *b wr IJg Lrl otr VNL\E: Q,r!.,, N\ F\Nb Tni 1* McGraw-Hill (t of crcNTrNueb)

Education.

g.CO continued J

g.Cs continueo I

Covrn>rEl*,

olrTp.JT

Pnosnnw\

< a L2 =400ntrn 1 11 = 0rltrn n= 5.985 x 10^-3 kg.6: Ix =

fll=2

(b) -1 L2 =300nm 11 =100 Ifin n= fx = 77.373 x 10^-3 kg'm'

1c6j

m=2

lr*O\^IFl,

\^ll+l (}\

APPr<)xrr.z(N-T€b 5iq4rcsfi

N=O.O4\

tt\

\= 4 rN. e: t rs., L= t1 rtr,, \: \O rs.

[^l| =

. 04 Lb/ ftr

h = 6 in.

w' :

.04 Lb/ftr

[c tN.

Lb) ?eos. 9. tB\ (c) ?*ce. 9. tB4

OururNe

rostxtq ..'EQ.s, (9,4ro)

Ar{l> (9.3i)

Ca= I*l.n "\fr*I*L). --I*l - I; -11^ Cs=I*l-rT.- I^T"t- l.,lr\ - 1.1*\- {x\I**t

HAvE--

AX cooR.btNNTEs

rHE

SouVe

eQul\srsN

T*8.

FTR:5T

?RTNCTP\L

\(e* (K,- C,)v<* (c.*KlFoR. TIAE ?R.rsgrPAL

sGsrDNs>(t<.\ MToMENT:

rRE

l s = K u +i n l

t ' - -h ( ' r - f ) THE-

(K.)

otr

Kr, Kz, F\Nlt\K3

(9. (1) of

inertia

SE6\n€NT

L, = \ rrxl.+Lahtt-- i"I'

*"ffi"I't fn .:ilTt$t*)\l-q$ ved. No reprod (cpsrrNr.)€b)

NND lprRb str INE\"Sl\r

VALuEs

(a) Prob. 9 .180 (a) moments The princiPaS. K1 = 0.01396 kg'6' K2 = 0.01430 kg'6e K3 = 0.02058 kg'6r

K,,

Lr\(,\ --o

K..3= i[- ( K,- L,) t

PR'cc-ruxrq oo-rprrt

L.,

IN€\"SI\,

otr TlrE'

CENTTR.O\b:

L€NCfitl

T'4OMENfi

(s€E, FoR eXNMPLE, Nerlncxr's MEs\sb u):\Nq nrrrb C-xNxue., N*ll"tER.\cAL Mer*cUs troR Gfxeerf EtEc',neess, L\o €\., KcGnxw-Hr\\, \9gB)," -T1+€ qgggQfftrC EQo{\F\oN SsgyE

?err.lr (tr,\r\

c*lB\C

C-c,K.*C2K-c3=o FOR

T\t€

INERSIN.:

Otr

NNb Ls. C, = lx* fl *fa

otr Pe{>lRAvt OutuNe Str &., L) \Nb \ lnrP*-n T\fE VALut\ \^l fsPuT n+€. VNLu€- otr C.b*,p,o-rE. rr*E \rsrDrrr\ ,\X otr Excll S€GMEI$T : NX= *tL-o) Foe E rg1{ SECTMENT S I S= \,2, -,-, \O tosnp.rte. otr THe rEF-T rt\E X cocRbtNA3i,es (Xt\ Eh$s: R\Gltl (xe.\" ANb Xu = O+ (S-r)lX

Covreufe-

MOMENNS

-nr,E VALuES sF T*E PRS\I.ETS Otr INEFSI\I JNPur 1*\. T*{1.,\J( T*E Str Tt\E CI>NSTNNTS Cr., C1' VNLue\ Covrv"rre-

(1x), ' = .(grns!1,)(srHD.)t:,*t_ql. s^.=H L= a ( E,Lr) t h(Ls s,*Er)'* q;1 a q;l ELL n(r* Y')t+

rHe

?Ro()RAM VALuE-S otr r*E

lx, Tr, fr

HnvE -- \e-\u = h(\= *o*\-\.'(t- ft.t =ah (i.- i*)

C,omprrre

lr\€

fnsnrr

(s-r)ox AK

Ke. = Xut

in. Ib'ft's

A gouX AN\ PRobrocrS ANb r-ra MoMElt\: oF rNEFtrrA. f*, Tt, fo,,I*.1 , fra ) ANb lu\ ?e.ruc,pqu MoMEht\s otr tH€Rst\ K,,, K2' K. As 'I1*€ SRrGrxr FoR Tts+E tssbY etr (o.) ?*.oe. 9.tBo

o.: 5 tN,, L. ZStu

Ye-Yu

\uss--

h = l - O i n.

\ trq., L=11 lN,,

(f,ur\.

(b) I J = t 7 i n. a = 2 in. lb-ft-s' 644.99 x10^-6 Ix=

\= = cos$, ! +srxrS,j

SREN{ENT

Wr='04Lb/tE

in.

h=4 2

inIr=11 10^-6 lb'ft'S

lrr{r\Er-t Qa

AsrxF(br:r

a, L, \., AN}: W ',,I-x

utuE

(o.)

[c\

o"rFR)T

61rRE -T\) \5

Frslb: tx

Pc<cgqnvr

(a) ;:l-in. 74 .OL x Ix =

:RrsqexnS;

vALu€s

Hor<o<'€N€q)s Grvexr:

Ix= H tk,t

trE

Pet*-r

, ( a )

(b) Prob. 9.1'81(a) msnents The princiPal K1 = 0.00414 kg'm' K2 E O.02978 kg'mr - K3 = 0.03225 k$'s1r

are

( cosarNu.b)

uction or distribution without the prior written consent of McGraw-Hill Education.

9.C7 and g.CB continued (c) Prob. 9,184(a) mornent,s of The principal K1 = O .2 2 5 8 L kg . mr

9.C8

continued (t")

inertia

are

0 .41933 kg'mr

K2B

K 3 = 0 .51610 kg'm'

ES,u, Nngue:

THNr TH€ Tr+E

AS rNEF<Sth, CrCORblNtrSg

TIAE

PRrxtcrpAL AxE\

OR\g\r\t

SORI4

AX€t

FOR.

*N' Go To 1r I**+o AN\> tT*-K.)^Tx\:O oR. ttr lt* + O p.rss I*f = (Ir- K.) = O S _ANr$,r 1.* - lf. = $ oR \F (I*-Kt)+ \1 = O: S*.= Sd \tE\\ ConaP.l6 AN\ \t t \x

\Nr-nA

Tte

BSbY

otr (o.) ?RoB. 9, \BO tb) ?gce. 9. \g\ (c) ?eog. 9. \94 F'O< ENC-r\ PR\N$PNL Mosz\Eh.fi OS INERSI\\ K., K.., sgT Kr) Tl+e oF EQuACTTONs FORMEb BY \^tO Eq:. (9. 34) cF TrtE E56f$roNS oF

(c):

\: +\i.*

bESER$.4INE

OF rr\E brRgc-rbN \^Jtr\c$ T\+E ANGLe:

Tl€

-l'

\*\''

\\ , = r{t>..* rx\

*N. Go To :o T\IFA(\ Rqwgrre Eqs. L9, 34) str r*E Xxts C-s€Ftr\crerfi\ CovtP\tig L\-K=)=$: ls

A\Nb E'D. t9. S-l) \*t* \t* \.3 = t SOLVEb

'xx= l-

(1^ - \()\x - fx:X\ - Tu(\a = o. - lx\ \* * q-K)\.r -1.1t\a Q -\r" \* - a.,. \r* (1*-k)\*= o

ARE

t\\ (c)

(e) oR C\")l \x=ffi

Axxuxsrs

VNLu€} r FRcf.4

(o)

CJ'RRESPCh$b\h\GI

gbS\h.LES q\x, \*,b\, D*,\, Sa NR.E

oc"

c\=

bgiFI.$4\N6>,

cz For< EAcr{ ?RrtscrPAu MoMENT

<>tr rxrRSr\

Ksi \=1,?,3 " D "

(smprsE

T'f (Tx. Ks\:- O \trrs f*r = Iar = o THEt\t \x= \,*\ - O ; \\ = \a = O t br"Sr =9o' '* Go ro B* (Beuow) IP

(fr-

Ks)= O

TrAEn

\a=

\x=O:

tBELour) Go ro " Nlx\+o nrNb (l*.Rr)=ta*=Q (c) oR \F (Tr:Kt) + o ANb T*t =f r* =Q (b) (a*^ K*)=Q (c) rF o DR. t\r\ lr* + 1** --9d = O: D\ T\\E\$ \ Cpr.,.rpur€ X)( Ar,rS \1,

(o-) oR tc) \x=

ffi,n ser

('cour,uuEU\

X.1= Q.\t Gs 1.* = O: Cr:vrPsTE Ct NNb Cz 1*r n r = lrt -z \1 -T;-IG T--\(t CosnP.rtg. \x, X.,,, NN\ \a t \ . .-r= q1+ff*a-ffl'

\x = C'\r *N"

Go Ts be-e'*vrr\.\€ \tr

THESF

EU\)ASTS\\S

\a =C.-\z $rrRls T^lO E'XS. Lg.S4\

NRE-

bePcs.IbE\sT:

x+=-bl:'' +Yt

\Nb i

Xx = C, Xa -ts * A*"

bx='ggo

\t,

\t=C'\a Xx= C.\* *li' Go -ro Ix\ = $ " CosnPq1'IECr \N\ Cz r = lrt C,: Jtl LL T*-\(.' 5]:R.,

Nr.\D f x\ = l\e a q)

S^=\a=9d T H E N \ ='\' -' ,B\ '- D ) \x=\2=s: Go ro p*5rb Tre. Ta* : g 1r (i* - Kt\= s \,..br = $, \^. \a = O, b^= b.t= qS Tt{Exr }.}= -*'B" G*> To (a) lP (1r- !(r\ + o ANb 1x\ = fax =. Q oR tF l'xr * o ANb (1r- U.=)= T.lr * $ ( t) oq tF i.* + O ANt> L.,. =-\\* -Ks\= D \C)

\x,

*,aH

G.Cov1p.y6

ANlb*:*= *

(NBDre)

C* C* NNb Cz:

(coxsr\Nueb)

ved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.

g.cg continued

lx-Ks

(^ l:s.

Cz *4" '

,\ B

a*r \-\

T.x\

Go To " \r" (laove) -i]+E VALuEs ,ctr N^. tosnp.nq \, '$rANb S* t = crDsi\t S.t = csft \1 bx = coi' \* Perur rr\E yALuc=S btr \x,, \1., \NN \ai $*, Ns.tu bL \,

?aoc=R.N\.\ s.nr.Jtr (a) Prob. 9. 180 The principal mdnents K1 = 0.01,396 kg-mr l<2 = 0.0L430 kg-ma K3 = 0.02058 kg.6'

inertia

are

K1 lamda x = 0.998L92 lamda y = 0.060108 lamda z = 0.000000

9x= oy= Qz=

3.40 86.60 90.00

K2 larnda x = 0.000000 lamda y = 0.q00000 lamda z = L.000000

0x= ey= Qz=

90.00 90.00 0.00

K3 lanula x = 0.059740 'lamda y = -.9982t4 lasula z = 0.000000

0x= 86.60 oy= 176.60 Qz= 90.00

(b)

Prob.

9.18L

The principal moment,s of K1 = 0.00414 kg-m' ,: K 2 = 0 - 0 2 9 7 8 kg -m : K3 = 0.03225 kg.pz

inertia

are

K1 Iamda x = 0.378602 lamda y = 0.171730 I an u i l a z = 0 .9 0 9 4 8 7

0x= 9y= Oz=

K2 limda x = o.8s1g2g landa y = O.3L9677 lamda z = -.4L4964

0x= 31.60 7L.40 oy= Oz= 114.50

K3 lanula x = -.361873 l-amda y = 0.931882 lamda z = -.02s351

0 x = l-11 .2 0 2L.30 oy= Oz= 91.50

(c) Prob . 9.L84 mqnents Ttre principal K1 = 0.2258L kg.rne K2 = 0.41933 kg.6z K3 = 0.516L0 kg'm' K1 lamda x = 0.8l-6479 larula y = 0.408265 l au u l a z = 0 .4 0 8 2 6 6

inertia

67.go 80.10 24.60

are

0x= 6y= Qz=

35.30 65.90 65.90

K2 larnda x = -.000008 lamda 1r = 0.7071L1 IanuXa z E -.707103

gz=

K3 lamda x = -.577376 landa y = 0.577347 l asu l a z = o .5 7 7 3 2 8

0x= 125.30 54.70 ey= Oz= 54.70

0x=

oy=

90.00 45.00 135.00

Chapter 10 Computer Problems A couple M is applied to crank AB in order to maintain the equilibrium of the engine system shown when a force P is applied to the piston. Knowing that b = 2.4 in. and l = 7.5 in., write a computer program that can be used to calculate the ratio M/P for values of θ from 0 to 180° using 10° increments. Using appropriate smaller increments, determine the value of θ for which the ratio M/P is maximum, and the corresponding value of M/P. 10.C1

l B M

Fig. P10.C1

10.C2 Knowing that a = 500 mm, b = 150 mm, L = 500 mm, and P = 100 N, write a computer program that can be used to calculate the force in member BD for values of θ from 30° to 150° using 10° increments. Using appropriate smaller increments, determine the range of values of θ for which the absolute value of the force in member BD is less than 400 N. P A

L B

b θ

C B

Fig. P10.C2

10.C3 Solve Prob. 10.C2 assuming that the force P applied at A is directed

horizontally to the right.

10.C4 The constant of spring AB is k, and the spring is unstretched when

θ = 0. (a) Neglecting the weight of the member BCD, write a computer program that can be used to calculate the potential energy of the system and its derivative dV/dθ. (b) For W = 150 lb, a = 10 in., and k = 75 lb/in., calculate and plot the potential energy versus θ for values of θ from 0 to 165° using 15° increments. (c) Using appropriate smaller increments, determine the values of θ for which the system is in equilibrium and state in each case whether the equilibrium is stable, unstable, or neutral.

90° a D

Fig. P10.C4

12_bee77102_Ch10_p017-018.indd 17

3/13/18 8:15 AM

10.C5 Two rods, AC and DE, each of length L, are connected by a collar that is attached to rod AC at its midpoint B. (a) Write a computer program that can be used to calculate the potential energy V of the system and its derivative dV/dθ. (b) For W = 75 N, P = 200 N, and L = 500 mm, calculate V and dV/dθ for values of θ from 0 to 70° using 5° increments. (c) Using appropriate smaller increments, determine the values of θ for which the system is in equilibrium and state in each case whether the equilibrium is stable, unstable, or neutral.

D θ B

C E

Fig. P10.C5

10.C6 A slender rod ABC is attached to blocks A and B that can move freely

in the guides shown. The constant of the spring is k, and the spring is unstretched when the rod is vertical. (a) Neglecting the weights of the rod and of the blocks, write a computer program that can be used to calculate the potential energy V of the system and its derivative dV/dθ. (b) For P = 150 N, l = 200 mm, and k = 3 kN/m, calculate and plot the potential energy versus θ for values of θ from 0 to 75° using 5° increments. (c) Using appropriate smaller increments, determine any positions of equilibrium in the range 0 ≤ θ ≤ 75° and state in each case whether the equilibrium is stable, unstable, or neutral. P

Fig. P10.C6

Solve Prob. 10.C6 assuming that the force P applied at C is directed horizontally to the right.

10.C7

12_bee77102_Ch10_p017-018.indd 18

3/13/18 8:15 AM

10.c1

10.c2

6: ?, I th, I =7 , d i n

Gtrnyt

a-=s&*14?t b = /do tuft, /: goo*1n P= /oo N

7/1 Ftrnoi ;?frz, #

yrqtuB

FoR

l-rZopr

oF ro

/8oo

/;/Ho

(Y")-.y

ConrLeSpo/ynl,va

lrhLuE

(ts" 77f

nT /ootuaeMe4 Ftvo: Qfttdt a/= l/ntJE

4,yQ 6 t:

rry

V4Lt6 OF

FrZc,^a3oo ro /go"

UStuc /Oo txeEE*t*-rys

foece fue

otz 6

taR

lFuolt s*'u

sini

b gn6 €nf =

(r)

Cos+=/'- ti"fo|a

(z)

.sine

OfrtaenznE

Wrs

Ea()

T6 O;

86ECT

co.e+ = #- *o'r.

b aar+

cle,

-13

{

c64

DFneeexz&zt

&tt6t-

zal

cas?

(t)

BD =(o'*J=-.

zot

cose)v'

(zl

gf=

e C. (,)

Z(Po) tea = Ra-6 srna

(si,

-eU-

5e = o

',,u\h--4=o

cosa+

cosQ

cos)

+ce{'srtt6)

(s)

l=Rom

t&o

U-St*4

(t"/e)"r*r

d^/o

Corzru€Fokq1v76

Fi nd

/oo t,vcrz.F,+tgvzt.

theta deg.

M/P in.

o 10 20 30 40 5060 .70, 80 90 100 r10 120 130 140 150 160 170 180

o . ooo0 o . 5483

vrttu|

'l = 'foo2rnt 4 =.fuo znq, I = /€o "n"r1/

1.0692 1.5369 1.9291 2 .2286 2 ,4246 2 . 5 14 1 2.5019 2 . 4000 2.2252 1 .9965 1 .7324 1 .4484 1.1563 0.8631 o . 5 72 5 o .2852

(M/P)max = 2.52

theta

Force in BD

30.oo 40.00 5 0 .o o 60.00 70.00 8 0 .o o 90.oo 10 0 . o o 110.O0 1 2 0 .O O 130.00 140.00

-436 .o37 -315.402 -234.736 -171.053 -114.121 -58.357 -o. ooo l 64.229 138.075 226.895 339.772 494.514 in

theta

Force

32.487 32.488

-400.014 -400.000

theta 134.327 134.328

2.520923 2.520929 2.520924 at

-33:13I------:331131

o.oooo

in.

p=/'m

atv-.r (il,, (z), (+) a*oG)

------139-93--------113:1!3-

(M/P )max and cor respondi ng theta 73.6 73.7 73.8

(s)

blrttuOze Zoo To Go Forz V4tue-r ar 6 o /So U St *6 tao t,Yar2tr*t{;* 6 O Foa GO= -+vau, ,tryo FoQ Gr. 1ao,Y theta

VE* a F

P E ' x-oo

AUTL//vE A F FteoAnAr+r lrVzgZ:

ou7llp? oF PHa6rcn*,t b = /-7 D, 1-- ?'€b. ENr62: Ftzot nnb, Ee, urtztb*s 0) G ) r+*o (z ) VnLUEsoF A fUntun-re ftr'O Pfz.hvr *n/f ftn

(+l

5c}

4o=-'# Ftzotrcnr-r Feuh',

/:/^/a

(s)

tgp= /' easg

(si, # cqG +cosi.<ia1t"

(srhf

5-Bo= #Es

b s ,a)Fa

- P\*o-P

cosQ

CogTvr$

b Cas O

(6O =(-4 sn(. b cat+ f

tno

A BU:

- 6 s;oe1 )o

sta4 #

li+o=(l

/ 4w

Force

399.996 400.01 1

------111133-------133:933For Absolute Fbd 32.5 deg.

= 73.'I deg.

Glven: 67=,{mvnt

6=16>?ea,

$ tven : Srrq,"t4 oF coNsTftNr 4- 7€/0/rh. uil f,mfr4tu uvH7vo =o & = /oti., w= /50 /t

I = Sao 7na4 P= /oo rv fuRcE

Ftua:

/=oa V4tU B Otr € /=Rorl 3oo 7z l,uoo

t\;

U S/ry(

l?WdE

FtvO:

VfiLuFs oF a

F)po: Pora"zutt

/A" tvCrzc^.tET-tTS;. oF

Faa

H?ottr

Ta /6so AT lso /rrcn1z-tx75,.

Atso

l,iol. r@H

F/,vD

vrltug5 otr

/€A. utrg Rr uP,

T^= L cdso

t*-

v4Lt*t

frB= ?aBc=l AC = L

($e

4B = zacos(r""- 9) E/o,u4:47/as or gP4*4 J = nB -6a.

*=asn(zsLf)

(z) (s)

#= -ocosQ!-il*

(+)

cl6

/. At*

BDZ= o^o/tZoA cq o Gz = (o"* b=- Z qb

oF casTslps

.fA (, ): s ?a/si,€ "(ea)f,eo alE:qa Dro=

{,) {zt

P//:fr+zAtnzt

7e- l.aoso :

&a [g

(s)

Ero= - y'-sinQ6 a

(+)

-PE7o -6o $o = a

V/?rufu Woep:

'rBo

O urt-nvf

b = /fo

*n,

/. = Soo

tu*,

?t14t

( z ) ( s), ( +),4Na ( s)

E a uftT/oxs

6O ForZ V4tues oF 6 u s hwd: /D' lPc rl€ ME'v zT

FtuP

-wacoso #=4s# a srny +vY *--- n(n #"e,

/:orz

Fuo = foo

theta

Force in BD

3 0 .o o 4 0. o o 50.00 60.o0 70.00 8 0 .o o 9 0 .o o 1 0 0 .o 0 1 10 . 0 0 12 0 . O 0 13 0 . 0 0 140.00

2 5 1. 7 4 6 2 6 4. 6 5 4 279.747 2 9 6. 2 73 3 13 . 5 4 4 3 3 0 .9 6 0 3 4 8 .0 10 364.263 379.358 392.994 404.924 414.947

------ll3:33-- - -- -::2' eo2 theta I 25. 68 12 5 . 6 9 125"70 For absolute theta

f-non

3oo TO

(tt

(z)

Ouru*F ep paodpnu t%;, E/77FR a=/Or4, F=ftv/4 4= Zr tros (/) zztaouarr (z) F/?odfrnpt /t/ saeunfl(1 f Vntunr€ fipD 1>Uat7 V Jufu6, 4A/o q/?ttftAz tua VeLuB oF @ /=r;ryv O 76 //So US/*g/€o/rvszQrvmV = o, U,SE s/6,y /:rt o Vqzoa aF 6 Forz Jv/lg 6F d"W6" To E-szaBl/s,q -ffrett/7y theta deg.

v in. lb

d Yl d T in.lb

dvldT in.lb

o .o o 45.OO 60.oo 75.OO 90.00 105.O0 12 0 . O O 13 5 . 0 0 t 50.oo 16 5 . O O

o.0 -278.7 -371.2 -355.8 -294.2 -236.2 -213.2 -236.2 -294.2 -355.8 -371.2 -276.7

-1 500.00 - 6 6 1. 3 3 -107.'15 18 3 . 6 7 254.81 168.47 -O.O0 -168.48 -254.81 -183. 67 107.15 6 6 1. 3 3

3 7 5 0 .O O 2654.48 1592.79 656.97 -73.56 -537.62 -696.70 -537.62 -73.56 656.97 1592.79 2654.48

theta

3 0 .o o

fUr+tua7F / Eoo

l/ = 14 s2- lNa sia 6

15.OO

f= 4oofr PAa 6frnq

c/e'

Pl<o6pnpt

a. = daa

f/vH:

(s)

Eae

EN/a"6Y

Po74/7/rtL

-375.0 34.227 -375.0 34.228 -375.0 34.229 S t a b l e E q u i : li b r i u m a t At

= theta Unstabl e

dv /dT

dv l dr

-.04285 13 1 2 . 6 -.01985 1312.6 13 12 . 5 0.00328 = theta 34.23 deg.

d vldT 90 deg. Equi I i br.ium at theta = 90. o deg. v

theta

- 3 7 5 .O 1 4 5. 7 7 0 -375.O 145.771 -375.0 1 4 5. 7 7 2 Stable Equilibrium at

dv l dt

dv l dr

- .02571 13 1 2 . 4 -.oo259 1312.5 1312.6 o.02040 = deg. . 7 7 1 4 5 theta

ror""-i;-;;----i----'

399.994 400. 005 400.o17

value

t65

G/Ug,v: /=,fu *a,, EooS n c nrlDOE AEt= Hcn oF Ltru(f/l /, = $-AOnq

uAtsTpt T4A/FD

Frrtp

o To )6" usru6 o

7€o

U ftpD / u/la /+p

l/etus

trvcrz7 /t41*zy

@ivnLLrtrs 6t= O l=on

tN 4,gn O = o

& = 3 -#/.//.h P=/so,v, 4=ZCo*-

Lu= 7f,4 P= ?oo,tt ( A^tOrtulQ ,hR zu: Urlzus oe e FrZoH ,9

{frZwt

@*:

otr 6

g g tal

FG)L|BRtupr

/+n

/ Bntuta rtttD sT4BtLr;rY ol= EaCH p^gtTray -

{Tnertl"T

/ ,t, cr2 E,t lEry z-s ,

V4 t u e s a r

EGurt -

FRaq o

/*;rYo

E/+0/

Posl vd,Y

AC = OG =L

/t;7t

a-= DlSThtcE ftB

F/ananT/os

OF siptztvl

u) (zt

l cue

(3) /"

G E z = 4 "t Lt- zatcas

A eBE_: /aw aF cod/,,rEs

BE = (a" t/,- z grpcerzetrtnz€:

Z(sd

! lWl -

aL cag g) ""

(t)

V= +As=+ p.(cosa d v=-ftt jt p-(sut6

?a t -sia s

cle

ry)d6

a - / .s r n e BE

SuQ Ra

.& S/oF=

cosa - iaa cos1 lE = a- BE r r d6 (tsr)-

Pare*mt at*m

Ou 7Ufr4 Of P@68nry

ENH|

Pl?A'Enfit /^/ ttguEu(Ct

E6t,

Pft.r,,rz \

BE coco - s;oe-'#' {os p

(/)

(= P L ec,,s rrfl(t -t <osa) F dufta = - pL sr-p(rt?/te) rtNt sia €

(9)

beta deg.

N.m

0.0 5.0 10.o 15.O 2 0 .o 25.O 30.0 35.O 4 0 .0 4 5. 0 5 0. 0 5 5. 0 6 0 .o 6 5. 0 7 0 .o 75.0

0.00 4.96 9.71 14.05 17 . 8 8 21.13 ,23.79 25.91 27.52 28.68 29.44 29.87 30.00 29.88 29.54 29 .O2

10 0 . o o 0 99. 768 99.138 98.285 97.433 96.792 96.524 96.732 97.462 , 98.719 10 0 . 4 8 1 102.709 105.353 108.359 1 1 1. 6 7 3 11 5 . 2 4 0

-flcosO

F=/floHt .(=O,?*r (r) zreouas (4), d2v N.m

o .o 0

3 0 .o o

-o. oo

5. OO 10.o0 15.OO

3 5 .O O 4 0 .o o 45.00 50.oo 5 5 .O O 60. o0 6 5 .O O 70.00 75.00

29.89 29.56 2 9 .0 5 28.41 27.72 2 7. 0 6 2 6. 5 4 26.27 26.36 26.94 28.12 30. oo 32.68 36.24 40. 73

3. 64 9.86 17.34 25.98 35. 61 46.01 56.93

-30. oo -28 .52 -24.13 -16.99 -7.35 4.43 17.94 32.68 48.11 63.64 78.69 92.66 10 5 . 0 0 115.17 1 2 2. 7 1 1 2 7. 2 2

4 1. 4 1 41.41 41.41

26.25 26.25 26.25

25.OO

AflD

frllD ,tzv/1O1, dV N .m

2 0 .o o

(/)

4 =3our/a

f vnl7ATE

V N.m

3 0 .o o

dV N.m 0. ooo - 5 . 18 7 -8.899 -10.206 -8.924 -5. 457 -O.479 5.327 | 1.406 17.364 22.947 28.010 32.476 36.316 39.528 42.124

dvplt

theta deg.

G) Ouzzr,ug'oF Pac6&.nr1 EN7E2: W=7d& f =/oo/1, /, = A{o PFOkt?al, kt Stioustc€, Eat (t)rareouat (6). Evntu*Zfrt,vQ PrU,qr V n*o Sttft{. theta deg.

el6a

(s)

SinO

(€)

cl O

^t?,, , L ! = 4 ('es\ ' * - ,tt -(#)

/. Aw oP 51tt€S eing @

(+:

Stable equilibrium

-2.57 -4.89 -6. 71 -7.79 -7 .93 -6. 96 -4. 76 -1 .24

Unstable equilibrium

0.001 0.000 0.001

52.50 52.50 52,50

theta =

41.41 deg.

at theta =

deg.

47 40

58 56 54 52 50

-0 .002 3 0 .4 3 2 24.00 9 6 .5 2 2 -0. OOO 3 0 .4 3 3 24.OO 96.522 30. 434 2 4 ' ,O. O 96. 522 0 . oo1 Stable egulibrium when theta = 30.4 deg. Unstable

equilibrium

theta

= O.

28 2S ?1

5 1rl 15 ?0 35 50 35 4fi {5 50 5$ 50 65 70 75 t he /o._',(de7'ees)

.9Plztttt tl

->P

fu:

UN-f72P7eHF0

10.C7 continued

lry//Fx

d=4 4=34,u/*,. P= /5a N , -( = 2oo 4n+,

V auoclv/le Foa

_rua: VAtues of

O l=tZoM: O To

)go UslNa

€o /rucanp?t*7s,

Foe-

-2O -25 -50

AH? aF 77./E POsrrta,.

-35

l/atuE or

/ic,,u I Lrc rzt u/v,

SZeLtTf

-10 -15

FtHo:

ALso

o -5

-tlO

/he to

F/ opa ft7tar

(dos r.'s,

)

.tp/2/'v6

(D (z) (e)

V--!4s'-

P (zl sno)

(+t

A'= 4s #

- zP4ca6

(s)

)Ztt

3;r=

^/-/3 ,7dsf\ *(E)')+RP!<ha -2(t H.

(t)

Pao6an'ut P=/3'o4 $o4zt, 4=3w"/*, .&rS, ()mrZouai (6) Pt1ol?r+tn,//V SEOuecE f,-Vntur+T4 #rD PE/*r V J u/ab, J'u/de1

ourAtYE

f/Y&:

t heta deg.

