PROBLEM (1) (20 points) m A block of mass m is connected to a block of mass by a massless string that 2 m passes over a massles and frictionless pulley. The block is connected to an 2
g
ideal vertical spring with spring constant k ! which is fixed to the floor. The spring
m
is initially unstretched and the incline is frictionless. The block m is pulled down ! a distance L stretching the spring the same amount. The block m is then released 1 3 from rest. Assume sin " = and cos " = . Express your answers in terms of 2 2
m 2 k
!
some or all of the given quantities and related constants as needed. (a) (6 pts) Draw a free-body diagram for each block at the moment the block m is released from rest. !
" n
! T
! T
m 2
m
! Fs
!
! m! Fg = g 2
! ! F g = mg
(b) (7 pts) Find the speed of each block when they move the distance L and the spring is unstretched again.
K i + U si + U gi = K f + U sf + U gf
Alternative solution: "K + "U g + "U s = 0
3mv 2f $1 1 m 2' "K = K f # K i = & mv 2f + v f )# 0 = %2 2 2 ( 4 ! % m ( "U g = U gf # U gi = ( mgL sin $ # 0) + ' 0 # gL * & 2 )
Take U g = 0 for a block at the lowest position it has. where K i = 0 and select U gi = 0
! !
! !
1 2 1 1m 2 m kL = mv 2f + v f + mgL sin " # gL 2 2 2 2 2 ! m mgL mgL 2 ! "U g = mgL sin # $ gL = $ =0 !m 3mv f 1 2 1 2 m m 2 2 2 = kL " mgL sin # + gL = kL " gL + gL 4 2 2 2 2 2 1 ! "U s = U gf # U gi = 0 # kL2 ! 2 2 3mv f 1 2 2k 2 2kL ---> v f = L = kL ---> v 2f = 2 2 3m 4 2 3m ! 3mv f 1 2 3mv f 1 2 2k " kL = 0 ---> = kL ---> v f = L 3m 4 2 4 2 ! (c) (7 pts) After!moving the distance ! L the block m continues to move up the incline through a distance d before its speed becomes zero. Find the distance d. Assume that the tension ! in the string becomes ! zero after the blocks!move the distance L and the spring is unstretched again.
1 2kL2 mgd m = 2 3m 2
For block m:
K i + U gi = K f + U gf ; K f = 0 and select U gi = 0
d=
2
1 mgd 2kL where v i2 = mv i2 = mgd sin " = 2 2 3m ! !
!
2kL2 3mg
!
! !
Phys 105 Final Examination
!
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Friday, 12-August 2011
PROBLEM (2) (20 points) m, r A uniform solid sphere of mass m and radius r rolls without
y R
slipping on a circular track of radius R such that r << R .The sphere starts rolling from rest at a height R above the bottom
!
g
of the track. Express your answers in terms of some or all of ! the given quantities and related constants as needed. (For 2 solid sphere: I CM = mr 2 ) 5
R
x Q
P
(a) (8 pts) Find " , the angular speed of the sphere about its center when it passes through the bottom of the track.
! Rolling without slipping ---> Friction force does no work ! K i + U i = K f + U f where K i = 0 and select U f = 0 1 1 2 mvCM + I CM " 2 2 2 2 ! ! 2mr where I CM = and vCM = r" 5
# mr 2 mr 2 & 1 1 2mr 2 2 mr 2" 2 + " = " 2% + ( 2 2 5 5 ' $ 2 # 7mr 2 & 10gR mgR = " 2 % ( ---> " 2 = 7r 2 $ 10 '
mgR =
mgR =
!
!
"= !
! !
!
1 10gR r 7 !
!
(b) (6 pts) The following figure shows the sphere at the instant it passes through the bottom of the track. The sphere is in contact with the track at point P. In the given figure show all the forces acting on the sphere. Label the forces and fill in the given table as needed using your labels. Indicate the point of application of each force. The first row is filled for you as an example.
! n CM
" fs
P " " F g = mg
!
! ! In the above figure, the forces mg and n shifted slightly ! from their common vertical line of action to make them ! clearly visible. ! !
Force | Point of application ------------------------------------â&#x20AC;&#x201C;-------------------------! | CM mg ----------------------------------------------------------------! | P fs ----------------------------------------------------------------! n | P ----------------------------------------------------------------| -----------------------------------------------------------------
(c) (6 pts) Find the potential energy of the sphere relative to the bottom of the track when it leaves the track at point Q after passing through an angle θ measured from the vertical as shown in the figure.
U = mgh where h = R " R cos # = R(1" cos # )
U = mgR(1" cos # ) !
! !
Phys 105 Final Examination
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Friday, 12-August 2011
PROBLEM (3) (20 points)
v
A bullet of mass m1 and velocity v strikes and sticks to the
m1
edge of a horizontal, stationary, uniform solid disk of radius
R
R and mass m2, pivoted about a fixed frictionless axle
Pivot
perpendicular to the page through its center O, as shown in
O
the figure. Express your answers in terms of some or all of
m2
the given quantities and related constants as needed. (For any 1 uniform solid disk with mass M and radius R: I CM = MR 2 ) 2 (a) (4 pts) Find the angular momentum of the system before the collision about an axis through O.
! ! ! ! L = r " p ---> L = rpsin "
Li = Rm1v
For the system: Li = Rm1v sin 90 !
!
!
!
! (b) (4 pts) Find the moment of inertia of the system about an axis through O after the bullet sticks to the disk.
I = I bullet + I disk = m1R 2 +
1 m2 R 2 2
" " 2m + m2 % 2 m % I = $ m1 + 2 'R 2 = $ 1 'R # # & 2 & 2
!
