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AP Physics C: Mechanics Full-length practice exam 3 — Free with Answer Explanations | Test Practice Hub

AP Physics C: Mechanics Full-length practice exam 3

35 questions

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Full-length practice exam 3

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Question 1

A satellite in circular orbit of radius $r$ around Earth (mass $M_{E}$ ) has total mechanical energy $E$ . Which expression is correct?

E = - \frac{G M _{E} m}{r}

E = \frac{G M _{E} m}{2 r}

E = - \frac{G M _{E} m}{2 r}

E = 0

Question 2

Using the work integral, derive the work done by a spring force $F_{s} = - k x$ as a block is displaced from $x = 0$ to $x = A$ .

W = - k A

W = - \frac{1}{2} k A^{2}

W = \frac{1}{2} k A^{2}

W = k A^{2}

Question 3

A torsional pendulum has restoring torque $τ = - κ θ$ where $κ$ is the torsion constant and $I$ is the moment of inertia. What is the period of oscillation?

T = 2 π \frac{κ}{I}

T = 2 π \frac{I}{κ}

T = \frac{1}{2 π} \frac{I}{κ}

T = 2 π I κ

Question 4

A uniform solid disk of mass $M$ and radius $R$ has moment of inertia about its central axis $I = \frac{1}{2} M R^{2}$ . A net torque $τ = 4$ N·m is applied to a disk with $M = 2$ kg and $R = 1$ m. What is the angular acceleration?

2

rad/s²B.

4

rad/s²C.

8

rad/s²D.

1

rad/s²

Question 5

A uniform rod ( $M$ , $L$ ) is held horizontal, pivoted at one end, and released. Using $τ = I α$ and torque from gravity, find the initial angular acceleration and the linear acceleration of the free end.

α = \frac{g}{L}

;

a_{t i p} = g

α = \frac{3 g}{2 L}

;

a_{t i p} = \frac{3 g}{2}

α = \frac{3 g}{L}

;

a_{t i p} = 3 g

α = \frac{g}{2 L}

;

a_{t i p} = \frac{g}{2}

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Question 6

The impulse–momentum theorem in integral form states $\int_{t_{1}}^{t_{2}} F (t) d t = Δ p$ . A force $F (t) = 6 t^{2}$ N acts on a 2 kg object from $t = 0$ to $t = 2$ s starting from rest. Find the final speed.

4

m/sB.

6

m/sC.

8

m/sD.

16

m/s

Question 7

The velocity of a particle is $v (t) = e^{- t}$ m/s for $t \geq 0$ . What is the total distance traveled from $t = 0$ to $t \to \infty$ ?

0

mB.

\frac{1}{e}

mC.

1

mD.

\infty

Question 8

Two particles with masses $m_{1} = 4$ kg at $x = 0$ and $m_{2} = 2$ kg at $x = 6$ m. Where is the center of mass?

3

mB.

2

mC.

4

mD.

1

Question 9

Apply energy conservation to derive the orbital speed of a satellite in a circular orbit of radius $r$ around a planet of mass $M$ . What is $v$ ?

v = \frac{2 π r}{T}

onlyB.

v = \frac{GM}{2 r}

v = \frac{GM}{r}

v = \frac{GM}{r ^{2}}

Question 10

A conservative force on a particle is $F (x) = - \frac{d U}{d x}$ where $U (x) = 2 x^{3} - x^{2}$ J. Find the force at $x = 1$ m.

- 4

NB.

4

NC.

- 2

ND.

1

J (wrong unit)

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Question 11

In simple harmonic motion, a particle's position is $x (t) = 0.3 cos (5 t)$ m. What is the maximum acceleration?

1.5

m/s²B.

5

m/s²C.

7.5

m/s²D.

25

m/s²

Question 12

A 2 kg ball moving at $5$ m/s to the right collides elastically with a 2 kg ball at rest. What are their velocities after the collision?