V N.m

dV N.m

d2v N.m

0.00 5.OO 10.00 15.O0 20.o0 2 5. 0 0 30:00 3 5 .'OO 40.00 45.OO 50.00 5 5 .O O 6 0 .o o 65.O0 70.00 7 5. 0 0

o.o0 -5.23 -10.41 -15.46 -20.30 -?4 .83 - 2 U ^. 9 2 -32.45 -35 .28 -37.28 =38.31 -38 .24 -36.96 -34.38 -30.41 -24.99

-60. oo -59.73 -58.77 -56 .90 -53.91 -49.63 -43.92, -36.70 -27 .92 -17 .57 -5.73 7.50 21.96 37 .44 53.67 70.38

o. o0 6.60 15 . 8 3 27.52 41.36 56.98 73.92 91.67 109. 65 1 2 7. 2 8 14 3 . 9 3 '1 15 9 . 0 2 71.96 182.23 189.35 19 2 . 9 4

52.226 52.227 52'.228

- 3 8 . 4 1I - 3 8 . 4 19 -38. 419

-o. oo2 o.000 0.003

15 0 . 8 8 15 0 . 8 8 15 0 . 8 9

Stable equilibrium at theta =

52.29 deg. ?

(coNT/nuEo)

Chapter 11 Computer Problems The mechanism shown is known as a Whitworth quick-return . echanism. The input rod AP rotates at a constant rate ϕ, and the. pin P m is free to slide in the slot of the output rod BD. .Plot θ versus ϕ and θ versus ϕ for one revolution of rod AP. Assume ϕ = 1 rad/s, l = 4 in., and (a) b = 2.5 in., (b) b = 3 in., (c) b = 3.5 in.

11.C1

A ball is dropped with a velocity v0 at an angle α with the vertical onto the top step of a flight of stairs consisting of 8 steps. The ball rebounds and bounces down the steps as shown. Each time the ball bounces, at points A, B, C, . . . , the horizontal component of its velocity remains constant and the magnitude of the vertical component of its velocity is reduced by k percent. Use computational software to determine (a) if the ball bounces down the steps without skipping any step, (b) if the ball bounces down the steps without bouncing twice on the same step, (c) the first step on which the ball bounces twice. Use values of v0 from 1.8 m/s to 3.0 m/s in 0.6-m/s increments, values of α from 18° to 26° in 4° increments, and values of k equal to 40 and 50.

11.C2

b A

Fig. P11.C1

0.15 m α v0

0.15 m

0.3 m

Fig. P11.C2

In an amusement park ride, “airplane” A is attached to the 10-m-long rigid member OB. To operate the ride, the airplane and OB are rotated so that 70° ≤ θ0 ≤ 130° and then are allowed to swing freely about O. The airplane is subjected to the acceleration of gravity and to a deceleration due to air resistance, −kv2, which acts in a direction opposite to that of its velocity v. Neglecting the mass and the aerodynamic drag of OB and the friction in the bearing at O, use computational software or write a computer program to determine the speed of the airplane for given values of θ0 and θ and the value of θ at which the airplane first comes to rest after being released. Use values of θ0 from 70° to 130° in 30° increments, and determine the maximum speed of the airplane and the first two values of θ at which v = 0. For each value of θ0, let (a) k = 0, (b) k = 2 × 10−4 m−1, (c) k = 4 × 10−2 m−1. (Hint: Express the tangential . acceleration of the airplane in terms of g, k, and θ. Recall that vθ = rθ.)

θ O

11.C3

Fig. P11.C3

13_bee77102_Ch11_p001-002.indd 1

3/13/18 8:15 AM

A motorist traveling on a highway at a speed of 60 mi/h exits onto an ice-covered exit ramp. Wishing to stop, he applies his brakes until his automobile comes to rest. Knowing that the magnitude of the total acceleration of the automobile cannot exceed 10 ft/s2, use computational software to determine the minimum time required for the automobile to come to rest and the distance it travels on the exit ramp during that time if the exit ramp (a) is straight, (b) has a constant radius of curvature of 800 ft. Solve each part assuming that the driver applies his brakes so that dv/dt, during each time interval, (1) remains constant, (2) varies linearly. 11.C4

An oscillating garden sprinkler discharges water with an initial v elocity v0 of 10 m/s. (a) Knowing that the sides but not the top of arbor BCDE are open, use computational software to calculate the distance d to the point F that will be watered for values of α from 20° to 80°. (b) Determine the maximum value of d and the corresponding value of α.

11.C5

1.8 m

v0 A

2.2 m

3.2 m d

Fig. P11.C5

13_bee77102_Ch11_p001-002.indd 2

3/13/18 8:15 AM

11,C1 continued

g=rRri't+fr*#)

g = gd, g- 9o'

Qr.lrcr<-R.ESrrRsr fu a*'. K)*,rwcf{nr\ Nrea*Ad\t\bwt SHD\IN} S. t B; !r 4,N. ;\>'?.,grN,r 3.O r N., i.5

oF Rcl> AP

REVouutrtlt

rso'

z"ls'e$s tbs, b. r,*li'(+4"'*3* )* s.*;

tN ,.

D vs. $ Aub D us, $ Fo*, orrE

eo'<.F.z;"",- b=rxs'({*#)* $=Z'ls', \=z?Do

tsoR$er. soR GRN*I oF g vs. $; Cosyrc.rgr urrBEL \'Kes FoA VAr-uEb oF + TNeREMEFITS.oS 4- bs,*t6 E: Gmwtr-

ffi

An-rxqv:r:

HrvE" ,frs=,*St,

Pusr ( +, 6)

oR. bco:S . I( sruf .s:b- co:$suS) b oR. b = \ ( sr*S - cos$ n"r S) !:rn{L-.b (r) oR rnnrga \ css.P A -u)-b\s"s+$) \ rtExr Se(,.SS = (Qcpsf SXQ,F>$)-(g,,slN$

oR 6- cos"g ffi

(\ cos+)?

b = 3.5in.

Os,rrc. E*1.(r ) - cDsb

sru$-b

T*en.-

b = 3.5in.

(z)

NorE; F.'n os6< m,xf'(1fr;)

Ereilf

E'a.(r) =) -9os D a D 'Mus[ rrs€ Jns, Foe THe:E vAuu€S oF S

h-otrtNG

Tt+E

[=\.8T, a.ltrs,oti

o.s

GR.APH,

ES>, flow r$e BALL

5rmruAr?uy, FoR 9.>'. S < zJoo. Eo..(r\ =) - 9'c'<\ < 9s'

,'. S: txts-'(ffi) FoR.

zloo< $ t3bs' .

Eq. (r\.-)

As stloWti

o(^ lBo, ezi aU"! r.\^:cors5NxT; XT Elgr\ Co*rE, (E\*nf ( \- L)(\),urrFrL r h.=o.4,

b-- rNrs'(q1*ii )-' iI'd \^IHEN

\fre$

BOTTUS€5 \ TTNN TH€

. reo'

5IEP5

FSR. EA T

ColqBru!$t$N

oP fir,

-9,s. bs O

pS -PRs(rR.xh4 !)ururNe otr b l..rP.t-i r/AL\IE BoRbER Fs*. CaR.APt{ otr S ',1.. 6', CoxsrRwt LAReL AXES . rN FoR. VA,uuES .cF + FR.aM b To 3bo IO SF I\\CR.E}{IENTI

CovrPurE D r

Fonst +< rxr.f'1ftf ), jbs' g = rANit(ffi,). Fon -^ni'(1frr)*

b-J

*.9o' (couTrNueb)

11.C2 continued

(rse1)1a (\-h)d Ftqvr Nei\€ -AJk. 1[ Srrttx T\rE ORKTTN otr A {aEcT\NG*AR. hl rt* (JlDRbrNNrE SYSTEi4 N\ Pt)l\fi o -\{oqraonmlU

( rrrtrssqm)

t-.4oT\ON

sP.- [ = Ys;

K-- )€o* S*t

MDTror..I (oN,F,

VgcargAu

\ * rGo*(&)1t - lltt

SuEsTrToTtNG

rr:s0,.

FoR

v= , !E{r

q -- ('is^)r' 3t

t -.

.'.

g$NbtTrSNISl

N: \ , \ = L, XrornL = $ CSRRE\eON\ \, Zr3r -.-, B TO STePt x\ctTAL \s Tt\E r:r,>r4 oc ANb A, B,,Cr - --, H 1r..lr\rNL

A\: (qJ1- 3fr

w$ER€

CoH:,ueq Tv+E MDTT$N str r*E tsALL ,1steR rT ON U\NbS A GIVE.N hiTEP l. Dgis.rqr-nrNE rF Tl+E BNLL *o*r^gltS TrNrce . OF.I :jTep A: r* \L

,>{ sfEPA' \=s:

Qs^\, = ( l-\) rf, cssx

,q : rf, srxrx Sgf

x' fr.'*

^5.xX- il

MsrsN)

/\cqeL.

oF. P(qrglAryt Oor Lr${e F's* rN\TrNL xtlGuEb t( i x= \bo, ?-Lo.ef Fon vALu€b sF \ : [: $.4. $,S Fos. rNrTrNL vELw\-trES Nol fo= \,8t, Z,4t.3,st Foq. EAL\\ COvtRrr..tAillrSsl OF K, k. Axrb rSo (.>mp.r,C. 6t* 6;.rb (r$n\:

r\st\LqrrmHu brsiNtc€s BE:T\^{EE\I T$E stoc{.€SStVE \1t1\fp.gT. POTNTS SF rF rAC. BALL B$n.>NcEt -Tv.l\qE \r{ DgieqrqrH€

Hlxx)o-ig="f ,>R (x^)o = t *^ (^sn)r

sr (,sx)r s o.'\S+ (rs-\)ts.i)- x.rorr\

'. " YnrSf BALL ..F\RliT '}n:\ STEP N.

(XH)o . Q, THE BALL Bs..,NcEs r\{rcE Jr DN tnEP \. TrxlrC€. sN I*l 6EJ.IER.NL, Tt+E tsAL\ BSuNc€t (xt= A.B.CI ---, \{) sirEp N \F

Conrs,SeR Axlb

T$E

TWrQE

BO$NCES

CoMBTNAI\,SN

bS A, h,

,'S. oN

\^l\\\Ct\

BAL\

tss$Nge5

Si€P

Tts\E

( x * ) o 4 ! * ( s r - r ) b X l *). 1--A

\^rr{en.e (X^)- a t t* 1sn)r

(s*\1 NND NRE GrvEu B€Lo\lr. AN\ X* Z. bereRh.lr^lE rF TFle L\NNS sN srep B: ^

FoR

R$ctf _ -(XA > o\,

.IAKTNG

nu\

\\rNE --

y . *iTt:i-F"V;ff-

tlr

.n\E

}q 1fieP

CrerqE**Lr rT Nr N

a(e/.sfX-x)\'r"

TF\E

\FTER.

Bsutttces

BAL\. oN

BouscES :Fep

oN \

rtr

uPbNrE

N(9,

\\E

^S*

NEtT

VALuRS

F$R

SfEP i

sr i (^si)r=(\-\)\q H - (*L)yl

.1,'5,r)\-1*

P$ogcr*A-

i :

\ r 1+l

ol>ThrT

akve

. O

(r*st\

xsb T\\NT Tr{Ectr Tv\E VERSTCAL CanstFotrtsuT (^Sts)\ MNGxlffUbe oF T*E VELOCTTy AtrTER THE BD$rsCe \s

(.s*\ = (r-tr)tqh-

tnsr)rJ

F. h, n\t t\

ctr

t{s\l B'Sr-rgcqs Tl\e B\LL -Tb\c. b$WN REMI\NING 1fiqtrS PRrNT : " BALL g5grrNue\ Nf B l.s -n\F. Ts Bs,.rxtCE bOWXr SfiEPS,

Fu.|\rIY, \F THE BALL BrsuuCEb on,.t SiEp B, HAVE 05rNG T|+E E)APRESSTSN \ER\\r'€h, ABsvE SoR $f, t'

[{NYE

C]\ECK

THE

bgienr..rrttE

= *\q t(r*)r *

-I.I{,XT

REFCJL;I

csN\r\ER

C.suBrxAr\sN

rnrr{ERE Xrq=

Notrs.rq

Xxb

KrcnAL i z g,

L XN s [* (\-r)b i=A

t4)f

+ (i-r)(o.s) nr16 x*orru > $.\s -Btuu itb Mt:f>85 tTE" \. ?P-r$.fiI

Rtsgi

Tt{e RALL ts,ouHc€SsN

NEIT

sl.l

ST€PS

)..t,rrrL l T.ncint . X-r,crr5.\- Xxl 0nspq6 \ q \. \+\ ,\hI\ \ 4 Bf csMFulE NEw xs 1s

?ssr\rvE

: tsi(re.)., + Lts^)*f- zgh ,'. Jr xA -( [*b, :rr€P B.

r*E

\tr

CoNsEqrrrtV€

-h=RlxA-itS

O x r t f f E PB , \ . - h ; 5sr-y61q"

Dn:iEqFarNE

6g1lgR6[-

- -

( courrnu$r)

18"

o o

40t , r . t o

1 . B m / s BaII 2 . 4 m / s Ball 3 . 0 m / s Ball t

Bal l

5 0 8 1 . 9 m/s 2 . 4 m/s 3 . 0 m/s w-a74 1 . 8 m / s 2.4 m/s

first bouncds twice flrst bounces twice misses step D c onti nues to bounc e

on step on step

A C

dow n the

BaIl flrst bounces tw l c e 04 s tep A B a l l f l r s t b o u n c e s tw i c e on s tep B B a l I f i r s t b o u n c e s JHlss_e3_9!3r-.I.-_ B a l l f i r s t b o u n c e s twice on step A B a l l c o n t l n u e s t o bounce down the steps

( cssnNu€b)

11,G2 continued 3.0 m/s Ball Ball Ball 1.8 m/s Ball 2.4 ur/s Ball 3.0 n/s Ball

50t .

11.C3 continued

mlsseg step B mlsses etep E mlsses step G flrst bounces twlce on step A flrgt bounces twlce on atep C mlsses step c

A sTEP ED., NlqGnrrN- HrLLr \9gg, ) wt\\ To Nu+4gq1cALL\ 5tte A,L = O.ODB 5

1,8 2,4

s BaIl

contlnues

= I

d,r

EG\J\T\ieI{5

gL\8o-"

\srsrb-Lrst

b)\st)'

t-\srNS-L^rt

1 . 8 m / s Ball flrst bounces twlce on step B 2 . 1 n / s Ball misses atep D BalL misses step G 3. 0 n./e B a l l m l s s e s s t e p B BaI I rnlsses step E Ball mlsses step F

40t

T\*G.

INTEqR.N$E.

St"i,s gi s Sr s S.

uNr\L

,{F

T}TE. MIbPI)IN'T

RE.SPEqT\V*UY,

T*E

T}+E VNLTCES ENbr

twlce on step A bounce down the

, wn€RE bi ANb b. NRE

XU\

FTNAL rrvt(,

INT€RVAL,

stePs 3.0 m/s Ball mlsses step B Ball nLsses step E Ball mlbses step G

t)se

urxtt\R

Ir{E

srr.tAL

-TO bgfERfYltNE

rr..FiRRFoLr{TtSt.I VELtCr"rY rt5t :

q ? {'* €!$

(se-^rJ )

- L - v l

k. \o, SlrHHl, r*\ oF ?*rr-$T .rrAE vALueb (*se. Z: Dese*.xrruE T*E VALr.rE oG b FoR ts FrRst aeRs \r.lAtcff T$E VEuostrY Eur-eR. wrErllsl> '161r\l \ obe T$e Mob\Fleb rs l\L. b,OOB s N$}1ER.\CALLY tfrqP srLE RQuxl$*fS r*-9g6aAiTE rr\E So,S,. tBq oR. . di.s. = l( Ssrub- hNt eo,b,>tso dE l- gs,tsb -h-ut b..lBDo, b. t r&>" oR. bot rgs, $,'tg5,o

GrveS: Log= tOvn \ -Qsrec.. . -hart. h. = O, Lrro-* *-'. *r{'.nA',, 4 S.=.'lo' \"o, \ir) Elg5:

ANt! The FrRsr d,rr* rwb oF S vALue\ FsR FsR. vtrfrctr tf.o EAe}{ c\)r4$,lNNtrsrl oF b" Ahr\ h

St=|"t

\^tn€.RE q

Axluv:rs TrS

lIr{E

TANG€N-II DrR.ESfioN |'n*E OF T}\E. COMPONENT $F rr+e NCCE.LERNTTSN

T*E UsE T\*E

A}RPLANE

ot = qsrN(\Bs-g)- h*rt = l sru$ - h*rl d_s Rgcnuurr.rrrTr+xfT of = JE HAYE = gsrNb-h.^st $f

S= lS'uS " hst ds = . MSTTSN

TTIE

NTRPL\XIE.

otr h oF S" otr THEM|NE ThlE VALue AT TlJre sPtcrFr€b vgLocr-tY AhrqLe

Ir.rP.rT VALuE lrtPur VAL\re (HSe l:

lsP"r

Osg I FoR. ExAMrLET Tl+€, MobrFrEb Euuer ( SeCoNt>-oRbEr. RuNGe- !(.vrrr Mgnlsb MQ11s5 -- sEE c*XrRl ANb (lxrlUe., Nrxnecrc,.*u MEn{osS , tbR -Eq{c'ru€E$, Zd ( Colffrr.rrXb)

Ty1E v1Lues

ft=0 16.23 12.13 8;37

for

attal ned

M ax l m ur n v etoc l ty

70" 100' 130"

L, bo, nx\ N+

otr --9n$tRRvt .. sr"JTR.rt

Ssr-nr.altv

3i +^t Tv\E

s{=sl * ffi,(sz-si)

?crHT

' l.l = (- g Fr CoNSr\\lT) t{AvE Now, SnsCE Tr+gnetrbRE) Tl+E bTtrFEREhTTTNL ErbvA'TrsNs .

bEFIxTE

otr S \T r+E rb Tr\E YALue TNTERYNL, UNTrtA ltVE NT ^Sz \:' Tt{E YELer\fi IFNERVAL, OF A TIME ENb L\N€NR. tFrTERPsLNttot t Ts bETtRrvtrlrtE . FrUN\- \NGLE .bg

BEGlruurNCr oF f. . D, \^rlrg\E

TNNGRNTTAL.

k =2 x 10t mt

a r el eas e

angl e

Vr,rrr 4G f -l x 10-' t't

11.6? 9.18 6.97

16.19

t2.7r 8.36

f = 0, theory t6.23 12.73 8.37

Ftrst t ( 6l rl and second I ( 4) al rest posltlons release .angle ( 4l k = 0

(4),

70" 2 9 0 . 0 " 100" 260.0 130" 230.0"

(&l2 ?0.0' 1 0 0 .0 " 1 3 0 .0 '

k =2 x 1O-l tn-l (Ao)

289.2' 2 5 9 .? " 229.9'

( 4 )r 7 1. 6 " 1 0 0 .6 " 130.2"

for a

k =4 x 10-2 m-t

(61r 2 2 9. 4 ' 2 2 3. 1 ' 213.6'

(6)2 t46,7" 149,3' 154.5

11,C4 continued Cx*. TRFi\rELrxrG..^9N AH EarT Rr\wrPi 5-=,j1D H yr\run-= Q; [q*,ro.l € r,so/rt i Rxtvrp \\ :srf<llrGrrtrT OR sORVEtr E\T\{LR

bys\\:

ANb

)\t

RESF€g-TNEL\|

,{t

\NTERVhLT hNb D\siAl*IcE,, T}+E

FTI*IAL TTME

INTERVAL.

(^r*ves

(?. Boott)l Sf \\ e\r\\€R

FrSb:

S\)NSTNN-T

LINE \R.LY

DurnsxrG

tUTEr?vAUS

r*E

VAR'E\

RAvrp

TRnrr(tt\

FoR

EACI\ RAMP TypE

C.OMBTNAiTTSN oF

nxr\ S \Nxu.(:rs JNr RAv\P, Et = qsNsrNr.tsT CA:e r : SteHrG*i Fsp. .n\r5 \)N\SsRlqLy bEg€Lgp.ane\ Rc(trUrucAR.

'*$'="f: -roE --

l+f\YE

F,rlrJlirQlN

1g= So + (-ro)t .o rSt= Sot + ?(-\D)(x-*) rr\Nf o. \5 q.o;11fis.s.tT xsb

T*E"r Nl\ IrbrrssG ANE

V{qg'RE trsR

cAR. Ttc

T\{€

TcriAbt-r}+e

ArorsL -io

sOrnE

THe

,1eFRES\ RY

qEIPEqTNELY,

RAMP,

.'. q^1 =o.,t+(glf

FDR. E-NC.ATIVIE IITTTERVAL. (UNrF. \CCEL. WrCn*Ss) O\. * C-otSsT\t\ l..lsrnr-6ga S, + qt tt-tt) Tr{st*r -t5) .

\= x, * q1t-t) . iq. (t-t)t

ry- = Go *Yh = Bb ftls

|\Ub

T\E

G1=-

rrvE THE_

cNR

\Nl

N.L!r),

( E)

(t) =) fu= s, * or

ttf

(Tt) (4\ =) f,a = \r + N, * lOt S*S Foe. T\*E FrxAL TrvtE TxITERVNL, Tr{Er.t, ASsumrsrG t, =s HFNe--

a =:l{

H\"E ANb

ASS\)MTNCT

ANy

Trwl€

tr+Ni

FOq.

0L=-rot|rt

At=D

(4) +

I q

= t} f,3tt-t,)dt {, ": (r) o R .r s a , . t r - f u t i - t , ) t S. = r-t

Ar 1=rr,, il^, '

q5IE-15: Cocveb f..Ar.4pr$g

.t.

= {:a^ I.:tu htt+,)'Jat

(t-t,)- frt,a -t-.)t (L\ DR. X =\r * r.5r Foe

5l:

\ S

\Nb

\AISEN t.te

(t\='> Nr=q-S ttl (?.1:1 Kr= \,+ risr- a -Tl+E

(FoR

tr\ +

Frxl\L

TrmE

, I{AVE -.

t= ts,NA{_ .J*\g*S\ F\'NE --

(&au-rru

\sSrJM\},Kn tr=

< t S\,

'l-s,ur*u= {g

(cot'nrxruq\)

1aN\r\t$r

-' HNvE tt+. (t-t,) o. u\t = at

r,r-=rsr' Ar 1=Lr, IJS:_= fiS.(t_t,!61 ,x

A; L'.t,.'i:x, .. [.,e* =I*,t",*

(s\

\t\tt+ERe XrrNxL \b -IrfE TrrNL \\SrNW.rtS,NrL -trM€ TAE sr\l}.lb$R.NTrctrt cfi: tAtrrftE

INTeR''AL

[r-:o, S= g

uL=\ s

oR Ntb (L) + KF,urL= x, * s,ts,xxL- tdrurL

ASSotqrNG

urrsEsRL\

FOR. AI,IY T\trtE (o1)q e$

oR 1$=sr * :Xt'

()

c.oNvENreNcE)

o=sr- fr (t.,,u*\

(\t)

Nor^r--S =o.\.=S,- (\.L.)

(t\) TNTERV,\L

Xsrut = Xr * ,\ts,r.rlL+ i.o.tlrNrl

^S, \Nb a.1 ,\RE lr\G. VELegt\iY p.rst> \rseRe A(f Tb\E tsEGrNNrt& SG b\srF\NeE r RES"EI:TNEL\ trrNAt- TrvtE "n{E t$TeRsr\L.

Arit=t,, r=rr,: Noru--

L5\

,trygQlf\L

.{AYE 5f= 0.=- R.tt--r,) (E)

f op. rS=O

,{r t-\rxr{.

rs)

lrtnNr&rry f{/\1r4}, tS

k-Ll

(4)

Arss wr\EN t= tz , HAY€--

Fer< n$.=\ S

r-il.A\Et-s-s>

b\Si\t\LE

\^lqeRg P= BU> R t\$1b lqna\*l= lo{lba F.sn EAcr\ rnvrE rNTeRvAL, O{. \b QO$\STANTANb SrscE TAE 0.n 15 b,4NxlMuvl r{r rrmE t, FRs$4 t, T$ \r, VeL*\TY bEfe€Ates

N{b

L!t)

ANb

too"

ss,Nr,.l- =s,

(5)

toot : R x-o-r*. = S

e csNSt\NT

Nour-. ot= qi* = Qf *

Rlrqp,

Trvre

\ :, brsl\NCE

,\Nb

{sTop

ARE TrA€ VEL(Sr'T.( ANb g€JITUNUA $F ]t\€.

oR K: xr+ s,(t-t) * tt

({,t) \:

rtR

Nbw--

t.t-t,)tlat t\.4,f (u)

at= qf *q'

? al*Ct,

( qohl't'rNuEb)

11.C4 continued

11.C4 continued -4

go> tt ANb \Qrnrrr\ = ,\)hr=t. W\+ERE e= Trvl€. rsTERvN\, , FsR. NNy \bw., (vl*EN nG. (qn)wtr* ccrrn.S 0i\ t-=tr VELCf.\TY \5 M$\xltq$t^) ' Qf,'luRS NT (\\tn.o. t-=t,4 Gtng. (O",\*rx^ Aff Trt<e\ *'.t .'. As:*t*E

CunrP.ne, fr.i

'Jesnirg r,.fie Ai.tb S"eebl ' {= t+t \ ,S,' tS.

(Hcrr€-. Foe

AS. L=Lr.

Tr\Eu--

oR. (\)r.

1=t-zr

E\NAL

r+AvE (8)

= Z(fr--rsr)

;ff;

+ 4\tJ- Bqsu* (A'.tl-r\b)re,

eornh'rqtr;;r

c*\E,

Gr,ap.rrg

.Ils.

Sr_> S

Oessre b\t;i.ANcE x;. xixlilsrr.hsxre .rrvrE lsb spR€b: FoR. r*E

{at+\ \ f,=lrSa Es;tA.l- TrtvIE i\sT€Rr/NL

C,cv\p\tfE L*,*ru-'

ts,Nl*L"

\rslANe€'

-rerrsr *-JR;,^!l

Covrp.xg }rtfiArrE XrrortL: Xnrsir\. t FnrsT rr\G. vNLues RS tsrop Ahlb XroTr\ CAs€ t'. Snnsrssl Lr\xt'trRL\ vNR\rNq H Fon ENc$ r\)cc€ssrvg lrME rr.I\ERyF^ltst.aP.r.fe \t t ,*L -- r.ir- 5

\*.L:

Covtnri€ -\rwrE trto" i ' tsrcp a {+ ts.HrL C.cr.aFrrt€ brSfANCf. Xrcrrr.f-', -

PF+cCtF.AF

L-;. *.'I''" tsrsp = \. +ts,ru\

Cromp*tre. TrmE tsrsp',

C*se l: Sryexr.rtt R.xr.np) ef = csrsirNt$t C.ob.tP.fiE -rrvrE tsop i {rtoo = t

\f rlruE

\gBB,) Nu,ue N.>O : (Ot)a. L(tSa:N,) Csmnrre (\)a \)psl:ie brtfTAN(E X,i X,.Xri S,* it\\e uesnSq TrmE nnb sFEEb'. x\,rNai Sz=0 \=t+\\ Fse THE Ftr.lAl- TrME TNTERvAL (\)r = L ( tS.--S") Cor,nxrte (at)z:

(u\ + Xtrrxr i x, + N\tFrr.AL+ {(ot).tlrrrL t$t) w*SSrE q Aub \r ARE Tr\e VEr*rT( xr.r\ \$STNxIC€, RESEqNVELYI AS \\G. tsgq\NNINCT bF .n+E INTTIVNL, trIN\L rIVIE

VELr.rcrsl

t+,r.rn-

lnErrrsb FsR. ,$,?. g51N(r lfqrNrsc.r't L:g,gr FsR. ErA${r\E\ Ct+.P*r Ah}b C*xfr-e, NrrndifrclU MstS: FOR.

E6Crf

*' ' tl

-n\E rtAUuE\ ?elr.rt vRr\n -rHE \rsrrr- L rtAuue\ -.$ t*roop ANb ANbl\Ton -SF tssp CnsE 4: Corverr RAMpr H u,qgARLy vARvrrS, Fon EAc$ snxgeb$we T\bnE rusesuNL souv$o TrlE essnsnsrt

F ,sl ^ -.'*^-:-'v*r As.t - B-[ -t. * (4t$,t- a,]o ) = o oR ;i " \ t Q..rArtrq- Eol\)NT\sN .\lr\r(rl bttr\NE\ f., Fon n\E- trrN\L 'trmE INTERVAL,tJ-=O N( L=tnuAt-, T*et r \S\Dl<truG t, = o rllv€ -. (\)z = a(rs.-'S,) \^rr\€trG. Eo..cb): ls.< o (s) =) o = s, * i(at)z tl..cat__ \ zhsr sR ter*^r= \(:)

tN\TtAL

),"

Krorr*.= Xr* \ \*,*r{* t, ottixr.L

o,lo =La(S.-s,\J'* I S

CrctSs,UeR.

(*"-*.

(ot rpsrE tslhuNLl ts,r.nr. Corrp*rrg trvrE tssp ; tttoo c,oxrHnt brsilr_tc.E l\nsrru*!

gr\ o KL = xr+ 6, r tto..). ( tt) (ouUr.rrHG E*\s. t1) rn\ tB) Ts ELTMTNsrE. (ot\2.

InfP,ory

TNTERVA\

nrqE

= -

F'ca At= \ S Ar-tb w$EN (S) + S.=\s:.+ i\qt).