! (c) (4 pts) If the angular speed of the system after the collision is Ď&#x2030;, find the angular momentum of the system after the collision.
! ! L = I" or L = I"
!
L f = I" --->
" " 2m + m2 % 2 m % L f = $ m1 + 2 'R 2( = $ 1 'R ( # # & 2 & 2
!
! ! (d) (4 pts) Find Ď&#x2030;, the angular speed of the system after the collision in terms of given quantities.
" 2m + m 2 % 2 Li = L f ---> Rm1v = $ 1 'R ( # & 2
!
"=
m1v 2m1v = # m2 & 2m1 + m2 ) R ( % m1 + (R $ 2 '
!
(e) (4 pts) Find the kinetic energy of the system after the collision. !
,2 1 2 1 )# 2m1 + m2 & 2 ,) 2m1v K = I" = +% . (R .+ ' -* ( 2m1 + m 2 ) R 2 2 *$ 2 K=
!
K=
( + 1 (" 2m1 + m2 % 2 +* 4m12 v 2 'R *$ 2 2 & ,*) ( 2m1 + m2 ) R -, 2 )# 2
m12 v 2 2m1 + m2
!
! Phys 105 Final Examination
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Friday, 12-August 2011
PROBLEM (4) (20 points) A cylinder is fixed at point A to the floor. A straight uniform rod of mass m
y
and length L is supported by the floor at point P and by the cylinder at point Q 3L such that PQ = , as shown in the figure. The rod lies in the vertical xy 4
g
Rod
x
S Q
Cylinder
plane and the axis of the cylinder is perpendicular to this plane. There is no
!
friction between the rod and the cylinder, but there is friction between the rod ! and the floor. The rod is in equilibrium in the position shown. Assume 3 4 sin " = and cos " = . Express your answers in terms of some or all of the 5 5
Floor
P
A
given quantities and related constants as needed. (a) (8 pts) Draw a free-body diagram for the rod. Label the forces acting on the rod and fill in the given table as needed using
!
your labels. Indicate the point of application of each force. The first row is filled for you as an example.
! n2 ! n1
CM
P
! ! fs
Force | Point of application ------------------------------------â&#x20AC;&#x201C;-------------------------! | CM of the rod m1g ----------------------------------------------------------------! | P n1 ----------------------------------------------------------------! | Q n2 ----------------------------------------------------------------! | P fs ----------------------------------------------------------------| -----------------------------------------------------------------
S Q ! ! F g = mg
! ! !
!
(b) (6 pts) Find the magnitude of the force exerted by the cylinder on the rod. About point P:
n2 =
!
# " ext = 0 ---> # " CW = # " CCW
n2 =
#L & # 3L & mg% cos " ( = n 2 % ( $2 ' $ 4 ' !
!
2mg cos " 2mg # 4 & = % ( 3 3 $ 5' 8mg 15
! !
! (c) (6 pts) Find the normal force and the friction force exerted by the floor on the rod. State their directions.
!
!
" Fext = 0 ---> " Fy = 0 ---> n1 + n 2 cos " # mg = 0
" Fext = 0 ---> " Fx = 0 ---> f s " n 2 sin # = 0
!
# 8mg &# 3 & f s = n 2 sin " = % (% ( $ 15 '$ 5 ' ! ! ! 8mg fs = pointing to right (or in + x direction) 25 ! !
!
$ 8mg '$ 4 ' $ 32 ' n1 = mg " n 2 cos # = mg " & )& ) = mg&1" ) % 15 (% 5 ( % 75 ( ! ! 43mg n1 = pointing vertically upward (or in + y direction) 75
!
Phys 105 Final Examination
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Friday, 12-August 2011
PROBLEM (5) (20 points) Represent the mass and radius of the Earth as M e and Re , respectively. A satellite of mass ms , initially on the surface of the Earth, is placed into Earth orbit at r = 2Re . Express your answers in terms of some or all of the given quantities and related constants as needed. (a) (5 pts) Assuming a circular orbit, find the satellite's orbital speed v. ! For satellite:
2Re G
v Re
! Fg
Me
M e ms
( 2Re )
2
= ms
v2 2Re
!
ms
v= !
!
!
!
" F = msa = Fg --->
v2 =
GM e 2Re
GM e 2Re !
!
(b) (5 pts) Find the time interval ∆t for the satellite to complete three revolutions around the Earth.
"t = 3T where 2" ( 2Re ) 4 "Re 2Re3 T = period of satellite = = = 4" v GM e GM e ! 2Re
"t = 12 #
2Re3 GM e
! !
(c) (10 pts) Find ∆E, the minimum energy input necessary to place the satellite in orbit. Ignore air resistance but include the effect of the Earth's daily rotation. Let T e represent the period of the Earth's daily rotation and use T e in your answer.
(
)
"E = E f # Ei = K f + U f # ( K i + U i )
"E =
where K i = kinetic energy due to Earth’s daily rotation
! !
!
2"Re 1 K i = msv i2 where v i = Te 2
"E =
GM e ms GM e ms 2 $ 2 msRe2 GM e ms # # + 4Re 2Re Re T e2
GM e ms # 2GM e m s + 4GM e ms 2$ 2 msRe2 # 4Re T e2
!
2
1 # 2 "Re & 2" 2 msRe2 GM e m s K i = ms % and U i = " ( = 2 $ Te ' Re ! T e2 ! ! GM e m s 1 1 " GM e % GM e ms K f = msv 2f = ms $ and U f = " '= 2 2 # 2Re & 4Re 2Re !
"E =
3GM e m s 2$ 2 msRe2 # 4Re T e2
!
!
!
Phys 105 Final Examination
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Friday, 12-August 2011