Online Physics Tutor In Virginia-AP Physics (C) 2021 Mechanics Paper Solution

2021 AP® Physics C: Mechanics Free-Response Questions Set 1 One-1

F

Que-2

m

=

-=

-

(a)

mp3 y

aNc>=

astae

mgh=↓Kz?

KE+ PF

3

-

=

I

faucet

=

S

consvast

seem

sedm

am S

ADVANCED PLACEMENT PHYSICS C TABLE OF INFORMATION CONSTANTS AND CONVERSION FACTORS

Neutron mass, mn

27

kg

10

27

kg

9.11 10

31

1.67

Electron mass, me Avogadro’s number, N 0

1

8.31 J (mol K)

Boltzmann’s constant, k B

1.38

10

23

e

1.60

10

19

C

1 electron volt, 1 eV

1.60

10

19

J

Electron charge magnitude,

kg

10 23 mol

6.02

R

Universal gas constant,

10

1.67

Proton mass, m p

Speed of light, Universal gravitational constant, Acceleration due to gravity at Earth’s surface,

c

3.00

10 m s

G

6.67

10

11

1u

1.66

10

27

kg

931 MeV c 2

Planck’s constant,

h

6.63

10

34

Js

4.14

10

25

8.85 10

12

e0

Vacuum permittivity, Coulomb’s law constant, k Vacuum permeability, 1 atmosphere pressure,

PREFIXES Factor Prefix Symbol

mole, hertz, newton, pascal, joule,

1 4 pe0 m0

Magnetic constant, k

m kg s A K

kg 2

9.8 m s2

g

1 unified atomic mass unit,

meter, kilogram, second, ampere, kelvin,

N m2

J K

hc

UNIT SYMBOLS

8

m0 4 p

1 atm mol Hz N Pa J

1.99

Jm C

2

1.24 Nm

109 N m 2

9.0

7

(T m) A

1 10

7

(T m) A

1.0

105 N m 2

watt, coulomb, volt, ohm, henry,

W C V W H

15

eV s

3

10 eV nm

2

C2

10

4p

10

1.0

105 Pa

farad, tesla, degree Celsius, electron volt,

F T C eV

VALUES OF TRIGONOMETRIC FUNCTIONS FOR COMMON ANGLES q 0 30 37 45 53 60 90

109

giga

G

sin q

0

12

35

2 2

45

3 2

1

106

mega

M

cos q

1

3 2

45

2 2

35

12

0

103

kilo

k

tan q

0

3 3

34

1

43

3

10 2

centi

c

10 3

milli

m

6

micro

m

10 9

nano

n

12

pico

p

10

10

The following assumptions are used in this exam.

I. The frame of reference of any problem is inertial unless otherwise stated.

II. The direction of current is the direction in which positive charges would drift.

III. The electric potential is zero at an infinite distance from an isolated point charge.

IV. All batteries and meters are ideal unless otherwise stated.

V. Edge effects for the electric field of a parallel plate capacitor are negligible unless otherwise stated.

ADVANCED PLACEMENT PHYSICS C EQUATIONS MECHANICS

Ãx

x

ax t

Ãx 0

x0

1 2 at 2 x

Ãx 0 t

Ãx2

Ãx 0 2

2ax x

a

F m

Fnet m

F

dp dt

J

p

F dt

m FN

DE

W

K

1 2 mà 2

P

dE dt

P

F v

DUg

Ã2 r

r

F t net I

I 2

I

r dm

mr

kDx

Us

1 k Dx 2

x

xmax cos( wt

T

2p w

p

Iw

2p

Tp

2p

1 2 Iw 2

w

w0

at

q

q0

w0 t

FG UG

1 2 at 2

2

1 f

Ts

rw r

acceleration energy force frequency height rotational inertia impulse kinetic energy spring constant length angular momentum mass power momentum radius or distance period time potential energy velocity or speed work done on a system position coefficient of friction angle torque angular speed angular acceleration phase angle

Fs

2

mi xi mi

x cm

K

w2r

t

a

L

F dr

mg Dh

ac

Ã

Dp

mv

Ff

t

x0

a = E= F = f = h = I = J = K = k = = L = m= P = p = r = T = t = U= v = W= x = m = q = t = w = a = f =