T\€

c+nrure oL, \r-

(-r)

o n h , . = ( o r '), :2+ ( *r " \ t ,

Sr_= l.5,+ O1

VJ*rue ,q>s Oeuxre 5153nrucEx, i {, =xr + q + tot

\,o{trbt ) 0.= Qnarx

&r= - ( tsrs - Si6n 1'f'

Csmtr*rT€.[h,,

rrrtnxr- :

t::*t*$'lN'|r

Ouresff

For a stralght hlghway and a constant of the speedl time to etoP = 8.80 s dlstance traveled = 387.2 ft

rate

For a stralght hlghway and a untforrnly change of the spegdr tlme to stop = L1 .17 s distance traveled = 789.2 ft

varylng

For a curved hlghway and a constant the speed, tlne to stoP = 11.29 s dlstance traveled = 581.4 ft

rate

For a curved htghway and a unlfotmly change of the speed, tlme to stoP = 20.?1 s dlstance traveled = 1015.3 ft

varylng

of change

rate

of change of

rate

?q,rrr rr\e VALLTES hhlb XtcrrL $.-t:nsr c'sNsir\hrt C.lse a: Crrev€b RAvt\ Et: -trvtg FuR EA0S :$eg€SWE. rr.IT€tNrfrl. ( crostrrsrl€J:) Copyright © McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.

rf, = *o\

E!E$:

; o( = Zoo lo

11.C5 continued Bat

TNCREMC.NTS

(q\

(b)

d*,**

ct)

(z)

s'oR. EAcrr \Nb

.tr.t ) \, B vn ANlb t l.B rn .Js.s Csmn-nt d: d: {sruZx ?*,*rrru\E oF A ntob s$ \NLug\ Nexr vAL\J€, Ottr O\ 'la.a > l.Brn ANb *ts.{ t rn (\,eeoa)Arrsve'. CrDMe\Jr€ (xreean)^*.,ue e \ooacat4[s,H* *

r*,r.rl trE 2.2m

3.2m

AsF,uVSts M.oTtor.t (sursoRvr)

t-loctaoNTnL

\ALuE\

s€

t\roeo".\4ecilE Nexr vALuE sF (or'4e,.lr€ .lr.,rrr -\.^xr I

Qcvrn>r€

X\***,

X=)4"* ( rs,(s: o(\t o R ' x t a r.tc.cJbK (Os,r. }rtei\\oN lglaircAL ( ^r" Sru K\t - iftt r *: \ SuBstrrr-lTrNG FOQ. t --

*.fffi \-- (rxssr\x-

Tr= d

Fsus'\ F.,

(4)

t\cgEL. MoT\sN) f,-\ = {" s,nro(- gt ,

s,r,rZx

"1.u,nx< \, $ m d:t\ruLK C o w r P * : r €d r ?c,*n rr\E V\Lu€S otr D\ ANb d Nql:r vAL\rE, os 0( a.e Yn 3 X.{-** S €,4 .tn

E=*t

-t d= * srnZo( oR Ar TL+e *nor,r-l TLreeReTrcAL T!+E WX6*, rr\ = 1). Tt+e.nr -. oR O = rS'.9tr.t!r - Styro^* Tr€n\ r-lrarr r = (,q

=i X\noo=

t s,nrlot

Covte..r6

(XH*ssn )ge.u*^ i

$qrs,n T*E

VALue\

ANt> \= g :

$,= (rNNu.)d- lf

AsS

(3) A

f,\.n**.

re) D( NN\

\\€\Cr\rT

.{rnr*r

(s)

tr.n^* = t \rxtd" ll($t'*^)L

(r'

t \.)t (1)

Otr x

Nexr vALuE otr o( Ls .{z.z = \,8 r\n

"THE ?cr,,qr V.INTER.r{tTS rrr€ NRBcR Xi CORNER C... Ne*r vA.LuE sg x, X*i-^* 4 2,2 vn oR lc \s.+ < t, b rn fS ANb Xy-,.* ? €,4 t{} Co*,.o.rd" g' d = B but 2'ot R.rr.lt IIIE YALueS OG D\ ANb d

\r\exr \ALuE- str D( If TRe \NrNtER. grr5 -n+g AR.BsR, *1: l. g, vrn AT TRE PorrtT otr tMPAcT. THE csRREStrcf.tbrNGTb{e sEvEN ?oss\BLE ten sccxoRre: rEsTe\ trsR \/NuuE Otr X \S Tr.lEN.- _z rN TRE_ ?RCCrnsM BEUSW, A\ae \LL$SiRAfiEb \.8 = (rHN o() X n -- :iq. 1-*

ETi,=

oR. x^ceo*. :

fF\trx t tt-r.ryo't'-

- n ' ' - - - t

*F?r=t

FcosA \NHERE rb\e t+) ygxiER. To TH€ ABDve

ANb

Srcsb c$R,R,EstroNb Axrb t- ) F\rTTrr.rG, Tt+e AR.BSR. trRsb4 trRsM BELD\AJ, REsPsiNeLy.

f.HfW luu

N rr{\vluvt ANb v\\x\Muhq V\LuE t s\ee otr INqREMEN\ t>tr Dr otr o\ Fon EACH vALuE C'or'nPr.rre .-1 Nt *a ?,LYn : g : 't TxNu" - o' oL+Z "la.e f c"Tr(-

oG 0\

rz)

(1)

(4\

(s)

t tr\

CovrHrre y sr f,-s.A t{s.4 o S.4 TNNI\ (-l)

( coswrsJuEb)

11.C5 continued hoq$rnm (o)

o.rlp*rr

tror c;t'iO.o0c, thc water hlt,s For. c'! i5,O0o, Ehe satcr hltg a For a 30 .00o, the water httg For o I 35. 00c, Ehe watcr hlta For a . {0.OOo, the water hlt,s below at x . 3.106 n For c ! {5 .00o, the waEer hlte belory at x r 2.335 n For a ' 50.000, the watcr hit,s For a r 55 . 0oo, the waBcr hitg For a a 60.000, the erater hlts For c !r 55 .00o, the nater hlts For c . ?O.0Oo, the natar hlts For c s 75 .00c, the uatcr hits above at x s {.55? m For c !.80.00c, the watcr hltg above at x r 3.133 m

thc the thc the thc

grourd aE O r 6.5s2 m grarnd aE d . ?.909 m gromd aE d r 8.828 n grorurd at d : 9.5?9 m top o! the etbor trcnr

thc top of thc arbor thc the the the the the

frqr

ground ag d r 110.039 m grodnd at d r 9. 5?9 rn ground at d - 8.828 m ground at d r ? . 809 m grotrnd aE d r 6 .552 m top of the arbor frqr

the tsop of thc arbor frmr

(b) For For For Fot For For For For For For For

the water hlts c a 46.200, belor at x r 2.202 m cr r 46.2Lo, the water hltg below at x E 2.201 m r 46.22o-, the waEer hlte o below at x r e.200 m c a {6,23c, the sater hlts c - 46.210, the sater hlts o c 4 6 .2 5 o , Ehe water hltg o r 4 6 .2 6 0 , Ehe water htts c 'l 4 6 .2 7 0 , thc watcr hltg c = 4 6 ,2 8 o I thc water hltg c 5 {6 .2 9 o , the water hltg o - 4 6 .3 0 o , the water hlt,e

the

top

ths

arbor

from

Ehe top

thc

arbor

from

the

top

thc

erbor

fron

the the Ehe the thc the the the

ground ground gr ound gr ound gr ound gr ound gr ound gr ound

at d. at d. qt d r at d . aE d r at d at d r at d r

t10.181 ,m t1o"18f m t10.18{ n t10.18{ m t10.18{ m t10.184 m m t1O.1S3 t10.183 n

Chapter 12 Computer Problems 12.C1 Block B of mass 10 kg is initially at rest as shown on the upper surface of a 20-kg wedge A which is supported by a horizontal surface. A 2-kg block C is connected to block B by a cord which passes over a pulley of negligible mass. Using computational software and denoting by µ the coefficient of friction at all surfaces, use this program to determine the accelerations for values of µ ≥ 0. Use 0.01 increments for µ until the wedge does not move and then use 0.1 increments until no motion occurs. 12.C2 A small, 1-lb block is at rest at the top of a cylindrical surface. The block is given an initial velocity v0 to the right of magnitude 10 ft/s, which causes it to slide on the cylindrical surface. Using computational software calculate and plot the values of θ at which the block leaves the surface for values of µk, the coefficient of kinetic friction between the block and the surface, from 0 to 0.4.µ

B 30°

A C

Fig. P12.C1

A block of mass m is attached to a spring of constant k. The block is released from rest when the spring is in a horizontal and undeformed position. Use computational software to determine, for various selected values of k/m and r0, (a) the length of the spring and the magnitude and direction of the velocity of the block as the block passes directly under the point of suspension of the spring, (b) the value of k/m when r0 = 1 m for which that velocity is horizontal. 12.C3

θ v0

5 ft r0

Fig. P12.C3

Fig. P12.C2

12.C4 Use computational software to determine the ranges of values of θ for which the block E of Prob. 12.60 will not slide in the semicircular slot of the flat plate. Assuming a coefficient of static friction of 0.35, determine the ranges of values when the constant rate of rotation of the plate is (a) 14 rad/s, (b) 2 rad/s.

14_bee77102_Ch12_p003-003.indd 3

3/13/18 8:16 AM

O.A

Ahlb

Ogrr

troq

l^ f O USTHG qlr.a O.$\ i : \r.t*rLe G^ , O A,Ss, Artt ', 0.1 w$r\E Og1*>S;

ANArvsr> KxrEvtarr\c5 l-{AvE-. $g " gx + gqn wrleRsSr.:rlFAG€

gqlA

Otr

ALoh(i

hrqcsqu

Trl€

lNcuNEt)

A, T*€rt-.

.rLGr=*:";::I". oR

I = $nc.\X- 0.c.) - nnc.(qi q"r^* o*csstci ) ^ (?)

mg9ryr N^. -15 -tf.l

*/fFf':

o = Fx, x' tnBo,^rI T- Fxs* \r.lg-5rN3o-= \arnr-\qAcos3o

oR T- E** \Oq sr*rji = rO Ash - \eqAcssSo" (3) N^*- Wgcss3d= - l^DA^srr.rrS \Aar: (4) oR. Nrs = log cos3J- \o\srr.riS

SurbrxrGr Frg.pNrs i. Fr.e= iol^ ( lcnsf'j - GAsrnSp') oR

(S\

5rrBsrrT\rrrr.rG Eq.s. (L) ANb (s) $fto Eq. (j)-. Z (9- Oglt+ qAcs53d ) - \ql. \g cs:r.rl-q^ sruio') +rsgsrrr 3d i I OO.gr^ - \OAr. eOStO'

oR g( r- Spcosj.j+Ss1s.1d) (L) = ta Gsre. - G5 (Spsr*3d*r." cs:ls') gEFoRG Norer Br-ocX A wlLL r\lcrr msrrE (qA.O) BLroclts B lfr.rU C WILL Nctt Msv€ (Ourr=Ot=s} TutesesoRE, TrfE SYsiEM WILL REMATN Nr

%,:#.b3i+ssrHiS) oR

p l D.BoB

=o Foe No

F4a;noNl

#r*LFx =Tq.crAl Nrr ttNfi - F^- F^b.o.Si o Tf\r0r >R t tg, (s,uAd - pc.o:S )- Fr r, Zo G;1 (1) tconrlNuG.b\

12.C1 continued

*rEF.r= o"J^ ,i^:i^I;h':^I,"Xr;:".(B)

bL\trNG: Fr=ANn oR. Fe.= pN6 ( css36 +As,x1 3d ) + zo,r*1 (g) a.rgsttT\rrrN(r EQ,.t9) rNTs E*t. L'\) -t'\h6( srr.r3o'-pcrstid ) - FNr.g (qos3S+;,rbrNio') -a,cA3 = ZO\ oR Nr.ettr-Of):ru3d- zl^cJDldl - es/^1 . a$q^ SueSfir.OTrNG FOR. Nrg L€q. (+'fJ.. ( lrcq clsb3o'- \o\srsrS )t( t-lr*) srr*jS - b'^crbbSd l - Lorrq -- LoOA A= (\-/*a) s,*BS - Z;r.Lsioo Le:r T*eu .. g ( Acsb3o'- a/J = (Z * Frss3s') q^ tctl

No-rE: Busex. \ \lrLL RE-FrINN Nr Res\ wwErrr q (\ Los3so-zp) = g oR t t r-^f1 srl.IlS - z}rcs\iS \c.s:3d - ],pr :. s oR ( | s,u bo' ) At * z( \ + cs\tto" );r - ! srubo' . s oR. p.. o. \?-\ BB FoR. Buoc.r< N rs REvrN\N " RESI l(r A! Now-- RQ\^lRrrE Eo. (t'\ qtyr, = U t3 ( t- sp qs:3so + € srN3oo) + Gx\S;* s\N3d+ l.e(s\is')] (r r) \ht$\C\\

values

ac c el .

For

those

tt 0.20 0.30 0.{0 0.s0 0.60 0.70 0.80

whtch

tt for

ac c el . of B w r t 4.307 3.599 2.891 2.183 L.475 0.767 0.059

B w r t, A , ml E r 7.358 7.L67 6 .975 6.780 6 .582 6.382 6.L79 5 ,973 5.764 5.553 5 .339 5.L22 4.901

the

wedge

rest

m /s t

Q s : \ J =\ \ b ,s = , o t ; osprso.4

qrz)

Oqn = ?,. ( t- sp.os3s' + S srNSd) \^tt\€N O.A- O Otr

of A, m/ e' 1.889 L.742 1 .594 1.445 L.295 1.143 0.999 0.833 0.676 0.5L8 0 .357 0.195 0.031

REbrrc€S

O.rrutSE

accel.

(to)

Nc^rS<r\-zrr

\Bffiq

tt 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11. 0.12

.r*tE Bbxr< b Nr \^l*\or \)Rtr'{E, \ LETVE: tTE = O, O.oS, O.l\), -... t}.+ F

PROCTeXt*f

f.rsP,..rf \N\TINL VNLu€. Str ),L1 /f,= S Covreune N : \ = (, -1l.. ) :rxr 3d- z1-lcss jd Cpvrv.rTE \: n &,* = \**"-z;r.

--rc:

\xl*rue or > o C:cmPu-re [en : O{x = tt 9\r- spcssrd+ S srsr3s') + Ox (SFbtNS" * \" cssSs'\l rRrtri rHE vALu€s sF lL, AN r $r.r-tsAerx 's,ol \ET:\ TEN-\I." + $. \ vALue (rop.[ A t\s3i \r.l*rue Ggh. > s hrnlT Tr+E, VNL$€! Otrb$(Te F:

F=p*

+ 3 srN 35')

Asl\ux-srs

5*rBrfrrrirTrr.,lG tgR

F ..

Norr-. ot= $f

$I_

s= (a\

Auso--'.i--..g cF. *=is elF lA

Nrrb

G"lr

S.l

DtFtrerlENTtr\L EQuNtroNS (r) lub Tu*rs, 5ESrxtE T|+E p1qfir$N CF T\trE BLC(K. A:

rr+E

ts1,-cnK LEAvFF

TFIE

g cosB- t'* o

T*us, bEFTNE\

TI+E

V\LuE

BLCCJ(S

LEINES

TT{E

( c-cn$rt.t*oG.b)

s$RFA(.e', N -+O,

,\T

y.6+tG}I Tt\E

ST..TRFAQ€.

Foa EAclt vrruuE otr ,trf Degrug Tr\E rr.trtrAL VALuE\ I'be

ta)

otr rt NNb g

tlE vrolrFrgr E*.rr.ee M€rttob ( sEs, TAE ssur.trrorrt r\s ?esg\-€$.l t l, C3) rrurrrl A sEtr ( cou.irN\rE!!)

12.C2 continued 5\IE At = O. OI S. TO N}r)IvIER\CALLY IN-IEGRNTE -nr€. EQuXtrsNS drf

'tL

g(srxrB- gl.Qsr$) ,ll.;? "

, . , ,1 - - . s

-^.

S= p^t

\^tL\€R€ p -- E St. Guprrr€ N,' ANb Na:

_t N, = cssg, - iL. 'tF Nr. csb$l r.s*€R€ g, NNr\ ^'i. ARG, Tr\E lxUreS oF e ANb

\\re

REclrurtrxE\

ri\ \lE r\ESPEffTrrfELY, .VilCCr'TY, oF t\ TIME TI.ITERNFL r NNb

TtrE v*ure\ r{t n-rE Sr ANb ^L r\RE oF T\ME TNTETIVAL. T'[re Ehlb Te Nlz r \ upbxr€ dr ANb S l rq ! s. ,, 9, = B. 1r Na ( o, r.)se uNENR rr{TERSDT-Ffi\S{ -r\> VAurrs. ot g Nr \r{}ltcu bEilF.r?.F4tNq nfE

NL5.:-- St ' ^.tr tg) Tneeefoq€, Tr{C.

p0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.{0

TrtE vALues

c>F tx

\rFfrERENnAL

MttT\qt

h = ,s.{ (\) (r\ - (4\

e\bJArnst{ir M\S5.

TF\C-

bEsrrrtE

Ncr^l-.

pg=ffi

*rsb fu=*^r-t

DEsrt.tg

Tll€

A$tl> brt?CfilrsN,

R.SbPEcF\\EU\,

,q56hrtnob€

etr rHE VELccrrY.

9.tru,*re

,',

QE m+r.*orr

te/o., Innvr vRur..r€ oF LE!1G1$ OF TI+E StRtrsG q f,n1er.rt uN:trR€T$\eb b\bTgs4 otr r-ttqrrs fup.rr coNb\\eN5: tN\-nArDgSrUg T++E (rrx)r=o, (r.F"n),.$ y,.oi Kr.G, t)uE TltE MobrFrEr> Euuee rvtETrcr> (see r*c, 'ro Pemugwr 5+\fi\)Nl t\. ct) wrTtt A tsTqp'

N=ot B. s,+ fiil$.,(s,--s,)

huw

12.C3 continued continued|

:tr"*=

lft.:;:*rrtsmGRr(ALLY

;

itt =-fl(.t$L -csxfr.)

dJx

ANsr B

L,l-----

.p\r-x-

S= 5-5.,(.tE+-t"Xfu) $t' rs1

Xil' s\ I

29.110 29.610 30.150 30.720 31.330 31,960 32.6?0 33.350 3{.110

GnEu: Bu,ex. oF Mh:6 vn lub 5ffirNG oF crohrsrAtq h i 'AT t=O, r*T:O ANb Tl\E sPR\NCr \5 W AxIb UN\$RGTC}€,b Frsbr (c) r A$b ^t \^lrtEN Tt{e BtqK hSSES '.hIbER

\^tu\ERE ( )\ AN\. ( )z bEr.rstE vAL\rE\ Nr THE BEfrrurtrUG Al.$ Er:l6, RESPE$T\VELY,S INTTERVAL' A ttt'lE rtureRFot-Agrror{ To \ElTER.}qrr.rEnlE Ub€ uN€AR VALueS otr F, q ANb S^y NT K.O:

Tl*E PwC'T o (b) Ytn v,ntel.t I!= I vn Sa TI{KI rr -+ \r.l}l€u rtte BLcsl( PAstes ulrbCft O

(-e,tl+

ffi((-z-r,).

n Ia s , + # , ( & - s , \

bs = (ss\,* f;*,f$^n).- (s^'),1

?nsn

(at

Tr\E vALuEs

15.00 le, lclm length Unotret,ched

otr

Y*,

thespring!1m

XL ! 9.001 m X2 Er - . 0 0 2 m r !r 2.?65 m v a 2.740 m/e the hor Lz ontal Angle v torms rlth

20.oo /e, k/m . Itnstret,chcd length

Fse atNS lnn\ or^a: Q- Htr-co)sruB

g-h({ii"F-q,Xft.)

F, f, ANNb',s

ft,

:! :5.19o

of t h e e p r l n g r l m

Xl r 0.001 n XZ - - .002 m r r 2.3?2m v ' 2.983 m/s Angle v folnre wlth the hor l z Ontal

0. 93c

12,C4 continued

12.C3 continued 25.00 /e, k/nr . ttngtretched length

r,hl(*cssb * $) T*er.r.. \l(:tH 9,- StcJo:p\ ltb $t,nr * ( ssrtrb) cssa'l .pt i\ oR [,"g s,ub . O.15[-u9cs$*St(rg-Ssrru$)s'u$1

E! 1 m

of the sprlng

X , 2! - . 0 0 3 m Xl - 0.000 m r r 2.L2L m v r 3.195 m/e Angle v forms with the horlzontal

ggg_1:

4.62o

THe

Buccx oF

ssRhR.E (E)

k/m19,11/a, Unstretched length

REST$.rG Acl\rNs:T

T$€,

buJT,.|

Cqd:

(t)

Tr{€. O\nEe.

bowN\^f,\Rb

MSss

tBs')

a 1 m

, 1 2G - . 0 0 0 m

Xl - 0.003 m r - 2.128 m v t! 2.94L m/g Angle v formg

rvrpENbrNqr of the spring

with

the

horizontal

-0.000

GrvEN, At|.55 -=: t

, S*e.s= \4 E

; Quor

: lo ,N. .,

\le = o.8 tb R.rxrc'e oc V\Lueb oF b [5g!r C

For< \^tr+\cil T\+e g\qpK boes

Ncrt

sLlbe \r.lr-$$l 15

TbENTTCAL To

b€St\}rxtG

Iil€.

Er\$SrOl\I

otr Cnse Z" rrls s\)r€R CASEJ: Tire BLerr( \5 RESirxlGF\GArNsr suRFr\SE Otr T$E SLCtti $9WARb Mo\oNr tt4PENbtNCr {. Qo't St rgd) I

o', - F'rcsgr+ gsrNg(-\

q,K:V;*:T:.1:H)H"l"f AEr

r,\-ifFra

'L( ( 0 . 6 * " o ) - r h = p'--+ rr:. ;,fti'-lt THenon= F'= p it Now c*.g

( \)

cpHs,dER, THe troLLOlnlrNC FouR cA\eS. \: RE:firr.rg oN TH€ rruN€R I : Tge BLqx :oe.FAc.E. oF T[+E suoT. tbpN\l\Rb vtci\\sN t: \MPEN\rNG (otSigo')

€Asu..tr{.\nf(s,No\+h*-,..-t l-\nve__ F*AsN

,'p.. (-rrHD- qFcs:S)- F'(- cos$.S-tllU) rS

CxsE

f:I

, :

Clse Z: Tr+e BL\)c}( rb RESTrh}g\ rrG\rxl:r 'n+E sLoT 5.rqprrfE oF i bowN\rAFh, \s rMPEhtbrnc. (Oig19d)

Tt+E O\rTeR. $.4cfrlsN

SAVIE

TI+E

NE}-T

T|D rl+E:E

ExPREsSEb Otr

IlEfeR.MrN€\. ANb

T*EAI

bESttrltN(r

oF A5

R3r.s.rs$\$$ trtrES oF

Bstrh\

- I .

EAcil

tr}R

vALuEs

NECELSNRY

tZ)

Ab n{€

Nnuut\puyruG

rlster<

Etl..rrrfrrr>N

T!+E

F .A, N cr-

.€?

ur= S -9d Now eL SuRvi rrr.rTrNG- - L cr.:(g -go"\* S s ru( O -9o' I =FtLilu(S-qJI*, $cos(b-9o")l cr

HNe--

F,r

* srnx)=As*\il( srnrn. cosx) (-'c.osot T*eu -. U.J ffi fF

\^JbltCH

srnr$) -- rnlts,'urj. {F c,DrE). F5. hJ(co:s THe,s fi * bt(\s ib srub)cosbJoR. t Ugsrr-rg - +tt13-s rrus)sx$l ( r ) - D,1s[u3cosD

o*o - N. \^L(srN - :t ..ro.l "

B-

Ar'lxuYs$ FrR.:nNorE-- p = n 12."-lq s,,.r$)tt= L(,f-ssrug) tt

. 9,-

$tS),

EqS. (I\

bOLVE

Rt}$r\srorrr\ ANb

THEN

AN\

(AN '\He

\^l[\qA SoR St$) - O cAN D btsSr115rNG FoR, q (ra.lt/st) SrvrPUF,YrnG,

FrN\_-

S,(S\ = (r-t.bz- rgbl).cosg- ( 4.5sga+\g3.?)srus + l,'ls6t.,sz$ + ?.S$ts,t,lZS t.(s\ = - (b't.b?+gSe)eosS+( 4.Ss 6t- tqr.e) srsr$ - \,.)ssasruzb+ z.Sbesru?S l.tore: FoR. rlrsgE vALu€S str b FoR w*rgt\ n+E Brm< tb

AT RES1 \^t\Tll p65FEST To nrE F n* =FfN

wr{ERG. N \Nb oF T*e EIS*I

PL'{ie r

F ARe 6rvEnl ABoVE FOR CAstS. \t-s'o, $CS) = F o - F

Two eorNTb oN THe TRA$EtrsRy cF A s?N(gSlrAFT' g. Ahlb \ o* % P\h.lb ltt€ RAbfAL btStrSCE, TO rsf Tr[E VeLsFrrTY Nr T\+E A|SFE sR Tr,\€. "eR\Gee ESS, T.r"re t GoR Tt\G. rH\Fg(nNFrt rg TRAVeL Btt\r.tEG.f Tr+E hrt{ts (o.) B nrls C os Filsr. \a,'\\S . i

b!rcst

Oururt{e os ?nqCR.F.sa aSr..)-r Vr\L\rE bF + : Coss,bER Grtes i nstr 4 Foe VALr.rt\ c>F S FesM INqRE}cENTS

S rr> flg"

CsvrPtrre {, tg) I , (S) ='(,J).\oz- \3$e ) c,csb- (4.ssbt+gra)srr 1, + t.fs$esr1.la$+ a.Sbtsrrrzg ComP\tre {, tS* f ) Csvreo-r€ {,tS\ * {,tB+to) To b€rERmrN€ rF A Rocrr LrR\ EFirhteer\t g AAI\ 1g+r" ) Ts {, tE)*$,1$+f) sO, sor-vE \.ts) FoR. b u$NC, Ner,.rtoN'f Mex$sb (:tg 'THE boLvT\oN To ?nsguerr \\.q4) h,xr ruE v\Lue atr So*, Ar\tb \N$EnteR. F.,.o- F Nr $ rs CotrrsrbER C*es FcE \/ALuE\

? cR. Eo Z Ar{b 3 5F S FRst-r

\NCRElr'lE$tTS

O TD \1f

sB = Bug.4E tb) A rxb B eq'' Rlss. la.trli NiA= z+r3'l\ t

Nnrlr-xs)s

n rvr

[lAvE-- +=tF(t*eccF-$)

LEu.(re.sg'il

h= f^***e {.maes = Gerr.ce Sr"*geg.t, Sxnc'ee. -'lBd sg"*oo ='O

rrrn€tr€

o*,

T*e*r-*-W

.(S.,*.h!*i*" ( fr.^"* ) sa=r*o,

u^,)

\t1 oR Q=cFt-q"lffi

C-sr-np.rre{.tS) : otr Tr{E T**r1 THE EccENTR\c\Ty _ +(4.sS+1rqre\sru$ cr\N BE btISRMtN€b. { .t$ = - fifl .Gz+rg$a)css$ - \:\s $'s,nst$* z.sdlsruZb cG rrtE TEtfi Frqsxr PAG€ bgg Covrvl.rrg $. l$+ t') ConnPur€ {.tS) " \.(S+\") f,r \.tst * [. ($ + to) g o, :rcuvE [.t$ F\iR. b usNG s,nerr\oil> NewrsN'b ?*rr.,.rr r$E VALTTE sF \""., lrr,lb F**-F l$ S \: vrlr\tr*qq ?. oR.€ b

f^r

ffi HD\YE-.

S= lh W*GR.E

coNlitANT.

TI\EN-.

t - tf,ox

dA = ).(r)tsds) w$ERe oR AA = lrzaB

ANb [ = iZrto,D wlrERE \o.\

Rat,e of ror,ar,lon =' z rad/s At 0 = 4 0 , F (max) F >E 0 0(1) = {.680 A C e = 1 4 8 0 , F(max)

e(3) ! rre.iz

(b )

R a te

- F =< O

= 14 r adle o f rot,ation At 0 = 1150, F(max) - F >0 (4) = 115.91

7 7 o , F ( m a r c )- F = < 0 AE0= 0(2) = 7?.63

Nore: l.u rltE ABove O\rrP\)T \ T+e i rN bENtcTGt T\\E cN\E FoR \^lr$cll Ms\\sr.l s ci I ra twrPEubrNG.

grvGN

Es. ctz.39!).

^' Sqi vALuE ss uS : uS. O.od $Nfib xH\ csHsrat{\: Nt r*€ \Al$GlTl\ER,.VALrrtS NSG. \(NS\IN OR. T\TC. PEF$GEE hh:fr€E o6' S^r t B^r r Q ( PEnr6€E\ \€T vNLuE (NFcqee) bee= \Bo'

Isput lNnyi

T$€. brtiTAr.KE Ge Ts NNb Tr\E vEL*rT{ Nr rF\E AYsG€e sR, TF\e "ER\GEE T\{'c. vALuE, oF S, FsIl. \AG. rre:s FsrtqT lNhtr OT{ Tt\E TRASCSF\n T Tt{E SECSNb brt{t lxp.ri \^lt\Ei\\ER oN Tr\E \\ \ES\NE\) BY Tr\E' VNLUC Ctr ffi oR. B\ Tt{E vo.rl)G. sF sr\s. LCxse t) $a (cXre, ?.) R,At$FL b\STA\trF. q \pcn

Iun:rr M/lt6^** conrrp.ne T\\€. c,ccEr{Tercrrrrg cs -n{8. ffi:

e = . 6 6-^S-\f

Frp$i

t .v.o..r)BR:**,

.1 'l J

12.C5 continued Cete t:' InrPrrr T*e

ls $..0,

vAuuE otr EL

, sEr lb=-5S

Fon VAL\r€s otr b FRo${ rrllc3€MENTs a.S str t )esrnE ARGA A'. *^

b.- s$

n =A+St,H:*"(r*e*,u{-' (GettAp)L L covrp.x€ Trt'4Ef: t=#* ?err.1t 11\E VAur.rEt

C.*sgz'.

qip, r$6p b.. $.,, r.us t ,

lsrP*;tr lt\E vNL\Je oF q. Sgi tr+E rxrrTrAL vxLue str b: S.b, V.l*,Ue t- 2 q rF \ > fz oR. Ngrgg C. q. rs fi<ct' r'!1 ^'1-\ JgR:^*r ' C'orqP'rrE t-: r=l \Ilsts

'-l,ffi(r*ecssb)]

6euxrn AREA \:

A= A.+ lstO,S

tjeunre.b: $=S.At

Cooqy.,re TrvrE { l ,L= ffi* yALues oF inp. r$.9, $r. C. ,, xrss t RrunrrrrrE ?eq'*nm (o)

brjrPuT

at to and the veloclty dlgtance The radial 3600 km and .8594 km/s respecEl.vely, g2 '3 2900 01 r 1800 s th18min29 Tlmet!r

the

apogee

are,

the

apogee

are,

(b) at to and the velocLty dlstance Ttre radlal {310 ml and 24X7L fL/a respecBlvely, 4035 mi 12 3 e1 !s 1O0o Tlm,etsE Oh33min30s

Chapter 13 Computer Problems 13.C1 A 12-lb collar is attached to a spring anchored at point C and can slide on a frictionless rod forming an angle of 30° with the vertical. The spring is of constant k and is unstretched when the collar is at A. Knowing that the collar is released from rest at A, use computational software to determine the velocity of the collar at point B for values of k from 0.1 to 2.0 lb/in.