A.Both move at

2.5

m/sB.First stops; second moves at

5

m/sC.First moves at

5

m/s; second stopsD.First moves at

- 2.5

m/s; second at

7.5

m/s

Question 13

A spring–mass system ( $k = 50$ N/m, $m = 0.5$ kg) starts at rest at $x = 0.2$ m from equilibrium. Find the time at which the speed first equals half its maximum value. ( $ω = 10$ rad/s)

t = \frac{π}{30}

sB.

t = \frac{π}{20}

sC.

t = \frac{π}{60}

sD.

t = \frac{π}{15}

Question 14

A mass $m$ is attached to two identical springs (each constant $k$ ) in parallel (both springs pull on the mass simultaneously). What is the effective spring constant and resulting oscillation frequency?

k_{e f f} = k /2

ω = k / (2 m)

k_{e f f} = 2 k

ω = 2 k / m

k_{e f f} = k

ω = k / m

k_{e f f} = 2 k

ω = k / (2 m)

Question 15

A rocket of initial mass $M_{0}$ expels exhaust at speed $u$ relative to the rocket. Applying Newton's second law to variable mass, what is the thrust force?

F_{t h r u s t} = u \frac{d M}{d t}

(positive)B.

F_{t h r u s t} = - u \frac{d M}{d t}

F_{t h r u s t} = u M_{0}

F_{t h r u s t} = \frac{u}{d M / d t}

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Question 16

A 5 kg block is released from rest at the top of a frictionless ramp of height $h = 3$ m and then slides along a rough horizontal surface ( $μ_{k} = 0.2$ , $g = 10$ m/s²). How far does it travel on the rough surface before stopping?

3

mB.

7.5

mC.

15

mD.

30

Question 17

A particle's position components are $x (t) = v_{0 x} t$ and $y (t) = v_{0 y} t - \frac{1}{2} g t^{2}$ . Which expression correctly gives $y$ as a function of $x$ (the trajectory equation)?

y = \frac{v _{0 y}}{v _{0 x}} x - \frac{g}{v _{0 x}^{2}} x^{2}

y = \frac{v _{0 y}}{v _{0 x}} x - \frac{g}{2 v _{0 x}^{2}} x^{2}

y = \frac{v _{0 x}}{v _{0 y}} x - \frac{g}{2 v _{0 y}^{2}} x^{2}

y = \frac{v _{0 y}}{v _{0 x}} x + \frac{g}{2 v _{0 x}^{2}} x^{2}

Question 18

A 0.4 kg mass on a spring ( $k = 160$ N/m) is released from rest at $x = 0.05$ m from equilibrium. Find the total mechanical energy.

0.1

JB.

0.2

JC.

0.4

JD.

0.01

Question 19

A block of mass $m$ is pushed against a vertical wall by a horizontal force $F$ (with $F > μ_{s} m g$ ). What is the minimum $F$ needed to prevent sliding?

F_{min} = m g

F_{min} = μ_{s} m g

F_{min} = \frac{m g}{μ _{s}}

F_{min} = \frac{μ _{s}}{m g}

Question 20

A 2 kg block hangs from a rope attached to the ceiling. What is the tension in the rope? ( $g = 10$ m/s²)

10

NB.

20

NC.

2

ND.

0

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Question 21

Two ice skaters (60 kg and 40 kg) push off each other from rest. If the 40 kg skater moves at $3$ m/s to the right, what is the velocity of the 60 kg skater?

3

m/s to the leftB.

2

m/s to the leftC.

4.5

m/s to the leftD.

2

m/s to the right

Question 22

A ball is launched horizontally from a cliff 80 m high with speed $20$ m/s. Using $g = 10$ m/s², how far from the base of the cliff does it land?

40

mB.

80

mC.

160

mD.

20

Question 23

A constant force $F = 8$ N acts on a 2 kg object for 3 s. What is the change in velocity?

4

m/sB.

12

m/sC.

24

m/sD.

3

m/s

Question 24

A satellite is transferred from a circular orbit of radius $r_{1}$ to a circular orbit of radius $r_{2} > r_{1}$ via a Hohmann transfer (half-ellipse). During the transfer, which statement about speed is correct at the outer transfer point?