ELECTRICITY AND MAGNETISM

m k

FE q

E

DV

DV

Q C

UC

r2

Gm1m2 r

i

i

= = = = = = = = = = = =

N =

qi ri

= = = = = = = V= v = r = F = k = P Q q R r t U

1 q1q2 4pe0 r

k e0 A d

1 Cs

Ci

1 C i i

area magnetic field capacitance distance electric field emf force current current density inductance length number of loops of wire per unit length number of charge carriers per unit volume power charge point charge resistance radius or distance time potential or stored energy electric potential velocity or speed resistivity flux dielectric constant

FM

qv

B d

dQ dt

1 QDV 2

1 C DV 2

2

B m0 I

dB

m0 I d r 4p r 2

R

r A

F

I d

E

rJ

Bs

m0 nI

I

Nevd A

FB

I

DV R

e

E d

e

L

UL

1 2 LI 2

g

Gm1m2

1 4pe0

qV

Cp

n

E dr

UE

C

F I J L

dV dx

Ex

V

e

Q e0

E dA

I f

A B C d E

1 q1q2 4pe0 r 2

FE

Rs 1 Rp

P

Ri

i

i

I DV

1 Ri

B

B dA

dI dt

d FB dt

ADVANCED PLACEMENT PHYSICS C EQUATIONS GEOMETRY AND TRIGONOMETRY Rectangle A bh Triangle

A

1 bh 2

Circle

A

pr2

C

2p r

s rq Rectangular Solid V wh Cylinder 2

V

pr 

S

2p r 

CALCULUS

A = area C = circumference V = volume S = surface area b = base h = height  = length w = width r = radius s = arc length q = angle

df dx

d f du du dx

d n x dx

nx n

d ax e dx

aeax 1 x

d ln ax dx d sin ax dx

a cos ax

d cos ax dx

s

x n dx

r

q 2p r 2

eax dx

Sphere

dx

a sin ax 1

n

ln x

V

x

S

4p r 2

cos ax dx sin ax dx

Right Triangle

b2

c2

sin q

a c

cos q

b c

tan q

a b

a

c q

a 90°

b

xn 1, n

1

1 ax e a

4 3 pr 3

a2

1

a 1 sin ax a 1 cos ax a

VECTOR PRODUCTS   A B AB cos q   A B AB sin q

1

Begin your response to QUESTION 1 on this page.

PHYSICS C: MECHANICS SECTION II Time—45 minutes 3 Questions Directions: Answer all three questions. The suggested time is about 15 minutes for answering each of the questions, which are worth 15 points each. The parts within a question may not have equal weight. Show all your work in this booklet in the spaces provided after each part.

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1. A 0.50 kg fan cart is placed on a level, horizontal track of negligible friction, as shown. The fan is turned on, and the fan cart is released from rest and moves to the right. The cart travels along the horizontal track and then down an incline. Motion detector 1 measures the acceleration a of the cart from time t *A 0 to t h 3 s.

Att A 3 s, the cart makes a smooth transition to the incline, and motion detector 2 measures the acceleration of the cart after t B 3 s. The fan exerts the same magnitude of force on the cart during the entire motion. The graphs below show a as functions of t. For each motion detector, the positive direction is away from the detector.

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Continue your response to QUESTION 1 on this page. (a) On the dots below that represent the cart at t wo different locations, draw and label the forces (not components) that act on the cart at each location. Each force must be represented by a distinct arrow starting on, and pointing away from, the dot.

FH

Fx

Fan

Fam

mg

mg

(b) Calculate the magnitude of the net force exerted on the fan cart when it is on the horizontal track.

Aten

(Meart)

=

(a)

(0.50) (08)

-

=

0.20N

(c) Calculate the angle ⑤q of the incline.

fantmg

sino-man

mgsinO= Man-fan 30= M92-ffan

mg 38

=

(0.50) (2.4)

-

=

9.40

Yngsio

(050) (0.0)

(0.50) (9.0) O

mycosef

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(d) Suppose careful measurement determines the angle of the incline to be 3° larger than that calculated in part (c). Consider the following explanation.

“The scale used to measure the mass of the fan cart was not calibrated properly before the measurement, and this could account for the obser ved difference in the angle.”

Does the explanation sufficiently account for the observed discrepancy? _____ Yes _____ No Justify your answer.

From the answer (c)

sino-man-ftan_Ma--M9,

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mg

to

Hence

the cart cancels

equation used the ayle of the

of

mass

out

to

the

find

Incline

Continue your response to QUESTION 1 on this page.