20 in. C

20 in. 30° B

Fig. P13.C1

2.4 m A

C θ

Fig. P13.C3

13.C2 Skid marks on a drag race track indicate that the rear (drive) wheels of a 2000-lb car slip with the front wheels just off the ground for the first 60 ft of the 1320-ft track. The car is driven with slipping impending, with 60 percent of its weight on the rear wheels, for the remaining 1260 ft of the race. Knowing that the coefficients of kinetic and static friction are 0.60 and 0.85, respectively, and that the force due to the aerodynamic drag is Fd = 0.0098v2, where the speed v is expressed in ft/s and the force Fd in lb, use computational software to determine the time elapsed and the speed of the car at various points along the track, (a) taking the force Fd into account, (b) ignoring the force Fd. If you write a computer program use increments of distance Δx = 0.1 ft in your calculations, and tabulate your results every 5 ft for the first 60 ft and every 90 ft for the remaining 1260 ft. [Hint: The time Δti required for the car to move through the increment of distance Δxi can be obtained by dividing Δxi by the average velocity 12 (vi + vi+1 ) of the car over Δxi if the acceleration of the car is assumed to remain constant over Δx.] 13.C3 A 5-kg bag is gently pushed off the top of a wall and swings in a vertical plane at the end of a 2.4-m rope which can withstand a maximum tension Fm. For Fm from 40 to 140 N use computational software to determine (a) the difference in elevation h between point A and point B where the rope will break, (b) the distance d from the vertical wall to the point where the bag will strike the floor. 13.C4 Use computational software to determine (a) the time required for the system of Prob. 13.198 to complete 10 successive cycles of the motion described in that problem, starting with x = 1.7 m, (b) the value of x at the end of the tenth cycle.

15_bee77102_Ch13_p004-005.indd 4

3/13/18 8:20 AM

13.C5 A 700-g ball B is hanging from an inextensible cord attached to a support at C. A 350-g ball A strikes B with a velocity v0 at an angle θ0 with the vertical. Assuming no friction and denoting by e the coefficient of restitution, use computational software to determine the magnitudes v′A and v′B of the velocities of the balls immediately after impact and the percentage of energy lost in the collision for v0 = 6 m/s and values of θ0 from 20° to 150°, assuming (a) e = 1, (b) e = 0.75, (c) e = 0. C

θ0 A

Fig. P13.C5

13.C6 In Prob. 13.110, a space vehicle was in a circular orbit at an altitude of 225 mi above the surface of the earth. To return to earth it decreased its speed as it passed through A by firing its engine for a short interval of time in a direction opposite to the direction of its motion. Its resulting velocity as it reached point B at an altitude of 40 mi formed an angle ϕB = 60° with the vertical. An alternative strategy for taking the space vehicle out of its circular orbit would be to turn it around so that its engine pointed away from the earth and then give it an incremental velocity ΔvA toward the center O of the earth. This would likely require a smaller expenditure of energy when firing the engine at A, but might result in too fast a descent at B. Assuming that this strategy is used, use computational software to determine the values of ϕB and vB for an energy expenditure ranging from 5 to 100 percent of that needed in Prob. 13.110. 225 mi A

vB ϕB

R = 3960 mi

Fig. P13.C6

15_bee77102_Ch13_p004-005.indd 5

3/13/18 8:20 AM

glvgNi

€J]TENJ coLLAEv{l= t? lb

CAP,ItlEt6HTr\P= Foe FrRsr 6cft "OoOlb At-c W6I6HTt$ O[l T-H€ Lna urHgEt-S whlcH Ae.g suP9ll&

SP E I N G I S UNSI?ETCH€D

lz@ +t , 6O'lo OFr Foe E€HAINTNG llgf e nT ls oN T H6'el5}e hrp{EErs WtTH stt Del0c thDgnrDl N6.

YELocITYNT B Fog &'= o,t lb/rr,r ro z. o lb/tvr.lN

o.t lb{r!_l t r0rn.----l ^r*rr**

ANrL\slqa a

4S=o,@

4n=j.8f,

AenoovNAHrc Dlhc Fd= O.OogEvz' UtrrnrJ ru ft/s Al.lo R tN lb . EIJ!.Dj ve{,c rTv AND srAps€D Trh6 \{rrt+AND!{lnrooTDPAO. FoB rue FtPsT 6c) fb ANtt Euelv 5 ft . Z\te.fuy .QopCrcele-e_

pEHAtNlt-fCJ3,bp

(to rou\i zd+zo1z(?o)b tili ANAI,-Y3LS USG WoLrL AND FNee('t tN lhke,€ilsNrs <atr ta,:= o.l $t,lrenruge.pLrH ANo Ctl rfl INTELVAL Q:o+AuF €oot4@ = \?oo r-.'c€r t]c' 14,64 lr4=l,zzo{t AL= + l=

[:f -t.

[3o: +.

-'l,

(tJ"=o

tr<)t' L=o)

0t*,

N -\rJ

'li t U.., :T'r+, f,-v"ui (rot r*) Axc-f,tu.i -a(rlr\ l (' t ) . - . , ( ru-Fo)Ar..l-= F;to,00q6u; b,*,=Lnf+ ?,9 Vs=VgtVe, w (cos -i[a(;l"Xtz -(zlb)(?€st\ :o) rvr/+tXtezo$ (-zA) ,t L - , 4 VB= tt)it"pur lr-,* tu/rv,) iZ + 8,q3zJ.)Ub Ve + t'?.

Tn+vn-TetVe o+. {ff*r,)

U; r,3z+ffi?L

t" (.ls\ (r) us = [qz,qs-d-?.qa: le .l (.P rHpuT

lR- lN (r) tN lb/tyr tN o,l tb/ru I AU D ifo p w*€ N fu_.2.o tb/rn

P0trur vAlues oF t.}s (rntftls\ NoTg:c.o(rnzrueveR_ EgAcHgsB FoB = r,Q3Qtbfin . h> sz, q f)/(4?,933; PLqaZAH OcXpor

0.10 0.20 0.30 0.40 0.50 0.60 0.?0 0.90 0.90 1.00 1.10 L.20 J,.30 1,40 1.50 1.60 1,70 1.90 1.g0

F+=Akht{,$S)W

Ay-c=o.t t+, $efi-,2 ++,b

tb \fr= ?ooc>

'T @,Oo14rW=0.36$/ 3: t

h5 Ttt€ t)€uOctTt€S lN Tp€ i fD€NTtpV A},. ANb\o lNreLVau wtTHouT ANb \.ltTH bP.{\G', WtrFt lJc'.Fo Ar..tO F+= O. f OnJ US€ A uOoP TO EoLVE rcE- tl,.+r Ar..lDro souV€ Foe, tL. SuM Alti To FfND )ci AND AT tr$b lt{rFgvALS Su i.r Atc To FtN D t L. PetNf oi,tc.yi tepe nt FoB g€HATNtNGr?rDft wtrtt Feao.36w. Pet xr 'flc r l|Js, ., t. AT qo +t tNTltEYAts 13.C2

13 . Cl

K (r,s/rN)

?AE.-

\nl"rtr.*) illlffi,nru.,zeJe]

vErJocrrY ( FT/s ) g.3g 9.13 8.86 g.5g 9.31 9.01 7 ,7L 7 .39 7 .06 6 .71 6 .34 5.95 5.54 5.09 4.59 4.03 3,3g 2.59 1.37

DISTAI\TCE ( FT)

v (Fr/s) r (s)

NO DRAG 0.719 13.90 5. 1.017 19.56 10. L.246 24.07 15. 1.439 . 8 0 2 7 20. 1.609 31.08 25, L.762 34.05 30. 1..903 36 .78 35. 2.035 3 9 . 3 1 40. 2,158 41.?0 45. 2,275 43.95 50. 2.385 4 6 . 1 0 55. 2,492 48.15 50. 6 0 F T . T O 1320 F T AT 90 F T . 3.984 72,65 150. 5.086 90.74 240. 6.002 105.?8 330. 5.803 118.93 420 , 1.524 130.?? 510. 8.18{ 141.62 600. 9.798 151.?0 590. 9 .373 161.15 780. 9 .9L7 1 7 0 . 0 8 870. 1?8.56 10.{33 960. 185,56 10.926 1050. 11.398 Lgd.t2 1140. 201.88 11.852 1230. L2.290 209.01 1320.

v (Frlsl DRAG 13.89 19,64 24 ,05 2l ,ld 3L.02 33.97 3G.6i 39 .19 {1.55 43 .78 45.90 47,92 INTERVALS 7L.76 89 .00 103.02 115.02 125.60 135.09 1{3. ?1 151.63 158.95 155.?5 1?2 . 10 1?8.05 183.6? 188 . 96

r (s) 0. ?20 1.018 L.247 1.4{o x.610 1.764 1.90s 2.037 2.161 2,278 2.390 2.496 { .000 5 .11.9 6 '057 6 .982 7.630 8.320 8.966 9.5?5 10.155 10.?09 LL,242 11.?56 L2.253 L2.736

GJggNi lNlflALL(

(.F1=O

1/,=1,1h4 P(^srt<- fltDAcf Br;rukEN Ah[b <.

RoP€ = ?,4 0t1tON6 lNrTtAL Y€LOctTf e6e0

Flr-r Dl Foe vALuas oF ]4$ilhutl T€N3|ON Fnn F'goll +a 'To l4o N lt\t SN

9:O

flr{ Dr

(A)Ttu€Tb &n lo @rPUF.I€<t€Ls:s (b l t/AcU6 otr k AF*rgLTl+glO

I NcE€r4€NTS..,TH6 (e) DtSrRuc€ h

(b), ursrANc6 d Feo*l

t.t{€ W ALL To rHF (roe THE Lr3 cvcLg) PotNT Wtl€Q€ Tt+6 ANRr-vsts BAG HrTs nte e, sou4nalu Tu Pt4P.Jt.tqtFLooB.

trE qQ*l(

BA6 HOV€ S AI.ONC A CIBcU(AE AP-C Ab ONTIL rn€ Q.op€ B Q,eAE s Q,lorus,e)

GF-r?esrftpAcr)

b+gvu,gttm,.(r)

@ Ac<+br.ATloN

coNsrANTr

ANO (t)

@-lt

?-'

StNo= h ilN

Otcr

(D^),=fri'. (L,-J,:ffi.:

I s r N e =h €=

FRoh

rHus Avs?A66 Ysto<rrr 15 (St.= 39h'\t

(nrru-nrupn r) lr @

(z) tt'-J-l,4qBrJr* 1

rQ:^) itTar: (AYAr^I i rhPuuse-Horug-tgfoHFoe- B

3 wtg

@!!.ESO U (PeorecrtuE TE A1:e-cT6BY) tJ6 - 4J s tN @'+ d- ( A-q cos e)+_tr*t tr^rto + g& /z Ur.r = {ufcose t 6-h):

(4)

- + o*r,^Ltl.ouour&l:+ W

(s)

brrrH Q=2,4wn\ Wt=5hq. $=q.gl wr/s.lN E'QuAIONCt\ . ANDFop.tr6arN sN lucewrdNTs FPon40ToHo N,,SouvE hg h. T.4=o V+ Fo?- eAcH hrsouvE FoR u (eq t), AND s(eq zl.. TatVl=T+ tV+ , souVE FoR- 4)s AUU q* (EQ6) AND Wrtrl rlrr AND h. SoLv€ Fogtp (aQ.s) ANb v.{rTH0, hrlu soLvF FoFd tN (ee 4\. PerNr h nuod Fop €ac$ Furn.

t3l q?i + o: f46

PPoePnrr e4x BUTFORCE (NE}ITONS)

H (METERS}

d (METERS)

40 45 50 55 50 65 70 75 80 85 90 95 100 105 110 115 120 L25 130 135 140

0.652 0.734 0.815 0.897 0.979 1.050 L.L42 L.223 1.305 1.386 1.468 1.549 1.631 1.713 L.794 1.8?6 1. 957 2.039 2.t20 2.202 2.283

0.503 0.585 0.668 0.752 0.839 o.927 1.01? 1.109 1 .'203 1.300 1.401 1.505 1.615 1.731 1.854 1.989 2.L17 2.305 2.501 2.75t 3.101

o+ $tr.rtr v ^ . . . =r z l Yri,=

\re

ftf,?c

Tlhe rcoH \= I (*pthsthtj

Va=trnn r+ut)gdl- unu{du=Q-a[d"=zgdi : o +=t drTL+Vr-ili+Va m1ci+o o dt=ftPt tl \ -zdu $ae)L- Gffi Tg= o

= o,1%enr( b \ (t -s;L=z-(+frf,)ruW (tl-?)i=

(€y+\ sr= a.n48a rlE,

tr)

\QorrrHOeb)

13.C4 continued TorAu'rtH€

13.C5 continued COhtseeJltror.lop MoHgHTOM rH rH-E ? DtPgcrrorl Fo q_A ANQ_9 ToG€THePh a 6U e s r N e 6 : N n e $ s t N € o t M g U d MA- o, 3co 0-g YtAB= o.1oo &.g 9^=4.f,= A vnls b SIN6o: od StNe.rt t AL Crl

To CoHPt'e-Tg TFe Lrt{ c?cLe

(z\r (61t(r\+ cs) FQe, t" : ($,-.)ut (tz-a)t r k -r)r-t (tr-+).

t i=(,4q I + o,?qgB +6,?488+f ,1t66){Ei {t=

(e)

4 . l Tz t r , . i

ouTrtN€ oF PPoq,_eAl,l Ser Tt,= t,f 0VI' (t = r\

n.}r-tlo -6bn/s (oa- o)e=Atistneo-n4! _, (a g =/JB EtNeo-aJ1 ('z) [uurtpcv (z) BV stNso AND ADb ro (t) ro GgTdi

c3) trEoH raQtlntro N (4) c*r-cu(AT6 n(-tu F o e L = | T o L = ' l O . F b B e A c H v R U O €o F 'EeuRrroN (e) ro D€r€e.ntNG t, (4) T- 03€ troB THg Lth cVcL€. suh t's To'obTAtN t*a ToTAc TrHg r*todatt t$g touqycL€. T A t = o \ A t t r DB A t r A A r R V e t o c r r f @, ,oBTArN p FoR Tr+€ TgNTH (ol FoK L:rO O F co ValS t{ rrs BAuu B 'V.lHtcH .lS AT O VlSUrtrY AYC L€ Auo tS NoT coNsreArseD bV Tt+€eotD.TnusrF PetNT TorhLTlrlc AND 1&rcB rHr; rcitscvcr-6. oNuv I{AGNTTUDSS Aa.g(tNsrD setrp ^JdAuooa HAVE

(e)

VAL06SFoP 'tto'< q>qd wHKHAosTtt€As Foao;qoo @

o'u:l-*p; -L(MrtrlLtMsuf)

METER' ;"Hlll';:i;l :::::: 0136?

U;' b' A F = !- to.lso)t0^-th1!to,t oa) N

Trupur so rNro €euArroNS Cb\ nuU (ql FEoH

€.u€.st'

lrne=?OOg , hnR= 3f,O 3 416=o t.Is=6vn/S oo= Zoo ro tsf tru lOo tNcE-EtleNTS

?-Oo To QOo tN tNC?€Hgt-tTi of, 5o Foe. €=1., €=o.?t Anrp e=o To oB-TArNui ANbud. subsTrrw€

aUnnuUui tN G) Tb oBfAlst 46. Perr.rt e.,Oo.(}i,OJ,t PgorrgRttourg,lr

FIND: uA AND r)6 AFT€e rMpAcT ANDENELGT LDST FoE.,

(o) €= I

Lb) Q= o,lS

ANALYSIS: 1g reEHS oF

€ ANb e.

13.C5 e

(*d)^=t4

<oNs€P-veD tNTrt6

n^^(tl^)t = nn()i \t Ttl$s Oi\=.> (Iar.i.c AND Ald' ts AuoNG Tt+s

20. 30. 40. 50.

1.00 1.00 1.00 1.00 1,00 1.00 1.00

_ -1 . 0 0 . 0.75 0.?5 0.75 0.75 0.75 0.75 0.75

.0,75 0.00 0.00 0.00

TI{ETA (DEG)

6 0. 10, 80. 90. 2 0, 30. 40. 50. 60. 70. 80.

90,. 20. 30. 40.

VErr A

(M\S}

-5.337 -4.667 -3.945 -3 .279

-2,Ti7 -2.325 -2.081 - 2 . 0 _ 0 0. -3.920 -3.333 -2.7A2 -2.118 -1.636 -1.284 -1.0?1

-1.000 0.332

I'EL B (M\S)

E LOgr

0.0 0.0 0.0

1.939 2,667 3.196 3 . 554

0.0

3.779 3.911 3.979 4.000 , 1.596 2.333 2,?97 3.109

0.0 0.0 0.0 , 0.0 , d1,3 3 8 .9 3 6. 3 3 3. 8

3.307 3.422 3 .482

31.8 30.4 29.5

3,500 0.959

29,2 9{.5

88.9 0.667 1.333 82.9 1.598 L.027 7?,3 7,,?77 50 . 1 .361 0.00 72,7 1.890 1.635 60. 0. 00 69,4 1.956 1.838 70. 0.00 6 7. 3 1.990 1.959 80. 0.00 6 6.7 2 0 0 0 . . 0 0 0 2 90. 0.00 VALT'ES FOR AI{GLES OF 90 TO 150 DEGREES ARE THE SAME AS THOSE FOR 90 DEGREES

Ut AXIS

1l.CO continued

Gtvg\ti lNrrtALet?cullB oBBIT

OF ZZS ullr, AgoVe fl

sueFAc€ oF Tuge'lE, -

zGMtt-#,J:

f,s,Qsxron f+'/s.

€QuATtoN. FoB $rg

rn$r\€ rcub srN$e , q = Sr'tlt[ra(+)c,". lro

AUp, TowAeD T$€ C€NTER.OF THG(aET*

FtnD:

()o AUUso AT.lN OF 40 }fNt FCP €NESCY EXPENOITOSE OF S TO IOO"/O D F TTIAT

R . 0960ml

us€D lN PDoB,13.toq fN sYe_l-NcB ANALVSIS

I NpuT co NsTANTs tNTo € QUATION ( 4 \ AND SOLV€ trOP. 4Jg Fo R. YALUES ol= Y- oF S % 'f o loo"/o AT tNT€ eyALt oF s't/o. FoE. €Acr-r VAuu€ oF' t)s ANO UsrNG T-HEetveN eou5fhNT VALUCSoF (Urhec1l.6 ANb fg rUe€ €rlrJ51 roN ts) To 6oLus ticE $g, PPI'NI' kr Og AND Sg.

peoozl,rrqrrpqT13,e5

R,39G0mt

fA= +rBs rI "q(e+22g:

vg=

=e:

GMfyt

[i;,?l3ff-U.*ArTh,e.= H q#

: Lh^qr ry I u,tLCu^;L+tq"$l (r) zcnth-ht ui= (^hi,.^r{A"^*

Eueqev,FtPeN u rrup9

rN P.Eo B. La,to Q

KG)

vB (r'rls)

5. 10.. 15. 20. 25. 30, 35. 40. 45. 50. 55. 50. 65. 70. 75. 80. 85. 90. 95. 100.

2696 0 . 27329 , 2 7 79 1 . 29245. 29692. 29L32. 29555. 2 g g g 3. 30414. 30830. 31,2{0 . 3L644, 32044. 3243 g , 32929. 33214. '33595, 33971, 34344. 347t2.

pri

(DEcREss)

79.5 75.1 7 1. g 6 9. 2 67.0 55.0 6 3. 3 6L.7 60,3 5 9 .g 5 7, 7 56.6 55.,5 5 4. 5 5 3. 6 52.7 5 1 .g 51.0 50.3 {9.5

6[ep.GY €xpENDrT.u.aelbl,r*r 9. PqgBrsH -lo 6NEP6Y K-g- ! tnn(A\)A\i wHeB€ k, rs rrbh€ 166 a us€D ( N PZoB 13.loq. soLv,tNG Foa(ntX)-Ar.tD ReN.qctNGE BY Eeo FT(oN (ar

Gu^i=K- Ltu^t;.-(0^\"1 (t) r

Io ,suBsrrrur€ (sr rNrocr)\

(uJl+ oB= lr"^f;s.* *" Kuh)i*- zen[t+lh, coss*urs:(tr;Li.: qi=

tCIdi.rta= G3'l.o] Itoo f+'/s' , A,,l Fbrnt3,torl , L-Ulit5(.f t.32xro5)'= l?8,t4xrou f{/5'

Chapter 14 Computer Problems A man and a woman, of weights 180 lb and 120 lb, respectively, stand at opposite ends of a stationary boat of weight 300 lb, ready to dive with velocities vm and vw, respectively, relative to the boat. Use computational software to determine the velocity of the boat after both swimmers have dived if (a) the woman dives first, (b) the man dives first. Solve that problem assuming that the velocities of the woman and the man relative to the boat are, respectively, (i) 14 ft/s and 18 ft/s, (ii) 18 ft/s and 14 ft/s. 14.C2 A system of particles consists of n particles Ai of mass mi and coordinates xi, yi, and zi, having velocities of components (vx)i, (vy)i, and (vz)i. Derive expressions for the components of the angular momentum of the system about the origin O of the coordinates. Use computational software to solve Probs. 14.11 and 14.13. 14.C3 A shell moving with a velocity of known components vx, vy, and vz explodes into three fragments of weights W1, W2, and W3 at point A0 at a distance d from a vertical wall. Use computational software to determine the speed of each fragment immediately after the explosion, knowing the coordinates xi and yi of the points Ai (i = 1, 2, 3) where the fragments hit the wall. Use this program to solve (a) Prob. 14.24, (b) Prob. 14.25. 14.C1

Fig. P14.C1

z d

Fig. P14.C3

16_bee77102_Ch14_p006-007.indd 6

3/13/18 8:25 AM

14.C4 As a 6000-kg training plane lands on an aircraft carrier at a speed of 180 km/h, its tail hooks into the end of an 80-m long chain which lies in a pile below deck. Knowing that the chain has a mass per unit length of 50 kg/m and assuming no other retarding force, use computational software to determine the distance traveled by the plane while the chain is being pulled out and the corresponding values of the time and of the velocity and deceleration of the plane.

Fig. P14.C4

14.C5 A 16-Mg jet airplane maintains a constant speed of 774 km/h while climbing at an angle α = 18°. The airplane scoops in air at a rate of 300 kg/s and discharges it with a velocity of 665 m/s relative to the airplane. Knowing that the pilot changes the angle of climb α while maintaining the same engine setting, use computational software to calculate and plot values of α from 0 to 20° (a) the initial acceleration of the plane, (b) the maximum speed that will be attained. Assume that the drag due to air friction is proportional to the square of the speed. 14.C6 A rocket has a weight of 2400 lb, including 2000 lb of fuel, which is consumed at the rate of 25 lb/s and ejected with a relative velocity of 12,000 ft/s. Knowing that the rocket is fired vertically from the ground, assuming a constant value for the acceleration of gravity, and using 4-s time intervals, use computational software to determine and plot from the time of ignition to the time when the last particle of fuel is being consumed (a) the acceleration a of the rocket in ft/s2, (b) its velocity v in ft/s, (c) its elevation h above the ground in miles. (Hint: Use for v the expression derived in Sample Prob. 14.8, and integrate this expression analytically to obtain h.)

Fig. P14.C5

16_bee77102_Ch14_p006-007.indd 7

3/13/18 8:25 AM

14.C1

GIVENI

yt4( fl f nfnWoUert 0F 1ur,et GrtT V!* STAyW Pen-oy7b?tvE P tTtl ' RenTtvz. To 0oAToF iletcHT Wg HAN tF VF Gft-l V6 Rpn91e ?ir u(tE: FRortt oTtrEqvEN! f'F BoAt' vtfir1 RELft?rVE VEr.OcrTv {r,

fl1v,

GRSl'ftFFvED t\|ED vEtoc t T y 0F gom ftFrr- R 00r* ssJrrvlfvt t\An, -DrveJ ( WorvtAtJ E5 b) FtRsr, Div FlRsr IFG) u5E w-= llo tb, Wa,= lg.o tb, Wr, Statb, ftNe (Pgob,ftf) t ,to = uar= tuft/i'

,t L) 4L = 14i\lo , 4ttr. tB {4s , = GA 4fw= (B frls, 6o,, 14ft/e hu4LYStg (al ww.wt 6rv(5 FiR:,, i. q = V t t . o F B o A r A t r T e ew $ r . t A NJ t v E J otVQp nlb = y'EL,OF 6oat Affeg DoTHsvtlmr4ens..HArurr \ / 4 j , . .

QvA'il cpx/ge

; wdd fi r'#-:irf.iilr,

Eo-

0 = -$(.r;ul)+

(ws+rV,r)"]

(wr+rvqhrf fu

g=#*

qlar t< - t,,Wd ui -D = \D W;FWFfrrb

%rj.

w^(Wlr,

(d5+u/* )rrr'= u/tq *il*(rtr+%) SvUsr trlTnu6 fr)R 1.1i;ft?oq ( r) :

+ ib=ffir-W

(L)@ lrvrr

" B l r t su8o til (e)

r tb,-#ffin*W ootuvE or PRo&RfiM

(r), 1 1 L 1 P t 'Y t r* , W * , W b , 4 / e r, U n n , k t / D E6ls,(?)Ao'D

?RPcR4M_or',112]/7

?R08. 14,1, J:1":l[i""?ril:l.rl':l .Boo ( b) M an divee flr et of boat = '0.229 Veloclty

(L)

r o

( a) Wom an diveg fir st Velocit,y of boat = -3.950 ( bl M an dives f lr et, Velocit,y of boat, = -1.400

a \

t\ t U) r

( al woman divee fir st of boat' !! -1.650 velociuy (b) Man dlvee flrst of boat E 0.943 lfglocitY

14.C2 | Gt/rnt

I c Les AL o F MaJ.t 4tt; 5Y STEMof qv PftqT ,

?INF'IVTI 1, - i , . utffn' VE'roCrf rr3 OFCbm /

Coneor/€hJT5OF Adreum{ 04oUEUTut4 0F SYSr-u t196r,'T T0 SouVe P{ut5, I h.g 0K(GIN O. USe PnocrRnr',\ f t N 9 l r f, 1 3 .

kN fit-vsrs

ao=t,!f*f uf tt({)J

(()

r;(q)r) -'t;'(drj

(a) (3)

oF7 (oGRAM d u T l r re- . 1 USFD Prurr'R %oBLEl4 Nul498R AND JVSfE,ut 0 F u^r,R tF J/ UN:T|, E^, i-ER ftn L = l - r i s L = " l t i

mi\il; ,1 , Vt,z,i(n)iP^)1,(rril;,(%)t U/t)

rlR L=I To'" -- n i l f U , 5 , c u : T o l ( | A Pui 'r t 1 5 ,e r - / T E R

y{;(6)', ,7t, Ui.,zL (+t) t. $,);, ( ,\)t, 6fa)i(frls) Y =

SHELL

plDrlttlG rrlr(ff VEL\CfTY

ttp 1.nr!\PntrirtT9

ffr , UA ,t, EXpr or;gi td i tf (eE Ftp a ltr.-i.ttii :t f Wt'w'nTs d F(opr WFrr-Lo Vl,, VVriW AT Pr)trtrAo Ar OISr.pnJCF A ' r ' i , l p r ' P D t v r 5A ' ( ; - - 1 , 2 , 3 ) r H r # r r l ' r ( i ,a. i . i = j r s 0 r 4 o x ' D , t " t p 7 E 3i t ; n t t J ; J i , FlttD:

gpetD

o F E p C H Y a . * G l , l E N r. 4 F r F i F r P l o S r u M T o s o ( v r ( a ) r n D D ,r , r ,? 5 , ( b ) P c o r , t L { , ?6

WL/ZZ,Z ftFrD cornPrv7E il.L . C D , -P 4 u T € 7 H E S t ) t ' t e( t ) , ( 7 ) , \ N ' DQ l ' pUMBER 0B f lett't ?Rtrv 7R TRrrJr Vf*u,Jes zlTxmlEg TIR, f+*, HJ,'H* , lf S l or{ tT|, RESuurS A(E EvpBr5g trD ,N ttg,rl/t, 4 t l R Y r . fl . lt TS , R[ S L ) L r s , ] RE l F l ) . 5 , C ( t < 7e E Y T R E t E eo r N f t , f b , s .