A.The satellite must decelerate to enter the ellipse and decelerate again to circularize at

r_{2}

.B.The satellite accelerates at

r_{1}

, coasts on the ellipse, and decelerates at

r_{2}

to reach the circular orbit speed.C.The satellite accelerates at

r_{1}

, coasts on the ellipse, then accelerates again at

r_{2}

to match the circular orbit speed.D.The satellite's speed at

r_{2}

on the ellipse equals the circular orbit speed at

r_{2}

, so no second burn is needed.

Question 25

Derive the gravitational potential energy by integrating the gravitational force. Starting from $F (r) = - \frac{GM m}{r ^{2}}$ (toward $M$ ), which integral correctly gives $U (r)$ with $U (\infty) = 0$ ?

U (r) = \int_{r}^{\infty} \frac{GM m}{r ^{'2}} d r^{'}

U (r) = - \int_{r}^{\infty} \frac{GM m}{r ^{'2}} d r^{'}

U (r) = \int_{0}^{r} \frac{GM m}{r ^{'2}} d r^{'}

U (r) = \int_{\infty}^{r} \frac{GM m}{r ^{'2}} d r^{'}

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Question 26

A figure skater with $I_{1} = 4$ kg·m² spins at $2$ rad/s and pulls in arms to reach $I_{2} = 1$ kg·m². What is the new angular velocity?

2

rad/sB.

4

rad/sC.

8

rad/sD.

0.5

rad/s

Question 27

A 0.4 kg mass on a spring ( $k = 160$ N/m) oscillates at $x = 0.05$ m amplitude. What is the speed at $x = 0.03$ m?

0.8

m/sB.

1.0

m/sC.

0.6

m/sD.

0.4

m/s

Question 28

A particle's position is $x (t) = A sin (ω t)$ . Using calculus, find $a (t)$ and identify the relationship between $a$ and $x$ .

a (t) = A ω cos (ω t)

;

a = ω x

a (t) = - A ω sin (ω t)

;

a = - x

a (t) = - A ω^{2} sin (ω t)

;

a = - ω^{2} x

a (t) = A ω^{2} cos (ω t)

;

a = ω^{2} x

Question 29

Two blocks, $m_{1} = 3$ kg and $m_{2} = 5$ kg, are connected by a massless rope and pulled across a frictionless surface by a force $F = 16$ N applied to $m_{2}$ . What is the tension in the rope between them?

6

NB.

10

NC.

2

m/s² (wrong unit)D.

16

Question 30

A simple pendulum of length $L = 1$ m oscillates with small amplitude. What is its period? ( $g = 10$ m/s²)

π

\approx 3.14

sB.

2 π

\approx 6.28

sC.

\approx 2

sD.

\approx 10

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Question 31

A 4 kg block moves along the $x$ -axis subject to force $F (x) = 6 x - 2$ N. Starting from rest at $x = 1$ m, find the kinetic energy at $x = 3$ m.

16

JB.

20

JC.

24

JD.

12

Question 32

A disk ( $I_{1} = 2$ kg·m², $ω_{1} = 6$ rad/s) drops onto a stationary disk ( $I_{2} = 3$ kg·m²) on the same frictionless axle. They reach a common angular velocity $ω_{f}$ . How much kinetic energy is lost?

0

JB.

14.4

JC.

21.6

JD.

36

Question 33

Derive Kepler's third law ( $T^{2} \propto r^{3}$ ) for a circular orbit from Newton's law of gravitation and the definition of circular motion.

T^{2} = \frac{4 π ^{2} r ^{3}}{GM}

T^{2} = \frac{2 π r ^{3}}{GM}

T^{2} = \frac{4 π ^{2} r ^{2}}{GM}

T = \frac{2 π r}{GM / r}

Question 34

A particle's position is $x (t) = t^{3} - 3 t^{2} + 2$ m. At what time(s) is the particle momentarily at rest for $t \geq 0$ ?

t = 0

onlyB.

t = 1

s onlyC.

t = 0

and

t = 2

sD.

t = 2

s only

Question 35

Using the integral definition $I = \int r^{2} d m$ , find the moment of inertia of a thin uniform rod of mass $M$ and length $L$ about one end.

\frac{1}{12} M L^{2}

\frac{1}{2} M L^{2}

\frac{1}{3} M L^{2}

M L^{2}