The experiment is repeated for several trials, each with a different angle for the incline. The acceleration of the cart down the incline is measured for each angle. The graph below shows the plot of the acceleration a of the cart as a function of the sine of the angle sin ⑧ q.

(e) i. Draw a best-fit line for the data. ii. Using the straight line, calculate an experimental value for the acceleration due to gravity g. (f) If the cart were replaced with a second cart of mass 1.0 kg that has a fan that exerts the same magnitude of force as the original fan, explain how the graph given in part (e) would change.

scope: ma: a

=

teflon= ffan

+mg

1052M52

5i8

gSio+an m

3

=

mx

(Slope (Sing + [Intercept =8=1052M52

+b=

Slope

Begin your response to QUESTION 2 on this page.

2. A block of mass m starts from rest at point A and travels with negligible friction through the loop onto a mg . All motion of ·m

horizontal surface, where the block makes contact with a spring of spring constant k

the spring is in the horizontal direction. Point C is the highest point on the loop, and point B is the rightmost point on the loop. Express all algebraic answers in terms of m, h, R, and physical constants, as appropriate. (a) On the dot below, which represents the block, draw an arrow that represents the direction of the acceleration of the block at point B in the figure above. The arrow must start on and point away from the dot.

Justify your answer

Rosentripetal component owards

-

adownward

once

=

the

g

-

grtas"

centre

aa

↓. are-

B

.

.

.

g

(b) i. Derive an expression for the speed v of the block at point B. conservation

Adbly

Point A 2

ITE)

A

=

at

point B.

(TE) B

mgR+ mos MOB => UB 25Ch-R)

mg2+mc0 m5 (2-R)

of energy

=

=

=

FC

ii. Derive an expression for the magnitude of the net force F on the block at point B.

FNex

=

-

-

(mg)" + (FTTh might mag

=mightgFeT2

FNGTo <

W

B

mg

Continue your response to QUESTION 2 on this page. (c) In terms of R, derive an expression for the minimum height hmin Wh necessary for the block to maintain contact with the track through point C.

CTE)

A

=

CTE) c

m=mg (24)

mgeAt

+

the

Mg-= Mg(2K) Me, +

g2 UC

=

18R= R

=

=

+

och

28 (n-2R)

I

-

R

8(2R)

2

B(n-20)

2

Ch-2R)

2h-4Q

2h=5R =

=

50+2.5R

For

point o, minimum to complete the

velocity

circular motor,

N=O

mg + N-moc

for

which

put N=o

Wo =

9R

(d) It is determined that h ⑭ 0.30 m and R * 0.10 m . If the block is released from a height greater than that found in part (c), what would be the maximum compression Ar xMAX of the spring? ·

mg2=1K Ceme 2mg2

=

k

mac

rmae=

=>

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<mg2 K.

anger

-

a

-

(4)

=>

=

(0.30) 2010)

0.35 m

(e) A graph of the maximum compression of the spring as a function of height is shown below. The height hmin is the height calculated in part (c).

i. Explain why section I appears as a horizontal line segment on the horizontal axis.

Block does not make it the

It

look can

at

this

height

not reach

through so

near to

spring ii. Explain the reason for the shape of section II on the graph.

high-ket a

ace

"

stence

barebollc

Begin your response to QUESTION 3 on this page.

↑

3. A triangular rod of length L and mass M has a nonuniform linear mass density given by the equation * l gx2, where g 3M and x is the distance from point P at the left end of the rod.

L3 (a) Using integral calculus, show that the rotational inertia I of the rod about an axis perpendicular to the page and through point P is* 3 ML2. 5

AT

I

dF= =

S

dI

I

=

dice dice

*

So

L

Ucedix"=Ucax

05 of(15-05] P 325 --M =

=

=

-

-2 -

5

dee

am =X -

are

am X de =

(b) Determine the horizontal location of the center of mass of the rod relative to point P. Express your answer in terms of L.

rem=Sance I

am

am L

(xceace

=

->

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The rod is released from rest in the position shown, and the rod begins to rotate about a horizontal axis perpendicular to the page and through point P. (d) On the axes below, sketch graphs of the magnitude of the net torque t on the rod and the angular speed w of the rod as functions of time t from the time the rod is released until the time its center of mass reaches its lowest point.

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