"Rodri.AM ArrlAt Y5 ts -l D E T ER r ' tttr t f D t R t C t 0 N C c ! l t ' l f 9 0 F TATH AoAi 7i=i,r2,)

P J'r)6P..ri Fl I u T P"-)T'

x'co4P:W,(24 tl, r \U74fz +dr(t1,uj = ({,tv,l2 r w)il^ €)

Problem 14.09 Hx = -3L.2 kg*m^2/s Hy = -64 .8 kgrm^2/.s HL = 48.0 kg*m^2/e

Problem 14.13 Hx = 0.000 ft*lb*g tly. = -O .720 f trLb*s 1.440 f E*lb*a Hz .,

(,.=ffi FlnStcoi,\eur€

(r)

(?")i= 4,llt , (At);= t;/t,t, (hr);=- Afl; (2) Tf+€N /\ : 0F L INEAR t\^0HENTU c0r-./5rR'/A1ror.t

d*I-r"li*t,t) =# 1,+ trIrr|lrVs e(t,rwe*doX ", f uz p: r -.?lt (h) I com w'tfg),9 +$ (t), *wrQhr\ =(tt,+\uz t1[

W,(2,\,tr,t W.(4).O+r,lr(a.)r ?-CuttPt 4=(\+v/'ls*u/t (5) T f f t s e 3 e o S . A R e s $ L v E D 5 r r . a u L T A N E 0 u sf\L. yR ' , t \ r 4 r t l , M

t< VrrI.u -S af n)L,\ ,', ) kv D d r:".tTeR gNrER,rfltu?5 of WLI ri, t, At f |R, L = 2,J onlgogM€s FRonEos,(t) er.t-o c0,v1Purq 1z) DrRrc-Tr 6p4gurE coEFf. rr'/ Ecls.(3),( h), ( s) pv D 5 orv[ rl? ,tt,1rr,TJ. 9Y QtLt ?uTtN 6

ftN-Ddt = Dr/D, ilz=\/O,

[r=Dt/D

OuTpvT ?nOeRFf-{ g) -

Pr obl em L4,25 VA = L678 EE/s VB !t 1390 fC/s VC = L23O tC /e

(-b ) P r " b L e m L 4 . 2 6 VA = 2097 ft./s VB = 1853 f,C/e vC = ?38 tc/e

{

'14.c4

14.C5

A F - N J h 5 6 o r l 0 - k B l ' L R ^ l r L A v r ,o5N c A ' R ( E Rf T / t l 0 K n f| ) t r 3 7 * t L t * u o r c st M T l e N D 0 F 8 0 - n r L o N c

GlVtrt/j A t , " - n g P t n t v EA A r u r f t r N sA c o r r J s r A rst rP e E Do F 7 l + l * l n

'Ilt'= 50k4/tn i : H A f N 0 f t l [ ' . - , r ' ? 1 : t lv r ? l f t l { i H iY/N..i -Bi,.l',*/ 1.\c K. I - r A t N c R E M E ' . t 1 'Tt * e J $ r A N c r @ ) v5tN6 p1 a r/E ftNU f H f coKRE< ?orJgtil6 T RnvE Le-D 8v rH F vdr/EE cF Tftt T/^{f , THt vri-'>:,'T/r Ar/D Tt-tz ilCC(f ^E R d 1 I o nJ of' THF Pf.,+^r: . (nf.:.,i,t.-r

.tll) S'lu,T:.

i . .I ' t q i " . i : i

AT AN htfGtE q = lb', W H I L EL u t s t g r F t G PLftNE SCooPstr'{ ArR Rr RArg o? 3OO fS/t AMr D fcJct+A Rf: g(- lr Ar ft R elh TtvF SPEED oF 665 n/5, q wfiiLg P t L o l T t + € NC f t n r J | € S A i l c L E 0 f C c t f v . 8 (i, gltN AUE Tt+E fl1r'.I( SE Mffrr{TAf EN6ttlE 5 F/ND ruP V4luE5 oFs( FRol{ Oro ?rJ uirNc l" rNceEpwrgt oF fuANt, (a) tNttrflL PTcCELERrtrlont (b) fr{A/tf'tuFf5P'=ED TktA T',ttlu p.E ATTA|,VE}, ( Assrtue DRA(' To B E PRo PoRTrurN n L To 'ty'.)

)'lr Fuq-=,)

fiNALY5IJ FRcPIE(i.(l+,St): N& Rnrt .1fieUon

t{t [q_

4,1do = (m + t't?) rt

tETTtNG tt = dz/dtt ' 4t4lIodt= (ilt+m'z)'lt

W="

L-=.1;.Wdz=W (,nt+nh)"-an' (t) l,nn on' lfo

(,) ftR 4I: t= 5olr/rru6

c L l ' \ t B r J GA T t r w s t e q, A MD s PFED /4', /2\ l(a) PcnuF

\J/

{ rs'17c-7 r o tz D t F f r R E N ' 1 1 r r n r u 6u(Jl )r r rF

+/a

7E = ntdfrl sindr=7t1Q Doea. =(Po-Du- ryn1t,rc)/n

+'ftt'4# o = ,'{r4r +(qn

-o

6fa/ft = t,'- ANe dt/rl| = &'! MnftNk ' 'tnt tI' "ffn7 fl.;= f f i , 0 -- 4lt'rli +@;.,n't)d

(L) l4Ax, e?EED w ( l L e C U H ' Bt t l G A t * w 6 r E 0 ( ' ,

OOILINE OF PI<OGRdM

F r u z e r 4 H = G c o o l + 4 , ' t r r t 'kFgofn t , D o =t & k n , f n = J 0 n f s f } R x = 0 T o L : 6 0 r t t n l v } u S r r u G5 - r n t N ( x f t t e v T i cn Lculrir€ L, u,Av.D & FRolt f0.t. (Z).,(l),U) n {D TflSrrlDlE

)

PR\G Rrlr't Du apl rr

hqet, u/r[L r*h/ B e z E R t )A v . D P U n v e f N E Q u l U t B R t t } }.' l

E4 f1=o : P-D - tn| Ei'r{= 0 (s) 3ul si !=

Suv

Dist,ance (m)

Time (s)

VelociEy (km/h )

Accelerat,lon (m / e ^ 2 l

0.000 s.000 10.000 15.000 20.000 25.000 30.000 35.000 40.000 45.000 50.000 55.000 50.000 65..000 70.000 ?5.000 80.000

0.000 0.102 0.208 0.319 0.433 0.552 0.675 0.802 0.933 1.069 1.208 1.352 1.500 1.652 1.808 1.969 2.133

180.000 1 7 2. 8 0 0 166.154 160.000 154.285 1 4 8. 9 6 6 1 4 4. 0 0 0 L39.355 135.000 130.909 L 2 ' 1. 0 5 9 L23.429 120.000 LL6.757 1 13 . 6 8 4 110.769 108.000

-20.833 -18.432 -16.386 -t4.632 -13.120 -11.809 -10.667 -9,667 -8.789 -8.014 -7.327 -6 .7t7 -5.173 -5.686 -5.249 -4.855 -4,500

5rr ', ,,t Pt-X, " " "S?Ez i ; ; Zp' to = , ( - s, t \ f f i )

(7)

eMEMTt Fon o< FKort o To L6, WlfH ;o 1r,lCR (a) use Eo,trrl To ctlLcuLATE a ( b 1 u 5 p E Q 5 . c 6 ) f t N D( i l T 0 c A L c u l n r e 0 * ^ ,

(coNlNuED)

14.C5 continued

14.C6 continuedI

fRoge Frqo' rrf $T

OVTIINT

alpha accelerat,lon degreee m l e^z 0.000 3 . 031 1.000 2.860 2.000 2.649 3.000 2.518 4.000 ' 2.347 5.000 2 .176 6.000 2.006 ?.000 . 1.835 8.000 1.665 9.000 L,497 10 . 0 0 0 1.328 1 1. O 0 0 1.160 12.000 o .992 1 3. 0 0 0 0.925 14.000 0.658 15.000 0 ,492 1 6. 0 0 0 0 .327 17.000 0.163 t8 .000 0.000 -0.162 19.000 -0.324 20.000

max v kn/h gzl .796 913"933 906.020 898.060 990.053 982.002 873.90? 965.770 857.594 849.379 841.126 832.939 824.519 816.166 907.795 7gg.375 790. 940 78e.{81 774.000 ?65.499 756.e81

tte Ftllt+rtrM€ =t+ ioreiie h =2oott/tt= Eos j.rrrlrF RvnrJ r O ft?.oH To AOs 4ft1 FOR.E COAPVE T (L F(uH Ee ,(4) C6r)AccetrRft'rro V (b) vaLoctrY F R oM EQ. (, ) "tl G l e L E V f r r l b N h f r q m E o . C 3 ) , D t v t g r N b R E s" . f 8v f20o ,lD oor$rry A tN f"l lr.Es. ?n oeR.-Ftq otlTpu r

RockeT

T o o D I b O F F u E u , l 5 F / R E p v E R T , c r r t ' . y F R o t , 6t R 0 , J N . Q , 17 co{|vf,4e| ?ue u ft-r Rftz e oF L>- tb/g AND eJEcl5 1 T e h r H R e r "f i t r { E t / f r - 0 c t r y o F l 2 . , O O 0f E / t , Ft x.€ qfrot1 Trfvt. o? fGA,trtc>tt Tct Ttr'E u4{?d t F's/ P e r . : r rr r 5 O F T u E t - t ! , C o N S v H E D rA N 9 A T 4 ' 5 r r r t E Itlt7R,,/ht5'. ( f t ) A ( C E ' - = R A ? r o d . . L 3 F f - r ' ) 'f< ? - r t , Y f t / r |

( . 6 )t r s v E L o c r r r L r , N ( t / 5 t s ) t 7 J € ( E v h r r o N h A s o u E 6 P . D u N Dr n Jf 4 l L E S ,

WE tg< Al

0.000 4.000 8.000 12.000 15.000 20.000 24 .000 28.000 32.000 36.000 40.000 {4.000 48 . 000 52 .000 56.000 60.000 64.000 68.000 ?2.O00 76.000 90.000

av fLls^2

10^3 fE/a

h ml

92.800 98.235 104.164 110.657 11?.800 125.695 134 .'16? L44.27L 155.300 167.800 182,085 198.569 2L7 ;8AO 240 .521 267.800 301.133 342.800 396.371 467.800 567.800 717.800

0.000 0.382 o .787 1.215 1.673 2,t59 2 .679 3 .236 3.835 4 .48L 5.180 5. 9{0 6,?72 ?.688 9.702 9.838 11.123 L2.s96 14 , 31? 16.376 18. 925

0.000 0.143 0.58{ 1.3{1 2,434 3.883 5 .7L4 7 .952, 10.628 t3 .775 17.{31 2L.639 26,449 31.921 38,L22 45.137 53.056 52.O37 72 .2L3 83.81{ 9r'.1{8

SArt'Ptf FtRa8. l'1,0 TaAT

(r)

tf=+c.hffi-g.E tvnFAl

PROCRRM

! = Lr/i', e i t2','olo'{ bls=

14.C6 , 4 r v e r u 0; F w E f G H T2 h O r i l b r t N c r u - D r v 6

A-/f tYrlJ '-,fpOh

EA/reR # = 3 t, I ft/s', il o = Ltlfi/6, flg= 2t0o/$r-

{f -- ^'/Et0cffy DF RDc'(87 AMg Ft)EL T r l o= f v r - t t t * L w E t G H T O t R u c t ( E T C o w s u ^ lE . a ?veLlS q =R.ftTga1 v17rt<t* .w=RELrltt./€vYLoctTrftrNHf('{r'tLt|-z'ECTED ^tvJ lurEFf ,i'ith(y

4 . F= d y / A C

L(trtilG

'h ' ' 't,L .Lf |' ' - f'Ao llo =,' ^dt' - Z l' h : Jroty= $ r\/F se r To cftLc.utArE Tfte tNrE6riA!-,

rAoq

oT0hj

(e) !4*

T rt ER,EFURr I

= ?

fvlD

i(-+a=)' =+Ve^*-4iE(-$ne-a+r) .Tf+us , EQ.Q)YtELD5 h = a ' n &\-{(hUu q

h =t,(

'Rrw{f rtNt.,I3;

*,) -+r6u"

L)- ixt" t -I r)

- gt

f \ m o b r F P = C F 6 J I I F I ' i ' / 6 : ' t ' t 1 tnf

SESPFCT TD f ,

va= s : --i . I d &

Q-= f1-,-*

1 3 = . -of r nno_f,b

(r)

(+)

cconlT;ifr/ ED)

Chapter 15 Computer Problems 15.C1 The disk shown has a constant angular velocity of 500 rpm counterclockwise. Knowing that rod BD is 250 mm long, use computational software to determine and plot for values of θ from 0 to 360° and using 30° increments, the velocity of collar D and the angular velocity of rod BD. Determine the two values of θ for which the speed of collar D is zero. θ B A

50 mm

D 150 mm

Fig. P15.C1

15.C2 Two rotating rods are connected by a slider block P as shown. Knowing that rod BP rotates with a constant angular velocity of 6 rad/s counterclockwise, use computational software to determine and plot for values of θ from 0 to 180° the angular velocity and angular acceleration of rod AE. Determine the value of θ for which the angular acceleration αAE of rod AE is maximum and the corresponding value of αAE.

15 in. P

30 in.

Fig. P15.C2

17_bee77102_Ch15_p008-009.indd 8

3/13/18 8:29 AM

15.C3 In the engine system shown, l = 160 mm and b = 60 mm. Knowing that crank AB rotates with a constant angular velocity of 1000 rpm clockwise, use computational software to determine and plot for values of θ from 0 to 180° and using 10° increments, (a) the angular velocity and angular acceleration of rod BD, (b) the velocity and acceleration of the p iston P.

P D

Rod AB moves over a small wheel at C while end A moves to the right with a constant velocity of 180 mm/s. Use computational software to determine and plot for values of θ from 20° to 90° and using 5° increments, the velocity of point B and the angular acceleration of the rod. Determine the value of θ for which the angular acceleration α of the rod is maximum and the corresponding value of α.

15.C4

B b

400 mm

C 140 mm

Fig. P15.C3

Fig. P15.C4

Rod BC of length 24 in. is connected by ball-and-socket joints to the rotating arm AB and to collar C that slides on the fixed rod DE. Arm AB of length 4 in. rotates in the XY plane with a constant angular velocity of 10 rad/s. Use computational software to determine and plot for values of θ from 0 to 360° the velocity of collar C. Determine the two values of θ for which the velocity of collar C is zero.

15.C5

ω1

θ A A

20 in.

4 in. 16 in. E C

4 in.

x 9 in. B

Fig. P15.C5

15.C6 Rod AB of length 25 in. is connected by ball-and-socket joints to collars A and B, which slide along the two rods shown. Collar B moves toward support E at a constant speed of 20 in./s. Denoting by d the distance from point C to collar B, use computational software to determine and plot the velocity of collar A for values of d from 0 to 15 in.

20 in.

12 in.

D E

Fig. P15.C6

17_bee77102_Ch15_p008-009.indd 9

3/13/18 8:29 AM

15.c1

Gtvtx: @c^k=?= 5& rf-, u)=o 5omm

BD= ?,t6 +qq,

Auo;b)

nNo Qeo

Q = O za

3b'

3at lilcnEHE+7T. (D 7'vto vALuEs or

/=or-uY"Ln$=? -. ,,.

6 -(48)

FoE

ugtv(

sina

(ns)"aso - ftte)ase 3= @o)cosf

l:r4ot-rA BOf , -tl 'L=2"c -(al)si"a]

p= srn(2, siap = c Fr?a,n

t'l)

-??ane

cose lf = -#,zso #

au n,o

BuT: t=

ott

#=@&D

(s)

case' u) tr t -- : &s n(\tr) eD C6(l

-(ao)stop'# +Oa)srnl tf (,): dD= Fnop, #= ob= - (*a)sirp -sof (ng)in9 4)

(+)

jf = a)=rficolpnn= €ao(g) ralfr

AD = 3.Sa ara

nB = ,fO ar@j OU7/J../E /.

2,'.usF-t?A(a) A

/=6Q

OE-EeA4-€ftQt

DErcZ(vl.uF DFH*auu€

7f = /64 4'r4/

€4cn

u4zu6

U.9/r/6 L=AZ)

OF 4

77/f'/ F tuBo, lvnatty 70 D8222-rtN-E d" ("1) ey us/,y6 EAk)

theta deg

beta deg.

yD nn

vD n/s

onega BD rad/e

' 300 .. 0 0 0 000 60.000 9 0. 0 0 0 1 2 0 .O 0 0 150,000 180.000 210.000 2{0.000 2 7 0. 0 0 0 300.000 330.000 360.000

36.970 30.000 2 5. 2 6 4 2 3 , 5 7g 2 5, 2 6 4 30.000 36.970 41.{27 50.643 53.130 50.6d3 lil,427 36.970

150.000 173.205 201.097 229 . I2g 251.087 259.809 250.000 22r , d37 183.539 150.000 133.539 135.234 150.000

1.963 2.6t8 2.885 2,6L8 1.649 0.000 -1.963 -3.531 -3.863 -2.6t8 -0.671 0.913 1.963

13.090 10.472 5.?90 0.000 -5.790 -7.O .472 -13.090 -t2.699 -8.257 ' -0.000 8,25? 12.699 13.090

Theta [for vD = 0] t{9.900 29.990 150.000 30.000 150.100 30.020

259.807 259.809 259.807

vD 0, 006 0.000 -0. 006

Theta [for vD = 0] 311.{00 48.592 3 1 1 ,{ 1 0 {9.590 311.{20 {g.5gg

132.289 l32.2gg 132.28t

vD -0. 001 0. 000 0.001

15.C4 continued

Qrvqr: o @rr= ( ,'Jb), ! ep= FtpO: (t)

ue.

9 -C;

Ano ,b"

dae

Qururus or fra&aom t l- usF Eeg,()anob) To F/^to F a,Yo AP. dP' ?o HND tP 2. aJE EQ{,P)ara(+) Dap BY ugrN6 ga,,(€) 3. OEryfrlut'tp . 1, usg rQ/ t) 7v Fttrq d; uttr fq.(z) Ta l=nrg

.47

lgo /Nc r?sptervz{, (z'l (o$)*iria,o, co e Q.t sPa\rlJlh

F6e.

AHg

6, vrl tu€

oF o,

6, DFnn.st2uvr

ffi(BP)sno (BP)cos&

(,)

'fuZF

a-p)

?oo.

flCt ElflUl7rdw:

{r/,

3rrytr= $dr(t-to) Nf S,= + cas(c-()zp 4p=GDa][e

9st

(v)

Wz nJ,[g xpl+l,rr,rh,F]

3^**/(A-r.*

(r)

e F fi=!p,+%tr

+=(er)uee;F !D,= (ar) und f

nqTAi usp=dradk 8f = l{)h, I A8- 3o)a

a(ap By

theta

bcta

dog.

deg.

o 16

o.o0 4.'80

30 4t co .76 00 to5 f20 f36 | 50. 106

I .00 la,c{ l0.ll

23.1i 2C.67 2 e. o 2 30.oo 28.68 23.79 14.05

f8O

0.OO

t-----------o-

--a

thctr

I---t--------------

o.000 o.712

2.000 t ,886 I .937 | ,850 l.tlf | .609 | .200 0.730 o.ooo -1.144 -2.8CO -{.02O

L AOS 2,192

3.818 6.728 8. 040 13.273 20.785 32.988 a6.t8e 40.298

-0.0OO

O,OOO -----------a

for ntxfmun alpha

thetr deg.

el pha radli'2

t 67.0800 I 57, 0900 | 6t. rooo l67.tloo

48.88093 48.6AC94 48.ba8g4 48.58C03

'('r) Ut

I?/6/!7Tantuu:t

= 3P' + 9P/?

+9c

A-/"t Gp)t= 4p At(notp-e) a,F + (nP)das= 4p cos(?oo+p-e)

- or.7 dea=#tapcos(?ao*F-u) (c*

(a)

Tl^tupo)

15.C9 continued

Gllt?,, ?,q'=/e6 ryt? il rtg=o

f?aO BDz

1 = /60 ena, /o ,,ro b s

l=navt

g€,v eJ

'd ao

Forz vAt uEs op 6 O -te, /&o a f

lOo lNar,?valf

FtlP:

(o) ?eo ara gsp

(t) % Apo *o ttlono.y oFO n p 4aac:e

/W:

Plnpe N/orlaN

&onL=!d"o (ooDn=l rio

{ t'l V€2actry: -i

Msn"ffiu Et

?s=3+%e ?s = *o+koldt

(r)

+(4oru)r,

df).Vq lo"4*l=looll-[-(o",

^ fit"J

/ o(*c'sp

Qg coc O

.lrIcToQ

Drc6r7nkl

*ffitL(

a-uc3sO

CotutpltrE^t7S

--(dr;r,brq

tra;*P

,ttD

P/Attf

y'ry,r;o^/

= 7711,*r,tulvr 0

Ra7*7tatv

ffgc:,ur -p

L-D

= 6 *{rb

+[ carrr,:*!+rzs

O"ZI*A VE+aarv! 7o 687flt*

feodAtM usq

z For? Encn

EQ.s (,2r?,

a*o4t'

* .(dsotirp

t,(tro,"h6 hl

Aolnscote +-(4?*f

Xrls;,F

a+f

VALUE es

lu t€eepcF

, fi,vo bao $ fZEcmtu rb F/xa nB, USF EA(€) ryiclF.tfr2ft7/a+t: -ry/ r-/zt>r''t EaS (d tNa (') *Bo e /= 14 Irtlcu{ *, F , oleo rtrto Qg,

[?op ABr

(a)

-( cos f

aB cN Q = oo'/%

ft7tau

- e7as9

lur*oSra(3

(wo|r*€

[q\el=[ %lJ.lrrro4f I

n c cetF2

\e]

aB=Jr; rle

(c*trtNurD)

(s)

ftlf,:

F. %,

tunB=/ oaun{hn= taao (ii) -(

= O. il

-zzt

rdls

OcoB '*+t

theta deg.

beta dsg.

omega red/s

al Pha radle^ 2

vD m/e

aD mle^2

o to 20 30 40 50 60 ?o 80 90 l0O t t0 120 I 30 t 40 150 180 t 70 l8O

0.00 3.73 ?.37 10.8t 13.95 18.69 18 . 9 5 2 0 .6 3 2f .07 22,02 21.67 20. 63 18.95 I C .E O l g. gE 10.81 7.3? 3.73 0.OO

3 9. 2 7 38.76 37,21 31,62 31.00 20,35 2 0 .7 0 14 . 3 S 7,31 O.0O -7.34 - |,f .35 -20.?6 -26. 35 -31 .00 -31,02 -gl .27 -38.76 -39. 27

0 -618 -1239 -1864 -2485 -3081 - 3 6 1I -1062 -4337 -4t138 -4337 -4062 -3618 -3OBt -2r[g5 -1864 -1299 -618 -0

0.000 1.495 2.913 t0.180 5.234 6.024 E .5 2 0 6 . 7t 3 C .C z l 0. 283 5.754 5 .O95 4.363 3. 0O2 2 .843 2.lO3 I .385 0.687 0.000

905 88f 813 f02 657 388 206 27 -l 34 -20e -302 -123 -462 -458 -{51 -197 -121 -41 6 -41t'

140mm

QlVttlr

E * lf,ou.v/s + *e=o (t)gfU rvurTg/ fu: Fan V/lL.u Gt OF 69 przqut ?ot 7q ./0o Ar

€o /,ucnpt'ets+rs

.@)e.auod *C'A,,na*.

AB=!

To =o AB=

fl9g' WF PtncF

arzl dttt AS €Hou^t AT OtSfnN6 D Ta l,[L'T ap C

, - tt)=A=S tro'a

6.-!s,oe d =(h--(s',t)? s/n'e) %)f q0-*

su26

%=1=+#e

gC =??rn,

&B= | /r, as,= /o ra J/S d t'Frpat = o (') !, tuP VnLuEs oF e trlTrw o 70 3,60"^n7 3o" lncazrtnTSi (2) rwo vALuEf op a FoQ u|lctl

llrtnr 3

(fre)r= (+,r,) cc'se

(ac)o=Un1 sinO

. EC=/?itl

G4y= ltb, +(ae), {gJs =/6utt- (n ey

tr [tc)e=vr8-h4-tuL1

P*=e

FB= D-hf-kos€

k"VO": # Hu=/ ;* a i

Bcz

l4)g=.X6,!casae =]casa ! ";ie nrd cas9 srn6 ( G) 4 t . 7=

-Q/B=bc)r: +(ec)ri'+ kc), Q vELoctTT

rl ,i, (ec), *krgr-t t &=(%)r[ -t' I Gdr ,Lo. (e9,

fu *--1 , 2

6t = *;

(+)

s( = r(*)stu"ecas€t

(?sh'6)

/2rz tiaca vhtuc op Oi oaTllNC aF PQ66aikt DSE-rc5 0) ,qro (+) ft c4Lcu1,476 4-t ,4no o1t (Cdg ustr EAs (Z) tvola) Tr: cr1tctil.47? (frL+xo 'l - - --tr, 1fB $8 Fl'vo Tl/g'u r^"1' -z , -t (&)y

Ua)sl-.z,tv,iUeh

0F Ci

q2= le *n{c/B=le+ ?e6Y(c/a

tl€ftloa4zry

4A=4='o, :r=ft;-#6' 4 n = o > .ffe ao; (-

(otdr= nlusioo1 (%)f=l{'cas6

du^=&)t*a%)i;f=toi'frj,

(B'c)r% -(aQa,* i (ocr) tu, - ( sc)? a, -(BC\\ tuy Gc), DgzmruuNn!7 (wr, tu!,w*) rc ,,uo. EG @) ut*zar , ,.r;= - 0,ily /ferl* E6 ( l) ytpcol,

Dy=

e) (b) (z)

lug=o'

1td"/tBde

7)tB/usE eq(c)t tjc= rc1)gut-(Bc)*ug thete deg.

omsge radle

al pha radls^ 2

fm/s

vy mm/s

vel run/c

gamma deg.

20 25 30 35 40 45 50 55 60 05 ?0 15 80 85 90

0.160 0. 230 0,921 O ./ t 2 3 0.53t 0. 043 0.791 0.803 0.9e4 1.05G f. i36 r.aoo | .217 1.278 | .288

a.121 0 ,22A 0.358 0. 5l I 0,073 0,927 0.955 1.O42 t.o74 1.O40 0.939 0.171 0. 5rl8 0.285 0.000

159.42 14f.18 116.71 8 2 .9 5 43.4l -1.83 -51.19 - 1 0 2 .E 0 -t54.04 -202.85 - 2 1 6. 7 1 -283.{9 - 3 1| . 2 0 -328.44 -33f.29

58.53 93.25 tll,35 13 9 . 8 0 | 62.78 lgl.g3 l93,99 19t.94 l92.8C t 70.53 | 5 5. 3 2 | 24.I I 86,0t 14.48 O ,O O

189.15 | 03.90 f . 6 0 .5 0 101.52 | 0e.tl7 f8f . gf 2 0 0 .c 3 222.99 24C,82 27O,22 2 9 1. 6 1 309.40 323.03 33l.af 3 3 4 .2 9

19.52 30.53 a3.90 59.1O 75.07 -99.12 -76,22 -C2 . 68 -5 t . 39 -41 .36 -32.l9 -23.00 -1t.65 -7.?l -0. OO

thcta ldeg, l 59.900 60. ooo 6 0 . 10 0

m l x l m u m a l p h a lrad/ e'2l | . 0?3892 | .073095 1.073Eg2

?u7ba1

Paa{4frpt,

Foa ufltl/fiL ,r/n1uE '6 =ot. ,S7nt27 lAt .'FasE;NeE 4!A r47 7-DP oF Sot|rtav EacH EauiTo^l P/2al2t7"t €4aut, EVtttutrE nilO s7t?E Desf1lvt 7?O BY 4'n" 6p g4c# E6 Urt7t61. /w*t Bm Lrrqla,ua op &, car"tPovpvrs oe Bc, llsr7 'L kltgs /uceawe v/tLuv oF a RY gob nHo € = 86o0" r?EPEil7 EaAL ultVeY ttN7rL

Pra*r '

(C*uvnaED)

15.C0continued J VFtocl ft

15.C5 continued theta deg 0.o00 30 , oo0 80.ooo 90.000 12 0 . 0 0 0 15 0 . 0 0 0 l80.oo0 210.000 2 4 0 ,0 0 0 270.000 300,000 3 3 0. O O 0 360 . OOO

Componentsof r o d B C xy z ln. ln. ln.

Velocl ty of C in.fg

-tE.'000 -14.000 -12.536 -l 2.0O0 -12.536 -14,0O0 -10.0OO -19.000 -19.404 -20.000 -19.464 -l g.000 -l E.0OO

40.ooo gt .221 23.430 ?.945 -8. 908 -24.339 -36.777 -39.97? -32.995 -12.649 14.293 33.951 4 0 .o 0 0

8.O00 |.481 C.OOO 4.00O 2,000 0.538 0.000 0.536 2.000 4. OOO 6.000 7.le1 8.O0O

| 9.000 t8.008 19.567 20.390 20.368 lg.4g0 | 7 .g g g | 5.885 13.898 12.649 l2.gg4 l4.olo | 6.000

Qf^= Puk E'

Componenteof r o d B C xy z

104.o34 | 04.035 | 04.036 10 4 . 0 3 7

-f2. l1g - 1 2 . 11 g -l 2.l l 9 - 1 2 . 1 19

3.030 3.030 3.030 3.030

tf,.=%Lo

theta 294 ' 2 8 1..002300 284.040

?,;_Fi) 4=eo,r/rY (r&)r= t6 ,ats (%)g = -'/'?ih/s

l/ELoCtT

ft)

(r)

oltr A :

%=%*%b=L+Qx

Veloc{ ty of C

2 0. 4 9 2 2 0. 1 9 2 20,192 2 0, 1 9 2

'6L- 3 - : s9

-ir

Determlnatlon of values of theta tor vC = 0 theta

otr B: t

i4 tDg

0 . 0 0t 0.001 0. ooo -0.00O

(ftt)e

lna)g

Componentsof r o d B C xy z

Velocl ty of C

(c)

- 1g .g g l -19.881 -19.881

-0.ot5 -0.0O0 0.003

4.989 4.970 4.970

l2 .192 12,492 t2 ,192

fuer:

A8=?-fth'

4fB --2o ln/s mtuftt?a Ponr E nt€ Diuarp 8r'J D/ f TrtNcF BC oF Ca tt 477 Fr?o/vo PanZ C p*tt

DefnzPtr^th7E oF (**ragrw) CdeSE u)Z= o o Eq. G) Y/ELos' a).4= *'+ EA, b)3 -(%)r= o @g)guz

rc *Eeo'

h)"= -(va)7/tnB) s

(6)

fQ,(p: ary t&)g= o +(ae)*a)7

'?fA FalZ v1laEr

(r)

t$=(Qy+@aluz buT/J*F Fop

pr?otnnM i

Vntutr J = o, Pt?oa,aau, E qua TtoNs 0) rtfeoudn (7) StQuFNcS /.FF7^.ydNo .nat++tgt*z oF Fnct+ EvntuazF .FAunn/dY a/Y/) PQ*Yr Vdtury. OF //v/z/aL

d) eok?tuilEltrr

l'tno Ga, vdLuF otr d

/ucaeffie f?GPtrrtT Paoc&,s

tlyr/L

1U-" BY

/'tm. cD = /€/a

Veloct tY

Componenteof AB dxYzvA ln. 1n.

/7/'a,

f?oo4g, hB=T

(frs)r=-Br,:- o,8J &di--7oin,

O.0OO O. O 0 O -0.800 l.oo0 -l.0oO 2.O00 -2.4OQ 3.000 4.OOO -3.200 4.000 5.O00 -4.0OO 8. OOO -5.0OO ?.0O0 -0.40O 8.O0O -?. 2OO I .O0O -8.00O I 0 .000 11.O0O -8.90O 9. 60O t 2 . O00 13.00O -l0.fOO 14 . O 0 O - 1 | . 2 O O 1 5 .O O O - 1 2 . 0 0 O

4N?

In.

ln.

tn/e

I 5.000 14,979 1tl.914 14.807 14.055 14.457 I 4. 2l I 13.915 13 . 5 0 6 | 3 . 15 9 12 . 0 8 9 12.141 | | . 520 l0.8Og 9.978 9.O00

-20. 00 -20.OO -20.00 -20.00 -20.OO -20.00 -20 ' OO -2O.00 -20.00 -20.00 -20. OO -20.00 -20. O0 -2O.OO - 2 O .O 0 -20.00

-12.000 -12.855 -13.?18 -14.593 -15.494 -16.427 -ll .4O4 -18n439 -t 9.548 - 2 O .7 5 4 -22 .088 -23.691 - - 2 6. 9 2 7 -27.991 -29.960 -33.333

Chapter 16 Computer Problems B

16.C1 The 5-lb rod AB is released from rest in the position shown. (a) Assuming that the friction force between end A and the surface is large enough to prevent sliding, using software calculate the normal reaction and the friction force at A immediately after release for values of β from 0 to 85°. (b) Knowing that the coefficient of static friction between the rod and the floor is actually equal to 0.50, determine the range of values of β for which the rod will slip immediately after being released from rest. 16.C2 End A of the 5-kg rod AB is moved to the left at a constant speed vA = 1.5 m/s. Using computational software calculate and plot the normal reactions at ends A and B of the rod for values of θ from 0 to 50°. Determine the value of θ at which end B of the rod loses contact with the wall.

16.C3 A 30-lb cylinder of diameter b = 8 in. and height h = 6 in. is placed on a 10-lb platform CD that is held in the position shown by three cables. It is desired to determine the minimum value of µs between the cylinder and the platform for which the cylinder does not slip on the platform, immediately after cable AB is cut. Using computational software calculate and plot the minimum allowable value of µs for values of θ from 0 to 30°. Knowing that the actual value of µs is 0.60, determine the value of θ at which slipping impends.

Fig. P16.C1 B

L = 450 mm

θ A

E b

θ h

A vA

Fig. P16.C3

Fig. P16.C2

B L

Fig. P16.C5

16.C4 For the engine system of Prob. 15.C3 of Chap. 15, the masses of piston P and the connecting rod BD are 2.5 kg and 3 kg, respectively. Knowing that during a test of the system no force is applied to the face of the piston, use computational software to calculate and plot the horizontal and vertical components of the dynamic reactions exerted on the connecting rod at B and D for values of θ from 0 to 180°. 16.C5 A uniform slender bar AB of mass m is suspended from springs AC and BD as shown. Using computational software calculate and plot the accelerations of ends A and B, immediately after spring AC has broken, for values of θ from 0 to 90°.

18_bee77102_Ch16_p010-010.indd 10

3/13/18 8:30 AM

C|rpy

G tvcty I nn = g$s afe: Ie*ft

W= dlt

RAO 4g RF/EFSF, FQoter /2 EsT (o) .ao A Nct g'172prrtrC

47 A) ru

{e ir'rq A AF fa:z,

pEteASE

l+ A['0 B

F= O ToSSo /NeE6ktlqv6.

l,Srxc

t)stvl

Forz

U e/,*,

//+ttutEotnTFLY

Tl,qz

rt*A

d= *q

mtd

/tvs7t4N7 CrE, eT

Acc ete Ra 7ro'N

on/.Zo+ + ry,(#n)

* l@urrL ll "1 V"l] =er+Vou,)o-,"1

I *rl?o(

ft =1,?*'F

It4=z(F)e6i

/,)

f" q !oru,,p =.-,t (2 E"*p)'iF ,F.= 7 *S

S/h F €asp

^rg)ror(I

1N= r?d3 (t - i *rH

6u r up?

+l Zrr=z(Fu)rp: _ Fl .-ea

P ((*'

= O ToSS'

4f ,f"/,+caf*q&\rzyt

pEzeahrtvr:

&, (s). ALSo aHo il (€ro/: EQ(il) l=/* - D?fenpu,uE f?traotnFD VALnF ot= 4= (a) u,fg ,r^4Atter? lNCn':ptENTs Ta FrnD ?wa oF coe46p4nQ/n/6 7a 4s = O, S'A. F beta

?sl {p?

O.OO0 5.000 f0.0OO | 5.000 20.000

0.000 0.326 0.041 0.93S 1.2o9

1.25O 1.278 f.363 | . 501 t.0gg

O.OO0 0.255 0.47O O, 6 2 4 0.?14

no sI tp no ellp no sllp sl i p sllp

---::-seek

i,, s : 3i .. 9i 5? 6 9 62.6t0

cnd of

range__ 'i.Bra

t.Boe

i:il;

| .gog

; : ; i ;g o.ot

o.Eoo

;:ffi

o. soo

sltp

ni'ii,o no el I p

g = att e

ll =*6- a) f=Aa,LZ

6 F P/2!?t Rrl&,t :

Forz

(z)

L auz

A = * k r * a s ) = l a st a=ffi

F-- +zd, .su13

{f rtr)=r(F)s6: N-*J, errd-cosp =*, *o Lp =,tt*/Ff

i',

|IYALL,

wt77

CINTT*CT

,uj

* *g I co-sy3. *

4T w*ret

dFa 2oSB

u).h

+)Ifu?) =z(tn")ro: *e(+ bsf ) = Io * .xad(*)

V4tt€ E^tO

i"*2,/2

a F --t

,fo lrYcrzE r.liENTT,

firG /EEE,

rtF7El:

r?oo PaTeTEs +Ba,uf Tltr D e p, nFm PrzaAsc

Wp tvazr

Noa+tnt ftFctPPs 47 EaR e=Oro,f1.

l = a r z4 3 6 , { \ PiaUe oF V4/atrs af B l^/r /tc* l?oO lt\r ltc.St/p

FOQ

/twru/ao/r?7Ftf

?a=o , EtNo'

l*lM)Eqr|7tity

€o

+) il,lh=t( bt)erft B0cose)-,*t3($suo)= -Io( -otdfsrQ

(+) valuFt

at =€&g, I'-o,?{n,, OUTzrug OF PRodunv,: Wl 'fta a) 4no d., Frtca vntaF oF e Avntaarg /,rtgr uSF @ 4NO A 70 Evatu,qz€ A two B, UStttt A/=

5/,44l1fl? Fat?

WHtcil

/Nce,F*?6tt6

1-t*o

lltrlb€

B =O.

theta deg.

omgga rad/s

alpha rad/sz

e nl sA

A N

B N

o.oo0 5.000 1 0 .o 0 0 15.000 20.000 25.000 30.000 35.000 40.000 45.O00 50.o00

3.333 3.346 3.385 3.451 3.547 3.679 3.949 4.OOg 4.351 4.714 5. lg6

0.ooo 0.990 2,O20 3.191 4.590 0.309 9, 553 11.595 15.889 22.222 32.O49

2.500 2 .629 2,617 2 , 7 74 3.013 3.359 3.949 4.549 5.501 7 , O 7| 9.413

36.550 36.406 3 5 .9 6 3 3 5 .l g 0 33.99S 32.259 29.805 26.309 2 1. 2 4 3 1 3 .6 9 5 l.gg4

0.ooo | . 4rl8 2. ?90 4 .o g 4 5,?-71 6.216 A.762 6,65? 6,O24 0.955 -g.l0B

---4 5 . 74 7 45.749 45.?40

Find theta 4 ,717 1.777 4,777

forB= 2 3, 4 2 0 2 3. 4 2 2 2 3, 4 2 3

7,357 7.357 7. 358

12.265 12.264 12.2O2

0.Oo2 0.ool - O .O 0 l

6:aro, h=6.*t. 3o-16 CYlbreH. /o -U PU7ftcq AF?Erz AB lS CuT, /=!yr 4. tDoc tnupt CYtttt0trl o& P67 SU? Faa $=O 7o3oo 1)Srnl W:

5o wcttFrfFvTs,

Tr4w fu? a,s

:OolOtW

lftltrep

l*tP&trr2g,

+t/ e4s6, cilC€t(

tLlrPtnf

b frpzapn clL/NOEe

16.C4

€lffe*t aF Ep4lvr 6twrl PttoE /g,C 3, lderyr), ?l*s=o ?ne= y')-,/6oan*t 3c6O.n* arp , ?^f$?t +v@-S&g Ftlr7: Cot-tPOPPA Zs eQ/7c Tant alunprc e Ano D n7 NpTs cls/ryi A= O ro /6 /^lceE\"rafl-8,

eFErd

Tr ?r.

OF otv eO FAR /oo

*rc3

rrra

We=

*P*

(*r*,)3saa

=TF€?

+\7t=

::;:;?" J/t*oar?t Irtl=

aflc

{

aF rc'ruEs 87 fr.+7$6tzlrr ExpnTED oN 7o CYUNOET? ,4c7-s Af 2/S7A*C€ 2< t=rZorvt tor2tYJr?.

Rgsat74&7

Jlrr=iltr)rf:

F14acasa

i tf

N - -.J.= 'rttt a. €rn6 N=de(S-"she)

Z(fr)6:

+)zkla-t(r7e)rr' )W( * -r) = h.o r"'a(!)+h"aine (*.'r) \(* -r)=A-; e)ia"d I $o"a)',s,na(*:,1 (+ -r/ t - su"e) = lsrnoass; (! -r)c*'e = t *, e cas,

+!a*s),p

c,E ?Qo6r(ftI4 /ATE /4, D/t4

/E .<tr

b =8u

frn qNo

(t)

l=ote

.Mttyt,,au+4

.FAca

t//|tuF

villctF

e/:4.<

(?")r=

,lt'vA

-a.B cia 6 AT

fttto dar.

a^,o 4 P")ga ea as6

rTrE TLaroolF

(z) G)

Pg ttn",

t*7ao

* t rr, = f( or)r*

h =6 h,,

|uo

wE Fr2.s7 FlNo

r zr= /'@)7 +l As=*(ou), * oo1 -

t 6 >€ef

cyt,oem nPs tF r <o i la,o > *=#

Og=

Fer.

(q)

.e poD

Noza: €tArna: theta

mu req.

?sllp?

?ttp?

BF4cfraNS,

0.000 5.000 10.000 15.000 20.000 25.000 30.000 35.000

4.000 3.739 3.471 3.196 2.909 2.601 2.269 1.899

0.000 0. 087 0.176 0.268 0.364 0.466 a.577 0.700

no sllp no sllp no ellp no sllp no ellp no gllp no sllp sllps

no tlp no tlp no tlp no ttp no tlp no tlp no ttp no ttp

F l n d t h e t a f o r m u = 0 " 6 0 ------3 0 .9 6 0 30.980

*t6r*f

(,{)

@^{_]=n

x=+(l-hta'a)

$-r=+#i

P/Zt,vT y'lo 'lt

sh7 -46 case ^( cas 14

4s K/*pzzcs

/3Vnt

Ito'

+l z4={ea!

(,0=f =

G(lTllrtE

q( =

BD=(

2,200 z.Lgg

TllE

gE€t<

7€ OYNtt,ilt< alvzt 7 VF tnAtlrT colv,v€crt.v/ 4^.2 PtsTav

Jpt

{?oo

(co*zzv

ueo)

---------.

0. 5999 0. 6004

16.Q4 continued D

po^

CottPrcnryl

M*:-Unraa4

i=i-uol'

Bht? er r.t4<S ,a ,Su PPc-e7FD tg y

(to)

S?@/v6 t

ot:

CavsT/t'nr

moo&,

ltAtUpJ

ftr?

TBoQ't

Er Eg

B €f.'/a,^r6

-t

a lB

e Fnopt A rb

/pt7l4c

5fra.Ftrl

satv€ Fart

Prwaz Bf I

Ktogrlq !

Dr'4+:tr-#r.*?ax-?a#

_' gr=orto,p-#p+ffdr'?iotanp IZrr=

0')

+ Dy= ?r7Bo 4v

t {. T F5=Z(rstt{: D g= aho6! ' %* OurulNl.

EY= @cod

*- D,

('z)

Uo= ono oA,

Ir s)

on rA ,

Unle:

/ooo fl)ar (H ) =

ap1 ?.{*g,

nao-

P/?o(nnt+, bez€

ANo Pftiarr

lhrlh?s

oF d,Y 7S:

B*, 8, , 4, fi /tr) /ry2otu4 o 7o /eao ,4 r f

= *r^;

o"3 cosa rT z*-)o,z'x ?C-osO

0 10 20 30 40 50 60 70 g0 , , ; i " ,9 o .*r,'100 ', l t o , Lzo ,. 130 li 140 ,;' 150 i,i 160 ;.i 170

ii 180

= Ior g(=

(a)

(ar\=ir= f 3tan? (+)

@ilv

(ae)y= do=Jfi taol F)

Br N

By N

0.00 109.19 L g z. 7 4 194.59 L24.Lg -33.5? -265.49 - 5 3 9 .? 6 -911.62 -1034.81 -11?{.96 -tzl7-.65 _1169.20 _ 1 0 { 9 .3 1 _g??.31 _6?5.57 -456.95 :230.09

4605.82 4 4 9 7. 3 7 4L77.66 3663.87 2985.61 2 t 8 S, 5 2 1318.43 447,34 -364.52 -1064.65 -t621. 34 -2028 . tt -2300.43 ;2466.79 -25s8.80 -2604.t7 -2623,56 - 2 6 3 0 .3 8 -2631.99

_ o .o o

()

Qa)r

Positlve dlrectlons of force DOTII{ and TO THE RIGHT

theta deg

I 4r= iS ton fi s-

+[28'','%'-HT:=-Z, , a; * s[ (") f<r*Ert artcd 2

/2att

ffis;G

3*g

J /N StcuaztcE,

tm6-

t o .ra i

#"o'Yt

/=o,o/otr EAs, (r) T.yaaut+ Oz),

/NC/26/e/

( r+

W'+ng !IIFr= E/Tieq,.nTt-

=o. lrnt,

/:v4l

Bftcnrcr

t)Y tL =T(u6)r4: (r*t")+

OF Prlodranu:

FrWnz

nc -ra

TG,r)e,f: B,

JtrsT NF7E"

2&s6

smp)

BnFa kf

+rg

* aaoor( {*tp)-*no(*

nF:lzrz

rlc

7f at g'/o^/r

+f :A,30 a ?Tcos4>'4713

+)Tt'tB=E(MsLr: ?rlcasf - DslsLp = 4e(

/runyVt1afFly

conponents

DX N

0.00 -279.5? -520.30 -688.07 -759.59 - 7 2 2. 4 9 -589.25 -387.68 -160.35 4 ?, 8 5 202,89 290.2L 3ttl.'47 292,25 2 4 2. 9 0 182.09 119.39 59.71 0.00

(t)

+t Pds=--al+$a=-$+b(9= 3t

are:

Dy N -226L.78 -2203.39 -2031.39 -1755.71 - 1 3 9 3 .{ 7 - 9 6 9. { 5 -515.59 -69.61 33{.94 665.41 906.22 1 0 5 6 .5 9 1 1 2 9 .3 4 ll{s .24 tL26 ,72 1093.40 1060.09 1036.51 1029.09

(t)

EvO A t

!6a)y l( v = ta n-t kdy'

O UZ-/NL: OF'

Pt?ianJqM I tN SFouswa4 P266EtMt

L-45. iltt,eoun

(t)

ryn(ut7G rtNo Puryz Ae,,F, on, y Vrttur oF € Fno*t O ro eiTus/Ati

fte

/oo /Ncatr/vEw6. theta

ta.S

beral

sannal

Chapter 17 Computer Problems 17.C1 Rod AB has a mass of 3 kg and is attached at A to a 5-kg cart C. Knowing that the system is released from rest when θ = 30° and neglecting friction, use computational software to determine the velocity of the cart and the velocity of end B of the rod for values of θ from +30° to −90°. Determine the value of θ for which the velocity of the cart to the left is maximum and the corresponding value of the velocity.

The uniform slender rod AB of length L = 800 mm and mass 5 kg rests on a small wheel at D and is attached to a collar of negligible mass that can slide freely on the vertical rod EF. Knowing that a = 200 mm and that the rod is released from rest when θ = 0, use computational software to calculate and plot the angular velocity of the rod and the velocity of end A for values of θ from 0 to 50°. Determine the maximum angular velocity of the rod and the corresponding value of θ.

y B 1.2 m θ

17.C2

Fig. P17.C1

θ D

Fig. P17.C2

17.C3 A uniform 10-in.-radius sphere rolls over a series of parallel horizontal bars equally spaced at a distance d. As it rotates without slipping about a given bar, the sphere strikes the next bar and starts rotating about that bar without slipping, until it strikes the next bar, and so on. Assuming perfectly plastic impact and knowing that the sphere has an angular velocity ω0 of 1.5 rad/s as its mass center G is directly above bar A, use computational software to calculate values of the spacing d from 1 to 6 in. (a) the angular velocity ω1 of the sphere as G passes directly above bar B, (b) the number of bars over which the sphere will roll after leaving bar A. ω0

ω1

v0 G

v1 G

G B

d (1)

B d

d (2)

B d

d (3)

Fig. P17.C3

19_bee77102_Ch17_p011-012.indd 11

3/13/18 8:31 AM

Collar C has a mass of 2.5 kg and can slide without friction on rod AB. A spring of constant 750 N/m and an unstretched length r0 = 500 mm is attached as shown to the collar and to the hub B. The total mass moment of inertia of the rod, hub, and spring is known to be 0.3 kg·m2 about B. Initially the collar is held at a distance of 500 mm from the axis of rotation by a small pin protruding from the rod. The pin is suddenly removed as the assembly is rotating in a horizontal plane with an angular velocity ω0 of 10 rad/s. Denoting by r the distance of the collar from the axis of rotation, use computational software to calculate and plot the angular velocity of the assembly and the velocity of the collar relative to the rod for values of r from 500 to 700 mm. Determine the maximum value of r in the ensuing motion. 17.C4

ω0 r0 B C

Fig. P17.C4

17.C5 Each of the two identical slender bars shown has a length L = 30 in. Knowing that the system is released from rest when the bars are horizontal, use computational software to calculate and plot the angular velocity of rod AB and the velocity of point D for values of θ from 0 to 90°. A

θ L

Fig. P17.C5

19_bee77102_Ch17_p011-012.indd 12

3/13/18 8:31 AM

17.Cl continued

2n 4

!!re!? t + SW O+o:'ae\f

tJZ rpl? .-z F 5. fv fi nu

A zne

anae&t,

, O - azc$e- + 47at@o* t'at en 6" (-"+ ane) ,l; = - 4rr77 ft u sL e,

Qf=-l(-ffi)+ (olt€zztavof

(s)

06 gypot, B

!t'-'t" 9

tt) a

l/s,a, g $ sio ao

l/, = *rg $ in e,

T=o

Ty a,

1=[n

oLz

f r = 1 * " 8 : + I I a f +* * o u a '

fr?- u,l * te'= ("", f,u she)\t* tr^=

Lo,,sia e+.

n{+

ryczse)L

*Azarn

. {f" [fcoer"Qf+ ft"erfi)1s)ae

ir' . L**. @7r'

(+)

i n*(cocrr@)Jw"

n= l{-,p*rn@)+f

(s)

T= {lko*o@)l'" T l'v

=d*tro' tq)

=,/fc"crr@ldf

o +tuqrs*t;",

(t)

Ou7L/PF 6P P&t6taftf4, f,/vzm OftTAt / = /,2a,

a2=54e.

Ptr66o,1", rp .e[euENcE Co//77,tN

t q,nef str 42

74F

7il26F

dnd= 34g,

t6ts, (e)., (+) rwo ls)

cd&FFlctcNTS:/

eaS, Q) irn ("2 Ttrrt*yol:trc

7.yAt

e = 3oo, wrrc+

Pnhl?ntq

Fil* 4Ho ('trdl

r e, @' % LrTu)r, cluEn'

3-&g

Qao aa

€- 2g

cAt?r

rzt;r EnsED

Hpvt

/?rc.

t*t/6rt 6=

7Oi

fuet -

d. r?xo lS O= 3oo ro USt^/6 t6o

Qecae*tetT.f ALSo, ptt-to

1fc

!a'{.

AB z L

. eoe

Tolcfr

Con?EPoror*6,

k'rqxtpau

RtPe Mrtzrcs

Poo

ftA - ?o"

refil)

t=d * *a,f,.e s)aof * + frL)y !r=W t,u

R@,

hb\-=k + Luirn af* (tzls'[r- u cose]t

(rB),1

Llnear velocltles posltlve Omega posltlve clockwlse

the rlght

and up vBr m/s

vBy n/e

0.000 o.66? 0.491 0.000 -0.691 -1.541 -2.504 -3.52g -{.555 -5.502 -6.279 -6. ?95 -6. g?5

0.000 -2 ,257 -3.359 -4,202 -{. 824 -5.211 -5,338 -5.1?7 -4,701 -3.910 -2.913 -t,475 -0.000

theta deg.

omega rad/s

vAB=9 m/s

vC m/s

30.00 20.00 10.00 0.00 -10.00 -20.00 -30.00 -40.00 -50.00 -60.00 - 7 0 .0 0 -80.00 - 9 0 .0 0

0.000 2.0a2 2,94L 3.502 4.092 4.621 5.136 5.631 6.099 6.516 6.954 7 , 0 76 7.154

0.000 1.157 1.699 2.10t 2,427 2.6?2 2.937 2.923 2.933 2.991 2.795 2.715 2.693

0.000 -0. 154 -0.111 0.000 0.159 0.356 0.5?g 0.91{ 1.051 1.270 1.{49 1. 569 1.610

-------

-----------------------------

F l n d m a x l m u mv e l o c l t y o f c a i t L9,70 2.0315 1.1760 1 9 .6 9 2.0325 L,L766 1 9 .6 g 2.0335 |,1772

to the left. -0.15{09479 -0, 15t09493 -0. 15{09479

(t) (2)

(cotr v ^tuE|)

V.)rr-r*

teFT = o,l€+1*t/c

tNtrlt @= /27"

.filg

L=fu--rtltQ, z 2e

f?ao ts

?>e,

fictprglpg JNr+t*, O = o Foa

llatuls

Frila,v

rc, ,ta'

gt /NcnE*tE:N?T;

l--,-J

N*,

Pua

a) er.D1l

fu,

& ugtt.d /}tsa

SllPPrtu{

W,TottT

Ar;.o

Concsru*tqlNl

- - - . ' K.Pftotvg flutv. s a,rp72 gnnf v.Fu.E /. v O -tn. SptiF+eE JrrrFfoc ltK(-ca E{rtBt f?a//

G&: lE-

V+WF op 6.

?a-

/br?/WS

, fu2

huO itssuttltt;

PFZFetz*

ftfi?c

614

O,C-th,

tncle6S+*29,

u-yN6

Pnor b) tNsT lntzrf,

THAT

, l(lOAt,v(,

,.i ',4

, , ,w

}

J= | rh,'

'-i

G &tses oraeeztY 4BavE a

( b) ,lutaBen

/,CnJ/g

ao y'laec

spuatr

utt ct

t =l - r' e-tl,

*J8". J , *i

TaE

{a @ = Sin-' %,

torrnt

(Qon# DG= A6- AD r o zL

L Case

;

c D =h d t o , @ = ^ # turoPttivvT

aF !4tt7v4

n=(q)s)ne

4c= # eBou7

Consnttztar

i =,+ntinlo'#-(+ s)as -a tq-"f] <bry€nzve

rr c*

(r )

/?,to

u)2=zTh I.

hrvl PON| 7?

( t)

b,' = - j+ # / r?P.+cT07 8? €ftlEW

r?o747FS

ngoLrr

o'gQ toqq)

-=VZfr J.

(z)

€o'

h,h),

Auo t^

Fan

nFnil. raPttz rt7 6

BErzrze tnftlcr AT B

tt?!.octzf ca 4, 6A= (nc) b av7iltr& otr_q?ein4? FfZoenm1 /2r sEau*tce) nDrD6, hrcD, Ac, f.t tr, Q, O

-,n | = I *, (E*r) wi *-(A'.r) {",(4^*')b,"'4'gl,

o+o= Lt.u?-*o h

7+vr=6tW

tYAtottlt

oJ+* *,&]=* "*n4'-! *, i'f; l *&?* uo' ")

ffiz

U,='"'gh i 6tuo = /*y, i

frtryn

(seo

t f.c42n

2t721gPS'Y

ev L+z^67

T=orVr=o, V"= -'W h = -*3h T=

Vo=6 j

/Ns\

--,1C" a)1+toefl 4r= i +

Ftzdu

A"=f

W-ng

'vr(,ae

+) lfu+tav7T

nBoo7 B ,

orinr,

: (orr5 cas ?e), = f -o*

ior+

atilr

?e hr, = -4t*r*orr'aro

+ lra f7*

b?=W*,

(t)

So tryTftzvlgls.

t?ozqa'nBoctr

"tnyfuF

Cotsea

lJtrYT'L 6

tS 4BauG

HszaY

Tz=r+ naSkg

a=200nn

t=800nn

h En

theta deg

onoga rad I I

is=f-g vA n/g

0.000 0.000 0.000 0.000 1.911 0.385 17.365 5.000 0.553 2.680 34.19{ 10.000 3 . 235 0. 693 {9.938 15,000 0.826 6,1.01{ 3. 648 20.000 (}.959 3.934 73.786 25.000 1. 088 1.O79 8{.530 30.000 ,1.051 1.208 89.389 35.000 i10.000 | .299 8 9. 2 9 3 3.811 1.330 3.325 82.8{3 45.000 1.255 2.592 69.067 50'.000 +++++++++++++++++++++++++++++++++++++++++++++ Flnd theta for nat onega theta deg

h En

onega}'.r radla

,l.0907731056 3 1. 8 1 0 86.788 4.0907735825 31.820 86.?99 4.0907?35825 31.830 86.810 4.0907?31056 31.840 86.821, r+++++++++++++++++++++++++++++++++++++++++++

Vz' - orryh i Vssa t dt

n = +'-(4"r')tttG,ln(-lf*r")rt

V z= T 3 t u s I

*^(4*)rn-+ash'

+'" /4'* t)'i

.l=af- #

(cornuana)

17.C9 continued INF lnve

faat/D

6/ueut 4n63E,SQc sPtuu: A= Ttln/a,

Uil STeETe.vm / rzvtfi, Y;= tftu*rr tfuP nnO HuBl IB= O,346, ^2

hrtutae

vEtocrzy

(fnr

,VER

'fril€tzE

lS tlautst

.gPaEpE

r7S ttt, ttc

pAScpS

hoerc*,

ptlurze)

ol= 2,5'*r

Rotts

(tt - r)6 U*

Fteoru' 728

17*

hil

.5&*tn

Fr?e

4tr6OLnrZ

A LSO

,voprEtvtor|t

laB

/4a

Ta #t,

6fl lun=

tuo= l, ti

4 =Iatt

-& *Ortt

t N c t e c r e E ! 'fvA7

ef= ro c

7oP

*.rr,.rtt,

CoHg€ey4llotv

(t\

Ft1t46Y

ltJo

+t = /

j /N Crz€Mefl7

/ooo

s I \._f/

N,={r}* Rsu/Aa4}h

SFf?kl Sfr?ttl

tiln-[t4"* r'cosz4/ (A'+r'S] u, zgt/(frz*rt"

Ds= { r,'' /

tug

Par,vz

//=

fus.o4

576P,

(+t

/s Na&4ge72

ts Agovg

at= Eon<

I? ot t E0 ov eh)

NFrT

alv afiaFFarrrvEo aFrantvEo

SfeIV€a

d"'"

€js

o" tp&ttl

4 =(f-0

T= *rn-|+ lnq"* { I"r}* I ni"}- *( L+aa5)aoa fi = * Iuuf+ / o,%"*tnrl'* i A o"

v, =6

= l lu^^* t *.1o^+ **{r' /r, *4(r- a)2 d= tft"+*f)ut'+ * o'oti T+l/, =72*vz: f(zo**t')r!= *(:td*r)u'+ Ia,tti+ t+'1r'n)

* 4(''trr = {*l {4* n cl,}- (rg n f) 0'^ "r"lJ'

1 0 . 0 0 0in.

o m e g a O=

1.500 rad/s

Dlstance between bars ln.

omega when G ls over B rad/s

Number of bars sphere rollg over

1.0 1.5 2.0 2.s 3.0 3.5 4.0 4.5 5.0 5.5 5.0

t .494 t .487 1.4?6 1.460 1.438 1.409 1.370 1.319 L.252 1.154 t ,047

491 169 76 d0 23 t4 9 6 4 3 2

l-dla

1.1

* 'r1 r'L

fiz r(r-aso)

VAtu6

*=M-*o

-t

v'alfg,

@= stn'' P4,)

r =

Q*a

AgauT

r a

Ior+ +aoCro

Y= ff n,

Fa 4T

*r.tt

* B o u a o t .-rB '2 r'P2 $ J u u-, , + o . f= , ^g I u ,* +- , + L r

o uTAtnE ap p/a$,anM

Eyn-eft?;

ano t/rU bo

,/UcrZa*tsr'zT.

O, {,

f =,ldo aia (t)or /O nIS

enQ. CowgSqVllTro

4r-/

/ttl7rquy:

gPrtkt:4

,l=?,n,

P osttroy

tp tvtt

Oonz.rrv€ ?F Peatnfi^^: ,€nren mTn , 7tr=?, t4g, fu= o.g Ayaj ct)o=/O rdf Ft?ocpan fa\) tL) r+ND 4

r+rw)&et fra

Ea'h),

VALUET oF

fiT O,e2{,n /*cn?r4e*rzt,

l=+tls

dgla,rE

t?Frqcu

r mm

omega rad/s

5 0 0 .0 0 525.00 550,00 575.00 600.00 625.00 650.00 675.00 700.00

10.000 9.352 8.757 8.211 7.708 7.246 6.920 6.428 6.066

Flnd r marlmun r nn 731.73 7 3 1. 1 6 73t,77 7 3 1 .? 8

(z)

qt 0,Cn,.fl.=Z6ua/ua r.t*t4 Evuoezp

,w0

fr?tttZ

? F80u o, f,aq 70 o,?tn

tH€rt 6&k

fuf

WRE

4ro

v radlal

m/s 0.000 1.486 I.962 2.22r 2,34t 2.346 2 ,239 2. 007 1.599 (where vr = 0)

onegs rad/e

lv radlall^2

5.645 5.645 5.645 5.615

0.001{211 0.000{968 -0.0004275 -0.0013555

1r r 7 . v ,r \c . ,5 |I -----

O ttVvp P N t N , l/,' 3 o i n o

Baas llBe c*enst,

x,*rji;'

s,,.10

FoR

VALu/+3 o/= A

Fca^'r

o Tz-?o'_oq)rc t1lnrl2Ertc*rs _ l/z

, 2/t

Y'-s

'7utr't

' ,l % +, h

Kffi-

v=qr

,\y'-l

tr=t' tia

w.d,? @

H lnEenzrcs oe Posrton 2:

do=(co)a

&'*=at W

gaB=b

ED =!2

trry

-{o,rt' 'lf,=Ltu

AB=sc=L

;7:#;::':,';;o* CO=?tsial

lv\acet tu A ct2t t

(,1

(ta,, oc caEne s) (ec1'= (co)z+ (taf - zko)("2)cos(?d-o)

Ec a lt rt"* HJ- z(co)('/z)cas(Fd-4lr Cotrsw

v4v orv

(z)

ar= E/YEQ6Y

l/r=s

T=o

Vz= - ?atrg(,

sia)

n= !.,q;*

f,f a6^+ 4*i6r'* {t.i'o

= - *3L

sin6

,r T " = L ** $ + * ( ' # ) \ * J * r ' * '

6= *r1". /2/ff)ont%' = T +V, 7z t'/a-

o to = - o,sl sina * S[t* ''(f)'l-"-' s tt t -f grg . sh *-1. t ffii,J'lt/z VEraazy ar o, O u TLtp{

G) (+)

fD = (co ) u Paa-a rztz,! t

3 g?,? fl&L = 3o ,'r, = 2,9 ft) EVTnT? 3 gEoueucl (), (z), (S), anra (+) f6tt. n fQo6eri,w, tD &na % Foe, tl+ttttr Evnt UO7€ fttvp PQJNT e Faon 6t= O To ?On .Usttr6 160 Ttvctzgr2erptrTT, theta deg.

omega rad/s

vD. ft/s

0 10 20 30 40 50 60 ?0 80 90

0.0000 2.4806 3,t277 3.3226 3.3302 3 . 2 74 6 3.2098 3.1544 3.1198 3.1081

0.0000 2.1537 5.3487 8. 3066 10.7031 L 2. 5 1 2 3 13.9945 14.8210 t5.?622 15.5103

Chapter 18 Computer Problems y

A wire of uniform cross section weighing 58 oz /ft is used to form the wire figure shown, which is suspended from cord AD. An impulse F Δt = (0.5 lb·s)j is applied to the wire figure at point E. Use computational software to calculate and plot immediately after the impact, for values of θ from 0 to 180°, (a) the velocity of the mass center of the wire figure, (b) the angular velocity of the figure.

18.C1

18.C2 A 2500-kg probe in orbit about the moon is 2.4 m high and has octagonal bases of sides 1.2 m. The coordinate axes shown are the principal centroidal axes of inertia of the probe, and its radii of gyration are kx = 0.98 m, ky = 1.06 m, and kz = 1.02 m. The probe is equipped with a main 500-N thruster E and four 20-N thrusters A, B, C, and D that can expel fuel in the positive y direction. The probe has an angular velocity ω = ωx i + ωz k when two of the 20-N thrusters are used to reduce the angular velocity to zero. Use computational software to determine for any pair of values of ωx and ωz less than or equal to 0.06 rad/s, which of the thrusters should be used and for how long each of them should be activated. Apply this program assuming ω to be (a) ω = (0.040 rad/s)i + (0.060 rad/s)k, (b) ω = (0.060 rad/s)i − (0.040 rad/s)k, (c) ω = (0.06 rad/s)i + (0.02 rad/s)k, (d) ω = −(0.06 rad/s)i − (0.02 rad/s)k.

4.5 in. A G θ B E

6 in. 6 in.

Fig. P18.C1

6 in.

Fig. P18.C3

D A

C 1.2 m

2.4 m z

FΔt

A couple M0 = (0.03 lb·ft)i is applied to an assembly consisting of pieces of sheet aluminum of uniform thickness and of total weight 2.7 lb, which are welded to a light axle supported by bearings at A and B. Use computational software to determine the dynamic reactions exerted by the bearings on the axle at any time t after the couple has been applied. Resolve these reactions into components directed along y and z axes rotating with the assembly. (a) Calculate and plot the components of the reactions from t = 0 to t = 2 s at 0.1-s intervals. (b) Determine the time at which the z components of the reactions at A and B are equal to zero. 6 in.

3 in.

4.5 in.

18.C3

3 in.

Fig. P18.C2 C

6 in. B

6 in. 6 in.

M0 6 in.

150 mm x

18.C4 A 2.5-kg homogeneous disk of radius 80 mm can rotate with respect to arm ABC, which is welded to a shaft DCE supported by bearings at D and E. Both the arm and the shaft are of negligible mass. At time t = 0 a couple M0 = (0.5 N·m)k is applied to shaft DCE. Knowing that at t = 0 the angular velocity of the disk is ω1 = (60 rad/s)j and that friction in the bearing at A causes the magnitude of ω1 to decrease at the rate of 15 rad/s2, determine the dynamic reactions exerted on the shaft by the bearings at D and E at any time t. Resolve these reactions into components directed along x and y axes rotating with the

150 mm

120 mm C

D M0 z

ω2

80 mm B A ω1

60 mm x

Fig. P18.C4

20_bee77102_Ch18_p013-014.indd 13

3/13/18 9:13 AM

shaft. Use computational software (a) to calculate the components of the reactions from t = 0 to t = 4 s, (b) to determine the times t1 and t2 at which the x and y components of the reaction at E are respectively equal to zero. •

B A C

θ 360 mm

Fig. P18.C5

r = 180 mm

18.C5 A homogeneous disk of radius 180 mm is welded to a rod AG of length 360 mm and of negligible mass which is connected by a clevis to a vertical shaft AB. The rod and disk can rotate freely about a horizontal axis AC, and shaft AB can rotate freely about a vertical axis. Initially rod AG forms a given angle θ0 with the downward vertical and its angular velocity θ̇0 about AC is zero. Shaft AB is then given an angular velocity ϕ̇0 about the vertical. Use computational software (a) to calculate the minimum value θm of the angle θ in the ensuing motion and the period of oscillation in θ, that is, the time required for θ to regain its initial value θ0, (b) to compute the angular velocity ϕ̇ of shaft AB for values of θ from θ0 to θm. Apply this program with the initial conditions (i) θ0 = 90°, ϕ̇0 = 5 rad/s, (ii) θ0 = 90°, ϕ̇0 = 10 rad/s, (iii) θ0 = 60°, ϕ̇0 = 5 rad/s. [Hint: Use the principle of conservation of energy and the fact that the angular momentum of the body about the vertical through A is conserved to obtain an equation of the form θ̇ 2 = f(θ). This equation can be integrated by a numerical method.] 18.C6 A homogeneous disk of radius 180 mm is welded to a rod AG of length 360 mm and of negligible mass which is supported by a ball-and-socket joint at A. The disk is released in the position θ = θ0 with a rate of spin ψ̇0, a rate of precession ϕ̇0, and a zero rate of nutation. Use computational software (a) to calculate the minimum value θm of the angle θ in the ensuing motion and the period of oscillation in θ, that is, the time required for θ to regain its initial value θ0, (b) to compute the rate of spin ψ̇ and the rate of precession ϕ̇ for values of θ from θ0 to θm, using 2° decrements. Apply this program with the initial conditions (i) θ0 = 90°, ψ̇0 = 50 rad/s, ϕ̇0 = 0, (ii) θ0 = 90°, ψ̇0 = 0, ϕ̇0 = 5 rad/s, (iii) θ0 = 90°, ψ̇0 = 50 rad/s, ϕ̇0 = 5 rad/s, (iv) θ0 = 90°, ψ̇0 = 10 rad/s, ϕ̇0 = 5 rad/s, (v) θ0 = 60°, ψ̇0 = 50 rad/s, ϕ̇0 = 5 rad/s. [Hint: Use the principle of conservation of energy and the fact that the angular momentum of the body is conserved about both the Z and z axes to obtain an equation of the form θ̇ 2 = f(θ). This equation can be integrated by a numerical method.] Z •

θ 360 mm

ψ• G

r = 180 mm

Fig. P18.C6

20_bee77102_Ch18_p013-014.indd 14

3/13/18 9:13 AM

GtVfl|

18.c1

18,C1 continued

F( Gu Re SHc,uJdl{+t.x oF qTfRE rUErGr+ to,f 6 E oL/tt t5 sI.lsPENP ED Fq'oM loR'I AB. f'o'\PuL-(E

N,,tD(tS) Srrqu LrA ^rA)USL)/ FoR c'J^*uo FOR,D] , We ObTn t r'/

dt=**, o

-rAt = (0,5 tb,s)j t5

lr|-lza

D*=tr-

(lg

A P P LZ T9 A r = 6 u r S tf l E o r ? R O E R f t M . NP $lrlggln T'rlY F 'n'Fiep 0,,hs ttupttc r ,isl( YrltuEs A , f t f f i * ' @ , a'fitt, ,=#r, Fat= 32,2ftls' OF 0 TlRoPtO Tu ,Hct rN I D" ln/c [ :T-{F vr s C o n p u r gf r e o t t t R , f r f l D an F*of{ Fee ' ( t) , (Z), nW (9 (e)VPcoctrv P1z6 , frsu ? Qs,(t+) aP1?vrEU)na *u/lfu ( b) Ani€,t/lttr. v€,LoctTY

ctrlnlTE(r)e,%)* *ND(Q* trouajs,(fi,(e),*oj) Cut"l?vTE I,IJ,*r,to lr ff;ou E6t9,(8)Aul Iq PRonEq,({)

A N . l F r L Y-3, t S , : t 4 f t 5 5 ? F R t J N r TL e N 6 T H s L E NCTH0F Roa AB 2A +- s (PPru5 OF Ef crt Rt'vG

C \ t ) p l - r t t - T ro n l o F M A r s E5 :

(r) (t) Ca)

kbi &ns=?a. ) E ac+tRt i'G" da= 2fl2 frl' gNtt| E F(&vt?Et ,t/t = ilftg f 2'n1 R noAE^,7J oF tNef.rt tI t

7 R 0 6 R At 1 o e T P v T

( , r ) @)

AB, En)nu=(fr\or=!*r,e',(It\r=o

(5),

= 4 ,,tnb'i- ffiRd' ; ;R ft*;)

tkcH RtN&: (f)r

ArlD PRIIJT &aPvrE 6 = fbb/n Fof g= O T0 0 = lAo' prrtDusrx 6 10"/rucRE^4EurJi cALcuLftTeH, ttD H* r<opt ?6b, ( tt) dt DJ,uvn d* FKong qrs,Ct5) #o rABvLnlE Cfttcu&rt , ,

vbar

(b) gj;:,;:;'Xtl;,"\:;r(!,,+a,)f;

Vel oc i t,y of m as s c enter - 19.O7 fE/s (direcCed upward)

T heta degr ees

e N T I P EF I 6 U R T I

*;:irt= (f,r)*a+e (r)n , IJ=z(t\n, {=(I.)r*lQf

t s oF /Mt-R? r 4 r , Ffiali.;c

-iE R0 fAolucit oF t^/e(TtA rrcE Qr)" i!ffi ot/Ly ,rVorU

:P'=1(4')^ t |

rE- r Ot-{F_NTu 14 FRrNCt ?LT ', ATfAt! TMPuLSE ftMD wlorr"t ENTLtt"lftF IE( tt"tPAcT

-._ A b= a l t

(fr|'Ji=+nV ( vrsar o) 1to) t- 7 n=: ry i ro?Arrt/pL

Angular velocity ( om ega) x ( Om ega)y r ad/s r adl s

0.00 0.00 -54.88 10.00 20.00 -108.10 30.00 -158.03 40.00 -203.16 5 0. 0 0 - 2 4 2 . L 2 - 2 7 3. 7 2 60.00 - 2 9 7. O O 70.00 80.00 -311.26 90.00 -316.06 100.00 -3LL.26 1 1 0 . 0 0 - 2 9 7. o o 1 2 0 . 0 0 - 2 7 3. 7 2 r . 3 0 . 0 0 - 2 4 2. 1 2 1 4 0 . 0 0 - 2 0 3. L 6 1 5 0. 0 0 - 1 5 8 . 0 3 160.00 -108.10 -54.88 L70.00 0.00 180.00

0.00 L8.29 36.03 52.68 67.72 80.71 9L.24 99.00 103.75 105.35 1 0 3. ? 5 99.00 9L.24 80.71 67.72 52,68 35.03 18.29 -0.00

( O m ega) z r ad/e 0.00 1.81 1,18 1 5 .9 4 27.84 42.50 s9.49 78.29 98.33 118.99 139.55 1 5 9 .5 8 178.48 t95 .41 zLO. L4 222, 03 230.80 236 , L1 237 .91

i ; j

fior'rEw0F lr.tpulrf Abov7G fttdl nu6utlA? fg,l,lHG AFrt ( l,rtQ+c 7 ( Uori THhr IU Er?E t9

tivo

rtrl9r/L irt.it

EXc-:i,I f)

X Fat I = tl6 a] +Liln6 t JX Fati p[o, -'cnsa)].. - L ' F a f s i n o! | t F A b ( l - u s i l E -casb) (, |) g {t7 2 Fdt si n , HJ= r, H}= c FBb(t f t0,7)frND R ccRllr rv6 rHAl lJr=Iu=O \ $r_*i ':+ lt

,ftri

- t, 5P, u, = l-l'

Uz)

fr ,*'Q'J*=\

tt3) (lr)

+i, d, = 0

'*tt

u ED) iCor.l'trN

GHGM; wrTH4n=2tuo8, PRogE

18.C2

' kf 0,?0n, kE=l,Obm,kr=l,D|tt r{Rrx'T}+R\rsrER €i 7OO-N 20 -N rltf<usTERt A,0, C,D

L:l - lb ftJsEflt,Bi hlulltNu !A 0F ulti fffi THr(l(t{E5S

c,*H ExPEL FvEt tN g OtQECllriN, PKogF Hhs Ail t. vELoctTY

FrslD: ,G) c*?onrEurs r

'a)=d*L+deb

f txP wHr cH TW\ oF Tk€ zo-N Tf+Rusr€PsSflouLpJ|EL6ED -tO t?EDrrrF *t'16, leLocrTf. TO ZFRrr ftMt frn ftOw LoMb thcll oF n+e4 ^srDulD .oe RCTTV'ATED I sss untN&

'a)

o0s(0,o+oro4s)! +(0,060rd/{k,

*s tN PR\g, tA3i,

R o r n t r N 6 IF x D l : oF rHt.DyHnmrcr TrtrN5ATA ANgb FRoM t=0 r0 t= 2s AT O,fL^/renrf,Lsr' (.0) -ruET|l'lE (U ffrI 3 s/6rt.tr:t(#[T Ftsuers)4r w'ltcn fr= S,DEoF

V Cor4PoVeNTJOF f#tsE

REtqCTronlsARE EOttAt ro t

j , t+srN PRID,lQ,3r1 A I V A L Y 51.5 O spz (0.06OruA/ii (0,0,tOraa/s)$ (D'ol,D rad/e) ^toHeil7 rrl + l^rERrif$h Pco-DqTt &, 0F. UlE Cor4ArrE Z'{E (c ) A , (0,06orad/s); fruD c e^r?RorDAc *rtG THt *SsEn 8Lr turTH (c.spE(T To - (o'ozoPfls)k' (il A -' (o,o6orad/s) "HC t'rtb Pf,,ooucTs ! aF#fi FtRsT(cl]Pu.lgf$rfitr1 NT' Dt|'r'.WE

- ? ' ' -(I,..ffi d- ),\(,[,,\*ri ros,e*(HsodoRf, ; (T^)**' .11';391 . z r r f u , ? A iol f t R l A iJ ff

ANrry-:rg

(I*t,Inrnl -- | '- ,-Xd f os,r:ActllRt Atrer€ , (4Lr t ?f 3t. O t'fttEA tZ iid, o=

lhltt t tlt- hN h, MoMeN 7u nt !

A e, l * d . Z * o * I rcd . t = ^ k : d r 2 + q n { ^ r t H,,=o ir= ,nk:4 Tt+rlsHz=^ rla,

(l) (4r\, fdR

f r r { 6 y r r l Rt r v l P u l s t0 F T w O 1 0 - N r H R u 5 T e F J t L E r u 5 & 9 r u l 4 e r H n r A A N DB A R e ,A c T r v R ' r E D . At-|6. tr'1Pur.sii6ovT G

t'ff ',,-:]

7r,r,=:i o'Gd#o)+ no'=-#dpr.sr9,ffi "o=io'i'E'. i':!

Err/rt nE h95EH EAY

(4)**=2(+f)+z(*': ) =f ;

rr'Jr*rn= ,'("ioo)= -/o'

(JrJ*; 2(-#t)=-# of .:

pttt;l .,,r c J

ftR e OSfnte O E f S O F fN F C r r A T l te l fAl 3 H t;r tF Ati ,tpp = !n r (- FarA)I t \ex (-FAur) 0 Ex?RESSlodJ 3)r n"F HfrSS m OF TtrE Huurr A-Vtd6 lrrse = (- o,, aL + t,lo'r]aE)r(-Faql)j+ 7t) gOj, ftgt€4hLy t+ut| atttttat1 nY FS rl(Efi ,'iltttctl lg FOuRl, (o,lai . +" t r,?o7ta I r((-ratr)j 'Anoi' ( l) I^= &ata', 1r,= ; m d-', Irr,= (2) t!6.tr = t.107ta, F (ot^+ at)! + o.f af (ttr- 6t)t E '&a *t'tp tls o e n' v" '"'- F- f I NPvts7 -HbHeNTUtl Pk N ( t PLcWe oeTEA4tN ravE tN ti.'ffi WE MuSr ftAvE Ae*'jlrJ6, irr F. = O sETrlp6 Dr-=Nt \rdr=o i 5 He,'= C o f l P o n / t V 1 HJL3Tr,fu , 0R,v51rV6 1^r,6 t +r , AtA+Ats=-# Hral,zo7laF(Afp 1 6t)= o - t,r,il: fr=(4!-'.r:t :4:'!. 2>tAte-afa= H* t O,5aF (n t6- atB): o # -,.,1+h)Dlqrl, ryt a t), =(,it =i,ito) 1 $x r K c.t(ts,?q: Fe SOLVTNG 7t+ESE

FCsufrTrorrl3

5r tVU LTn ilgOuqr-Y :

Aro=W,otu=ry LI

)raq,t,+ t ln i^ln D',,,,t (. a!, (J,!-A_!*r,i(lrii ,,t)'t (3)-Dlj^= Lr,*jb t a!' Lr,,ria

A S : ' . , r r r r r e n /t t C , t i f : c I A J t + o u r Dg € u s e g . , lf bt^(Q, k5':titttTrN !:, t.tJr.r;7i(. ) C Sffgulr2 3 Z U'.,71;.At":D l l c - r r v A 7 € Df r r : L t c = f . : - L n l , S t U fL t t R L y , t f A t g ) O r B s H g r i t . br , € i ) . l f y , P . \ ' I Dl f a f p . , ( O BE usFo NrTH A[=lAt"l, D Srfour.O

--(J, *,ifry,l -- Jrq,B irr,b)'r )cr It),o-i i Jr*d t i.- lr:t lp,:'! Itrr':-'1 (In,o'= !^4! (L, !^x! + (I^

tFAtA)o,

\s,o()l'(\,o'r{rY)!k \r,t()l-

r or,lS n / S gff t l D i tto} oN r J A T!o o F frtoi EdUA_r te.\--

il o ur llNlF,.o_FCSj_c:_E-t EN'1LR, P+nf i A r b, Ct oR d"

ZA --s = Z { , S o ) o f l ; . , ^ o - . \

n;i:i"-tx(hllrftri)=16 .. A,L-'a*i-* qa.Asl'ij*j; qrF ru:il!'(}I;'LTB , F r u e0 i l r r v E C T ' R J t E a v r t l N b T F + EC o ? F f A

bbr: lAtn I

( F bts) o, ? rt47 Ate lf NOT,FFIHTA L o= l A t r l ) ? R 0 & R A r t t 'D U TP A T

( a ) C n U D B ; A t r = A ,1 6 o E i A l o : 1 , E U 5 s ( f i A A r DD ; A 6 : / , B f1 s i A 6 , 6 , 8 ? ts ; b n= 0 , 3 1 8 8s ( c ) C A r u oD ; A h . = t 1 , 6 S V rA ( i l # A U DE ; A d ' 1 , 6 1 4s ; b - t s : 0 , ?r 6 8 J

{

No=Ird

FRtrH w* tc|. v/6

c(s \fi* oBTArN

d = 0( b

A, = (\ r,4)1-Jrr#)/+a AJ i (\I, c^l'+irg)/ +a

Z-f=>(f[r=0: n+B=a T ttuS:

3t=-\

8r =- Ae \' ED) (couT I FJ

(z) t))

(+l

(r) '(6)

18.C9 continued

19.c4

oL/TLtN E oF PRocR A!!.

g,ttlt d=

@) EN?EI Ao-- o, og lb,{bt Wr 2,7 lb, tu-- 0 , J f b COt't?t)TE rm = W/32.2 CoraPu."€ I^, J^g,, Jro*, FRom eos. ( r )

, .,*ii

1l3K i tft:2,5 hS,q=ffiv,,, dt= 6 o tut/g frr t=o 2;69, DEcR€fr$6 Ar *rne o.tr t 5 rad/s7 Ar t = o, frr=o

120mnr

150mm

ffego,in,n13 iyyte 3: $Jrwtiei;g;X;il:ii,

FRortte. (a) CofttPvri q on!;; ---- 'r FrnrD; F O g t = O T O b z 2 s A r O ,l . j r N r E p v R ( S : :-r- i \Co4eWt c,J FRoA Eo. (r) (a) eqworvzurg+(of\fa cou?uzf A* At, gg BE F(pia eos ,(4), (9, ft^lD($. tf+f Qp7ftltNi t( *^/D AxES OF 771Tb7nl.+MrC RFR(lrou5 kr , 4 hus T*gilt-tre Vs E D FnlD f ItROnt =0 TD t=45 *t 0,2'5 ,NTEsVt+tS, ( D r U e - r f f v r e t t t A N D t ( W , T + t3 S t 6 u f? t c 6 v 1 7F t 6 u Q e r ) (ul DETFRn4rM: 6v ftt-,pec.Tt6r.t T+tETttt€ tNTeRvAL tN vtftt.cfl ftr w++tcl| E+ hNt a, *ee Reef€sT,u"_r:_EuvRr To zERa, AVt E?. C Xa , /it f St 6 rt Atvg R r.rrvZ,+F pe 0(-.^' rt tt ov?p THfrT

t r t t E N n t r V S t N 6 O , O I ' S l N c F : ; t t Et ) t 9 , , R E P f n T r r t r Sl " r < o c e ' , v R t ) Auht^{it5 y$t;;t0 0,CIo1-5 lt:tcrl-r,teuT5, rtft -DEstf?"D VA tOe 67 t t , T r t A r F o RW t i t c H f& l f i ^ , r l S o l A R € f d 4 n L r € S ? .

PRO6^At'tour ?oI

(c")

t\lf

hlD' v ,

, t r. *"o' .

,l ll ** ff tr , , .td ... ' . tfy. ,5,, ; r. I

F\ vl

1,, t|.:

Ay 1b

Az lb

By Ib

Bz tb

0.00900 0.00851 0.00745 0.00552 0,oo2g2 -0.00066 -0.00491 -0.00993 -0.01573 -0,02230 -0.02964 -0.03?75 -0.04664 -0.05630 -0.06673 -0.07794 -0.08992 -0.L0267 -0.11519 -0.13049 -0.14556

0 . 007s0 0.00796 0.00935 0.01167 0.01492 0.01909 0.02419 0.03022 0.03718 0.04506 0.0s387 0.05361 4.07427 0.08586 0.09838 0.11193 0.L2620 0.14150 o.L5773 0.1?489 o.t9297

- 0. 00900 -0.00851 -0.00?45 -0.00552 -0.00282 0.00066 0.00491 0.00993 0.01573 0.02230 0.02964 0.037?5 0.04664 0.05630 0.05673 0,o7194 0.08992 0 .L0267 0.11619 0.13049 0.14556

Az Ib

By 1b

Bz 1b

0.00282 0.00250 0.00218 0.00186 0.00152 0.00119 0.00082 0.00046 0.00010 -0.00028 -0.00066

0.ot492 0.01529 o. orSga 0.01607 0.01649 0.01689 0.01731 0.0L?74 0.01818 0.01853 0.01909

-0.00282 -0.00250 -0. o02tB -0.00186 -0.00152 -0.00118 -0.00092 -0.00046 -0.00010 0.00028 0.00066

Ay 1b

Az 1b

By 1b

Bz lb

-0.01818 -0.01923 -0.01927 -0.01832 -0.01936 -0.01941 -0.018{5 -0.01950 -0.0195{ -0.01959 -0.01853

0.00010 0.00006 0.00002 -0.00001 -0.00005 -0.00009 -0.00013 -0.00016 -0.00020 -0.00024 -0.00028

-0.00750 -0.00796 -0.00935 -0.0LL67 -0.0L492 -0.01909 o. soo'oo - 0 . 0 2 4 1 9 0.03022 0 .?0000 0. 90000 -0.03718 0. 90000 -0.04506 1 .00000 -0.05387 1.10000 -0.06361 1.20000 -0.07421 1 .30000 -0.08586 -0.09838 1' 4 0000 -0.11183 l. soooo 1.60000 -o.L2620 1 .?0000 -0.14150 1.80000 -0.15773 1.90000 -0.17489 2 .00000 -0.19297

Ay lb

-0.01492 -0.0L529 -0.01568 -0.0160? -0.01648 r'f , Efrf1^ 0.45000 0 . 0 1 6 9 9 o.46000 - 0 . 0 1 7 3 1 0.{7000 -0.0L774 0.48ooo -0.01818 o.49ooo -0.01863 0. 50000 - 0 . 0 1 9 0 9 0.40000 0.41000 0.{2000 0.43000 0.44000

'."1a

s 0 . '18000 0. 4 8 1 0 0 o .{8200 0. t[8300 0.48400 0. r18500 0 . 48600 0. 4 8 7 0 0 '0. r 1 8 8 0 0 . A

'i& . 3 9 9 0 0

.'0 g o o o

r_ ,

=, aiNr g + +nL"4rf - /'ona!D,q!

0 .00000 0.10000 0 .20000 0.30000 0.40000 '0 ' 50000

',b) ib)

i = Imtott! +iuiu, ! i +f*.u* W; (B,tz), =GfJty+gtsin 4Q. .4 (t"f^,4-i ntar$ !^=:^nia,t 1 |fr-r'6,t + dar<x

0.01919 -0.00010 0.01823 -0.00006 0,oL82? -0.00002 0.01832 0.00001 0.01836 0.0000s 0.01841 0.00009 0.01845 0.00013 0.01850 0.00016 0.01954 0.00020 0.01859 0.00024 0.01863 0.00028

trr r orrg e =?t (-u)tq!,* t o,s

mE--tl (Xr>rgNrd:gr/)= /*xrt Kdi-h j)- mffil.-6J) =,,r,(dotrj r h NzL-dr^):! ! h dz'l ) (ddr+ha.//j (Z) il& ",n (hYr-d'n:)! ++/t R d t n - t i o r !0, :f u o r t o N

4v\ot%+.nd:)i

dr-dr{}

\rfn {tn

f u,b

1$o'EAJ*,

lmtapry

_sbixtt'il'r,r]lti j;:,t\:I?i;!;:;i;l:i6r = 2bqi+Ibgi tMrF,- !ntt'ila,g+lntt",il+i* L'x'! - (ftib(hdz= Aq V +nI (dxr*hCj +{ndtlot+hDfik +nh(hV dD:)y OF IHE UNIT UECTOR'I EQUNTCTdE COEFE

=^(++jtd'+h')q, f,fo = n h t ; t d - + n ) % er= # @ DH , .,!u=ffi n,+bh4 - bd'J:) i E =#(- tLu d;) EJ=ft ( I *^,6rt bda,,+bt z I = z ( f ) 4 0 : 2 + 5= m A rr! +rdi ;\!

ll-j r'

(s) (+) (5)

r E,ti: tn(Au,'id;)! + dn(dno

F}WzE r*e coFfnir

j 7tE u^t'r tlE6ont

D2:'tn(hxz-ddr')-\

- r yz a v t ( d a r + h d ; ) - \ '?}}e FRoF'l GlvvJ oare Tw'frr We RECRLL Jy= l 1 , 5 4 3 , l , : 0 , Mm , b = 0 , r l n , d r i l , t ZM , h : 0 , 0 6 n

tDo= 6omdfs (r = - ,S ^Alf nND ttoTlTflA7 At 'TtHE E LDz=4zD dl = eo * d, b Mo,0,5il,n

' I t : ,

(r)

(s)

ft)

( Corvrt r-lu E-D)

18.C4 continued

JtsK u/ELoFsto dger rct 8rFr{frtf .ffi

O u T r - t t . loef P c o e n $ m g Prrns *owNt N ( s)^3N PRevtov5 PAee (e) EMTeR

Fvts tt tfta F

cDr'lPuTE o(a fRol't t ti. (3) FoR E=O TD t,= 4S A'T 0.2-s lNfERVrtLS Dot*9UrE cJt tND d, Ff,opt EO 5,(9)

pg"ftnD DEKFQEE ToR A CJ S.Ifl-PI*# AOOUT To Qat*TElreuT rRIk

FRoH Eee. (,r) nr.rp C5) crs.ipurE qc P/ND \ conlPuTE DzAuo l, FROM EQ5. G) Ar.tD(7) V; E , hlr|O Tft1vur*re (b) To .F,tvD rH F r,,i4 E bt aT n/rtteH Fe = c)r ' DET?RH tM f, Bv rru$pF(r ronl TrtF TA(F rd leRynrtM u)ftt(tl E, Cr+fr1\t685SttsrJ hUD r?urV 'rff?

, rt1ftqf, g=00r6 =0,-

P R | G R * t t o V e R T H A T | N " E R v n L , U s' N 6 O , O t - t , , v t R f ^ i e N T 5 , R F ? e n Tr * f s t r S t t C g s v R E u S t n l 6 .SPcEe:r rt)R \ ?tlE Ttrlt? 0,oot-J tN.cQFHFn/rr kt'uvrltcil l€*l ts s'qd(LEsT. n SrArLtt( PeOCFOr/eE tS USfD TO .??TERHt r.tE = 0' TrtE ?rF{€ t, hT Ytt+rcn *

-Et!tL: f^ln

.DuRtuC,EFtSu,r,to

Ar/$ Ttt4E f,POorfto rue,', RtTtlRNTOOo (pctrory

(D m{e'1tst.'4 rcRy*lrrs oF lF?o^,q'oqn tatst$e?NcRff coits tgERSucce€sf veLY 7{+rElN rTttl Conrorttr'N5

b 7

(b)

,!,

Dx (Nl

0.36s3 -O.2594 -2.L337 -5 .2574 - e.63os -15. 2532 2 2.1253 1 .z o o o -30.2469 1.4000 -39.5181 I .6000 - 50 . 2386 1.8000 '-62.L097 2 .0000 -75.2282 2.2000 -89. s972 2.4000 1 05.2158 2 .6000 2 . 8 0 0 0 -t22.0838 3 .0000 -140.20L2 3 .2000 -159.5681 3 .4000 -180.1846 3 .6000 -202. 0s05 3 .8000 -225. 1659 4 . 0 0 0 0 -249. s307 0.0000 0.2000 0.4000 0.6000 0 .8000 1.0000

Ex (Nl

Dy (N) 1.53b6 4 .9,150 9.6576 12. 5685 L6.9775 21.5848 26.4902 31.6939 37.1958 42 .9958 49.0941 55.4906 62.L854 69.1783 76.4694 84.0588 91.9463 100.1321 108.6160 LL7.3982 L26.4786

1.1653 0.5406 -1.3337 -4 ,4574 -8.9305 -14 .4532 -2L.3253 -29.4469 -38.8180 -{9 .4386 -51.308? -74.4282 -88.79?3 -104.4158 -tzL.2837 -139.4012 -158,'168L -t79.3845 -201 .2504 -224.3658 -248. ?306

LRsT s-TE" tN nrlrixr'rrNflrroil E (s)

Dx (N)

o .2700 0.2?10 0.2720 0.2730 0.2740 0.2750 o,2760 0.2710 o.2780 0.2790 0.2800

-0.7733 -0.7aL7 -O.7902 -0.7987 -0.8073 -0.8158 -O.8244 -0.8331 -0.8418 -0.8505 -0.9592

Dy (N)

6.2105 6,2289 6.2472 .6.2656 6,2939 6.3023 6 .3207 6.3391 6.3575 6.3759 6.3943

Ey (N) 1 .5305 -1.2591 -3.0975 -3.9846 -3 .9204 -2.9050 -0.9384 L.9796 5.8488 10. 6593 16.4411 .23.16{1 30.838tl 39.4640 49.0409 59.s690 71.0483 83 .4790 96 .8609 111. L942. L26.4786

Ex (N)

Ey (Nl

0,0267 0.0193 0.0099 9..0013 -0.0073 -0.0159 -0 .0244 -0.0331 -0.0419 -0.0505 -0.0592

-2.0107 -2.0206 -2.0305 -2.0403 -2.0501 -2.0599 -2 ,0697 -2.0795 -2,0892 -2.0ggg -2.1096

$ rzlfu.,zx

1.2700 L.27tA t.2720 L.2730 L.2't40 1.2?s0 t.2760 L.2770 L.2780 L.2790

-24.8258 -24.8654 -24.9052 -24.9449 -24.9847 -25.4245 -25. 0644 -25. LO42 -25.L442 -25.1841

Dy (N)

28.2716 28.3034 28 .3292 28 .3550 28.3808 28 .4065 28 ,4323 28 .4583 28.4842 28 .5100

-24.0258 -24.0655 -24 .1052 -24 .L449 -24.t84't -24 .2243 -24 .2644 -24 .3042 -24 .3441 '-24.3841

(N)

1,0,

4il5 h brxTS

INTO?ft?K

oF 4t[6.f4ctt.Atql/?Z coN,gFrRqA,Ttur,{ srv(E TFe p$Rce5coNStsToF REnczroN tlT A ftfro v/?-/6ffr

(+) ( 5)

frnD Qa = Z'S1n'to Y *rf T'% r r . o Lvro Foa P = (Qo/A) +" hno 5o i- ; co^t7F?vnnpN o? tNeReY

tt'r: e^d;{ ofi lcai il(ta *0. T +y'=F=(o,us? i I - E -l $'+lr;n'g+ 3'6'r !i'r*'?) - ent a coso 'lnt-a = 2E cose i. tl'f S'gih'Or r cas'a)

Rfc*rr.rn/6 (l) +u D sv $STr?r rr rJG FoR + z:e,olr(g) .

(o;.#/q)+r'i'' toLvrNG EoK it':

0'=* (2F * r:

Wt*tctr fs O? Ta+e F0Rr4 0

u h e R Ef ( 9 ) = # ( e e t ' fVi(c'tron/ OF 0 D"HNflD rU (3)'

Ey (N) -0.0253 -0.0114 0 . 0025 0.0165 0.0304 0 . 044d 0 . 0594 0.0?2{ 0. Og55 0.1006

USrr{6 T'tt RoTfiTrnlt?lR'Ahe Azg' v a?: fsin 0!-lt+i.'sh

L A S T S T t P r { O E T E R MNf q r t o i l D F D z Dx (N)

--. -

NAlYst.r

AttD O tg file c (s)

' '#

Q q= %1,$r: idh,(tL) 0u:?0',4- fontld@)0; {afti iifi

P ( , O G RA f ' { o u r P u T

(c)

(a) mtvtPlu{ v1.u€ O^l

T*t

a;';n4otite 0 -fbrb =Q y? -L-U U) o*-t-bo ,6 (tD) ;3 Jao fa-deil76i

Co), Ti*le rtfie tt I/?]DDD tuR 0 r0 ,EcR?rrs e To q1 {3 oB-rlrriraD.Tttt0ueo MlrFRtE}L NrECC*Tloa/, Aa 8,8,|06. DFnirtD EY rW ?ftlcT TtthT .f(Q,)= O ,G Cua.+tec= s teil) (0'rcc rftcH oesrRe;r v4tus oF o, 6r^nrre Q rcrfi EqC$)

'ffro + feom'80. (5).

u ED) (cohlTrhl

Ot']LINC OF PROGRAS E N r f e & : 0 ,l E n r , 1 = 1 , ? l n l s ' , A 6 9 U u e ' } n : | . e N T c R f N r " t A L c o r u o ' t T r o N: S 0 o * N P 9 , E$ TeR DELqElAe\rrdS Ytxr v/tSH Tu us € Cost?uTe I AMD I' FRo"{ (t) AND (a) ConPurF Qa FRoM (4) flr+JD E FRoM n) FoC 0= 0oTo O;Qo, (WHeN 5Ce1CttRr/4rssrsN), Auo UStN6 OEcRc\Ervrs dg : OoMPUTQo FRoFr(3) PUrE J rc) 7RDr4(a) Cor.f tN c n R R v o u T w u F { F Rl .C f l L L y7 H Et U T t S G pt H ( l o ) t ^ t t M Dl c A - r t D ' frf Lo tutE Rtl r+r-5,coq Pdr€ + 7eoi4(j) nruo PPrrrT Tk? VtAr uFJ nF 0 /r{-D f T*e R(,lD9 oF Tl+g 6scl tLA.tlorJ fN 0 .BV fi;u BvIJO Tf+E VALUE oF t tTg Mt Nt Muotl r/rl Lu€ 0q . ?RoA(fllu\

(d e g re e s )

E * t L A r J os c t c r . : r

l N t r r ( t L r - Yg,: FvD+ =ho.

A ?+ -

Do, 6=o

FrlJDI qO; l r .'l tl ttl | t't

-/ \ go0rnr

gitgotr.)<,f,!oTrr.,N 4ND?pRroo(rrrqs Reoru'ReoFoF 0'To a's'rurinJTofD) (b) + hl 1i. + Foq vhlust or o FRoMiluau Ot uS lnrc 2" tNcn?ggs'rT9

T\ Lr\ Ys

r J i t'- ') F - 0r , O!3 0

\^ ,..ffi

C o n J g t D ES( u c c e s s r v Eu Y T t l E f N n P r L c-oND lt ront,j

r= lgOmnr

(L) 0o- io"

r'r4= Fo-raA/s, *;

Litt) Oot10",yo.= loraAll, 4o-- J radf

( r ) O o = s 7 f, r : 0 , 4 r = f r a d / s (rii) A, = Eo",Tr= ,; radft, {r= 5 rad/s ( LL)

90.000 5.000 88.000 5.00s 86.000 5.022 84.000 5.o49 82.000 s.087 80.000 s.137 78.000 5.198 76.000 5.272 74.000 5.359 72.000 5.460 70.000 5.575 68.000 5.707 66.000 s.855 54.000 6.021 62.000 6 .207 50.000 6.415 58.000 6.647 s6.000 6.905 54.000 7.t93 52.000 7.513 s0.000 7.869 48.000 8.265 46.000 8.708 44.000 9.201 g.752 42.000 40.000 10.359 38.000 11.060 36.000 1 1. 8 3 s 3{ .000 t2.705 32.000 13.683 g ' s g 13 . ?04 lr . T h b ta mi n = 3 2 .,0 degfgeg Period = O. Zlg s

+'= o' 6o=rraait ro{s,+r=;:l^l:= l : , o - F ? A t V E o (iit) 0o= itr,t=xo ,1'" t

wt+Eh, 0 r*rpcrlFs

0) Precession RaEe ( r ad/g )

D i : K t ' ! t - L D E Dr ' ) RoI' l+6 t9 NEGLT'. rF),-t 'r'' P 55 9uP roctk

!^1.-

^rr.t'iQrsl

THO= 90 P HID O= 5 DTH= .1 TheEa

wr:"

19.c6

continued

Tll0= 90 PHIDO= 10 DTH= .1 Theta (degrees )

Avn L'(s, s

U : r r V AT * r E o T A ' t r r . l (F' e R r r f

Arge W rTR b nffs Qrrtr'rrr a/(r tvrg Tl+B P *?ee, . "

9=#ino!-6i+ $+ $cno)t $ o =i r d r i - * t % [ + r r d * b = I'f rin0j-I'6ltr(,i'+$rxe)!

-bl

Precession Rate ( rad/s )

90.000 10.000 88.000 10.011 86.000 10.043 84.000 to . o97 10.174 82.000 80.000 L O. 2 7 3 78.000 10.397 76.000 10.545 74.000 10.719 72 . OO0 10.920 70.000 11.151 68.000 11.413 66.000 11.709 64.000 t2.o42 62.052 1-?rL9.-4 The[a min E 62,L degrees Period =

\ z7 l l

W * e R EJ = * n d

(t)

a)"=**o' r'= im a-'+,rn'Q

(J)

f R( t r o d o P f N G / L A A { d r t E _ A r l g t 4 _CoNSE

5 (ruCg Tr{ P otv LY 8,.K1€ri rt {i t- i=o( ( r'-" ARE T* e REFcrr oJ f+.I G, u/E +{A/E A / \ n r o r F E u / E l C r 1 l - ?V / s - 1 ' r . 1 5 tnllF Z t9 ?r\r<r c)\= A ZM Z= O Fil/D X/i4A =0. 5 -1 .l $grytur/r rr r..l tt'tl. Rt',t pF REFERT rJ c-C , I T F0 L!-u,d5 t' rlT .1t(v DF j.f r11 = P.,g(AUSg ir TFlf A CON r5V OF X r{retf , H t, = corrll-rAN-l (sef t '+? Dts f( ,T dcSo l=oltmt,s Tt+*, , \ rll

l ? r i o 6 t, f, i , 1 3 9 \ , u / r u / R r T t r

i..LL) TH0= 60 PHIDO= 5 DTH= .1 Thet,a ( degr ees ) 60.000 58.000 5 6. 0 0 0 54.000 s2.000 50.000 48.000 46.000 44.000 42.000 40.000 38.000 36.894 Thet.a min Per iod =

Pr eces s 'i on Rate ( r ad/s ) 5.000 5.181 5.382 5. 606 5.955 5.133 6 .442 6.787 7.L7L 7.501 8.082 8.620 8.9{5

(tt+-cps7)=9,_ ,, .r. ? . [ = . :'FRoH , u . t iIFrlT, ', = E'(%++rctg%) (tn) cur./g, f Wtte?e P O,= d' ' -H.t 1" u * cot''6r' Ht= \ ll ?t I t + s r ' n ? D+ I ( V + + f t 6 0 ) * t 0 = c / 01 G) 6 (3) ryg*ftv1 I'+si""0, + /3:tO= RrcArLTrrr a:s6o(6) q = i FAot-ttN,:.tt+'-conp,iiJ-V-s {'frult'go+p v g ) L v rN G ( f , ) F c , R + : + = / = { 5 - - * ? = = 0 (7) Lt sin'v -L!onr,^r=CorltqEvA t i.I-r " T--r(4Dr."n1.,9* J*o*') l i !' b"+-I-( f ++ coso)"J =; t r'6--"si,t'"b

5u ssit rurr= FoR (

) reolr (-l) :

0V = - 4n{(1a)Qs f: i(t'+'s in'0 + i' d'l' *tl acob' =E T+v = E | ,(t' $! 6;1rzrs r rr i,'+g) - l,attX ', E = d'u)s00 f O Fecnl1,i//,6rr'.rg, *(t'{s;ffi.+ ft ) rng (ro) S o L . i t t u ( o ) F o R , 6b' i" = f c o i ' 'lr t '' C ,tY VrtER jG)

+lrwt-. a c,,bO)- i"rio"4 i,,(le-€

Ul)

(coV rl N (.lE D)

f . . r u g lI T u - i / r n r G F o R

Feo^a (7);11_10(fl),uF HAIE

(r -(. f(q. i (e- ft +Lrfvrtacoso) rrrr,%f

vtetr 6 f Rott fin. (to) ule

(rs.

,=$:#

4=d-v@-; d,t

Fo( a T0 DFCF6:n1? To Cn it rcAL r rVTt 0RA T'OMr lg Ob'T*6rs'D frtRor)Grl N/unlFrl S Q^) =o, C, Ser^/c D tFtMF.D 9Y T H e ftv<-r TrlAt ^tFteFD

@\rne14c

,rfiqi r., :,xA'l

f,(d)

( b) Fo R, 4e(

DEs, a rD

<tt^il6E3

srfrfJ

Fol

c -or2

v ALrt E aF f) ) (bn..teutQ $

P'?,,oH E6l, (Z),

? L I T E R c L - - O ,l g m , g = f , g t n l t i * r e u r ! / r e = t, T P v r F R t N t T t A L c o ^ / D t 'fr. o | r s : e o r % , A v . D + o . P N r E q O e c R E r . l E \ r 7d 0 7 o u u | t S i - - t o O S e C orqP\rrg f Ar.re I' FRo^r ( l) Rrv9 (Zl cot-rPurg Ff(oM(4),q7="oH(il/Arv-D E Fsotl(g) P F o R g = O oT b 6 - - O m ( w * e r u J ( O ; c * r r s v 6 e . r f t e u ) , ANg

gct;!t

D€CRFt-tf-t)19

d€s

CoHpurE & FRopt 0) F R o r ' t r .l t ) cot,Pvl€, t(0) g -trotlDEflueo CARrrY OUT l.Jur',r tr (?.r eALLy r t+g rtVTt6r.. ' t^,Fo.(13) * - T 2 " N T e * V r l r , 5 , ? Q t , v T T H e v 1 u t E5 o F 0 ) + , t r N D , P Q o t 4( 3 ) o F {, = T1 : 4 , c r : s t 7 { + F ? E a t o g 0 F T R e o g c t L t d ' l r o t tJA l0 l E D ? u l A t n z gB Y Dcrugl1r',9-li; !-- ., r-)Lu E gF t coRAEsrbNDn,.tc?o O = gnl . PR,ogRAMo\JTp ur

THO=90 PSID0=50 PHIDO= 0 DT H=0 . 1 0 Spin Pr ecess . T h e t.a r ad/s r ad/ s d e g re e s 90.00 8 8. 0 0 86.00 84.00 82.00 80.00 78.00 76.00 ?4.00 72.00 70.00 58.00 56.00 5 4. 0 0 62.00 60.00 s8.00 ' 55.00 54.00 52,00 50.00 48.00 46.00 44.11

trrEtlli-in Period

0.00 -a .2t -0.41 -o.62 -0.83 -1.05 -L.28 -1.51 -L75 -2 .Ot -2 .28 -2.56 -2.87 -3.19 -3.54 -3.92 -4 .33 -4 .79 -5.28 -5.83 -6.44 -7.13 -7 .90 -8 .72

= .0.668Jl_

50.00 50.01 50.03 50.06 50.12 50.18 50.27 50.37 50.48 50.62 50.78 50.96 51.17 51.40 sL.66 '5r.96 52.30 52,68 53.11 53.59 54.t4 54,77 ' 55.49 56.26

TlI0-90 PSfD0-50 PIIIDOI 5 EIIH.O.10 Theta Spln Precess. degrees rad/e rad/e 90,00 5.00 50.00 89.00 { .90 {9.93 8 6. 0 0 49.68 { .61 84.00 {.{3 {9.51 ' 1 9 .{ 1 tr.26 8 2. 0 0 80.00 {.10 19,29 ?8.00 49.18 3.95 75.00 {9.08 3.90 74.00 3.66 48.99 72.OO 3.52 {8 . 91 70.00 3.38 {8.94 68.00 3.25 {8.?g r l 8. ? 3 6 6, 0 0 3.12 54.00 3.00 48.69 6 2. 0 0 2 . A 7 4 8. 6 5 60.00 2,75 {8 .63 t18.51 58.00 2.62 5 6. 0 0 2.49 4 8. 6 1 ' 18.61 5 4. 0 0 2.36 5 2. d 0 2.22 4 8, 5 3 50.00 {8.66 2.08 48.00 1.93 {8.?1 tLB 4 5. 0 0 .?? L.77 44.00 1.59 48.95 42.00 1.{O 48.96 40.00 L.20 {9.08 3 8. 0 0 0 9 6 4 9. 2 4 . '0. ?o 4 9, 4 4 36.00 3 4. 0 0 0.39 49,67 3 2. 0 0 0.04 4 9. 9 7 30.00 -0.38 50.33 28,0A -0.88 50.78 26.00' -1.'{9 51.34 -2 .26 5 2. 0 6 2 4. 0 0 5 3. 0 0 2 2, 0 O - 3 . 2 1 -{ .51 20.00 54.24 6 18.00 55,92 ,23 -8,62 58.28 16.00 14.00 -12.09 61.?3 1 2 .0 0 - L 7. 4 t 6?.06 10.00 -26.30 75.90 8.00 -42,6L 92,t9 6 . 0 0 - 7 7 . 8 2 L 2 7, 4 0 5 . 62 -_g9l-q3 13I . 60 Thet,a min - 5 ,6? 4egEggF Pertbd .. 0.542 g l-

(rL)

/t L' \)

('f,u)

(tii)

18.C0 continued

TH0= 90 PSID O- 0 PI{ID 0'- 5 DTll=0. L0 Theca Preceeg. Spin degreee rad/e rad/s 90.00 5.00 0.00 -0.17 88.00 5.0x, -0.35 86.00 5.02 -0.53 8 ' 1. 0 0 5,06 -0. ?1 82.00 5.10 -0.90 80.00 5.15 -i. og ?9.00 5.23 -1.28 76.OO 5.31 -1.49 74.00 5.41 -1.?1 72.00 5.53 -1.94 70.00 5.56 -2.19 68.00 5.92 -2.44 55.00 5 .99 -2,7I 64.00 6.19 -3.01 62.00 6.41 -3.33 50.00 6.67 -3.68 s8.00 6.95 -4.07 56.00 7.27 -4.49 54.00 7.64 -4.95 52.00 9.05 -5. {g 50.00 I .52 -6.06 48.00 9.05 -6 .7X 45.00 9.66 -7 .45 44.00 10.36 -8.30 42.00 11.17 -9 .27 40.00 12.10 - 10.26 38. 23 13.06 TheLa min ' lg*.?_sleglcg.g = -Q..-6.0J_^g Period

T H 0- 90 'D[H.O.

PSfD 0 rt O

-r.ih;;;-1 0

degreea

sprn p$ -;

rai/o

90.00

S.00

88.00 85.00

{ .96 | .g{

90.00 ?8.00

{.9{ 4 .97

76.00 ? t l. 0 0 ?z.oa ?0.00

5.01 5 .06 5.13 5.21

.9., g:n E.l g.

8{.00 ' {, gr 82.00 { .93

5.30 5 .42

gii ?. gl

6 4. 0 0 6 2. 0 0 60.00 58.00 5 6. 0 0

5.55 5.71 5.88 . 5.09 6.32

7,5i, 7.?,'A 7. 6.?g: 6. 4 7

54.00 52.00 50 .00 {9.00 46 . O0 {{.00

6.58 6.89 7:23 ?.63 g .08. 8.61 .

6.1$, 5.?f # 5.35 ",!4 | . g o h- q { .39 3,9!.',l

. 68.00 66.00

4 2 .0 0 40.00 38.00

9.21 9.92 10.?5

35.00 34.00

LL.72 L2.8?

0.52 -0.6?

1 4. 2 5 32,00 3 0 . 0 0 . 1 5 .9 3 L 7. ? L 2 8, 2 3 rn;;fmt.@ Perlod

9.655

3.15 :: 2.10 1.53, j l

-2.09 ,: -3.79 '"'i :5.60

(tru

T H 0- 60 PSID O - 0 PH fD O. 5 D T H = O .L 0 gpi n Pr ec es s . T heta rad/e rad/e degreea

T 110''60 Pgl PQ rS O P H f D 0 - 5 D T H = O .1 0 Precegs. Sptn Theta rad/e rad/a degrees

50.00 0.00 5,00 -0.26 58.0d 5.20 -0.5{ 56.00 5.43 -0.84 54.00 5.69 -1.18 5.98 52.00 -1.56 6.32 50.00 -1.98 49.00 6,70 'l -2.46 45.00 .L4 -2 ;99 7 .64 44 . 00 -3 .61 I .22 {2. 00 - g.r?? {*llt 40.33 = g-0-r_deglges Theta mln - .Q,-O€L€. Period

5.00 50.00 F0.00 19.8? 4.96 5 8. 0 0 19 .75 4 .92 56.00 .l.90 49.62 54. OO 49.49 4.89 52.00 { .89 49.36 50.00 49,22 4.90 48.00 49.08 4.92 46.00 {8 .93 4 .96 {{ .00 48,77 42.00 5.02 48.59 5.10 40.o0 48..[0 5.20 38.00 48.19 5.33 36.00 47.95 5. {9 34.00 47.67 5.70 32.00 17.31 5 .96 30.00 46.95 6 .?8 28.00 46.48 6.70 26,OO 45.90 ? ,23 24.00 45.16 7 .92 22.00 {4.19 8.84 20.00 42.90 10.10 18.00 .11,10 1 1 . 8 6 15.00 38 . {9 1 , 1. , 1 { 1{ . 00 34,47 1 8. 4 3 12.00 27.92 10.00 , 25.06 15.59 3 7. 2 7 8.00 -J.0J5 6iJo 6..-01 Theta min - _6. 0.1 degneeF - -9J.39_€ Period

Chapter 19 Computer Problems 19.C1 By expanding the integrand in Eq. (19.19) into a series of even p owers of sin ϕ and integrating, it can be shown that the period of a simple pendulum of length l can be approximated by the expression

τn = 2π

2 2 2    l   1  2  1× 3  4  1× 3× 5  6   1+   c +  + +  c c      2 × 4  g   2  2 × 4 × 6 

where c = sin 12 θm and θm is the amplitude of the oscillations. Use computational software to calculate the sum of the series in brackets, using successively 1, 2, 4, 8, and 16 terms, for values of θm from 30 to 120° using 30° increments. 19.C2 The force-deflection equation for a class of nonlinear springs fixed at one end is F = 5x1/n, where F is the magnitude, expressed in newtons, of the force applied at the other end of the spring and x is the deflection expressed in meters. Knowing that a block of mass m is suspended from the spring and is given a small downward displacement from its equilibrium position, use computational software to calculate and plot the frequency of vibration of the block for values of m equal to 0.2, 0.6, and 1.0 kg and values of n from 1 to 2. Assume that the slope of the force-deflection curve at the point corresponding to F = mg can be used as an equivalent spring constant. 19.C3 A machine element supported by springs and connected to a d ashpot is subjected to a periodic force of magnitude P = Pm sin ωf t. The trans missibility Tm of the system is defined as the ratio Fm /Pm of the maximum value Fm of the fluctuating periodic force transmitted to the foundation to the maximum value Pm of the periodic force applied to the machine element. Use computational software to calculate and plot the value of Tm for frequency ratios ωf /ωn equal to 0.8, 1.4, and 2.0 and for damping factors c/cc equal to 0, 1, and 2. (Hint: Use the formula given in Prob. 19.147.) P = Pm sin ωf t

Fig. P19.C3

21_bee77102_Ch19_p015-016.indd 15

13/03/18 9:27 am

19.C4 A 15-kg motor is supported by four springs, each of constant 60 kN/m. The unbalance of the motor is equivalent to a mass of 20 g located 125 mm from the axis of rotation. Knowing that the motor is constrained to move vertically, use computational software to calculate and plot the amplitude of the vibration and the maximum acceleration of the motor for motor speeds of 1000 to 2500 rpm. 19.C5 Solve Prob. 19.C4, assuming that a dashpot having a coefficient of damping c = 2.5 kN·s/m has been connected to the motor base and to the ground. 19.C6 A small trailer and its load have a total mass of 250 kg. The trailer is supported by two springs, each of constant 10 kN/m, and is pulled over a road, the surface of which can be approximated by a sine curve with an amplitude of 40 mm and a wave length of 5 m (i.e., the distance between successive crests is 5 m and the vertical distance from crest to trough is 80 mm). (a) Neglecting the mass of the wheels and assuming that the wheels stay in contact with the ground, use computational software to calculate and plot the amplitude of the vibration and the maximum vertical acceleration of the trailer for speeds of 10 to 80 km/h. (b) Determine the range of values of the speed of the trailer for which the wheels will lose contact with the ground.

Fig. P19.C6

21_bee77102_Ch19_p015-016.indd 16

13/03/18 9:27 am

€rygNi PCQTODOF A SInPLF FEilDUtU},t OF LSNGTH O' I5

cn=';W'['*tr,lt;.(H}.n*qffi Y.'*.. d

hND 9un tS T$+g

N$eeE e:= stN Le* D6 NHPCITU

Eun:

T F f € r u H O F T r + € S e Q . l E Sl N B e A c K € r S o S ( N G Succ€'S6,t$$l-?l.?.,4,8 ANU 16 TGB}ISFoE VAuugs otr €h^ tr€.ol't 3ooto lzoo ustNc 3Oo f NC gF H e NT5, eX Peg:SS [gSUCf S ur tT H l cA t tT.-fj-99JLq=$FlyE-_E 16_Nl_E

P € . w e r T q .G t v € - N S g e t € s ! N r e e H q

l/l= t.1-3,.'.

,Gf4$,i:fu)?-"'1 /rt r'*[r L,{"::: r=="f lve NnY

coPPa?E

f,s rakcLoti

tt 't:

u=LtrWJ

h= 2l

B=V+

,,=3;

B - - t ,+

47 E*n

67E?

7n*AunntTr

I x#-)'I qEovf

/,&a 7rc au*Tr

)s 7EC&tr6JtN

(#Tr)

/s nt*ondr

nNlr| oetroTtQ gf

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