---
title: "AP Physics C: Mechanics Unit 5 Practice Test"
description: "A timed AP Physics C: Mechanics Unit 5 (Torque and Rotational Dynamics) practice test in the AP format, drawn from 1,022 multiple-choice questions and 19 FRQs written for the unit. The standard version takes about 88 minutes."
canonical: "https://fiveable.me/ap-physics-c-mechanics/unit-test/5"
type: "unit-test"
subject: "AP Physics C: Mechanics"
unit: "Unit 5"
---

# AP Physics C: Mechanics Unit 5 Practice Test

## Overview

A timed AP Physics C: Mechanics Unit 5 (Torque and Rotational Dynamics) practice test in the AP format, drawn from 1,022 multiple-choice questions and 19 FRQs written for the unit. The standard version takes about 88 minutes.

## Test Format

- 20 Multiple Choice Questions (40 minutes)
- 1 FRQ – FRQ 1 – Mathematical Routines (24 minutes)
- 1 FRQ – FRQ 3 – Experimental Design (24 minutes)

## Unit FRQs by Type

- [FRQ 1 – Mathematical Routines](/ap-physics-c-mechanics/frqs/mathematical-routines?unit=5): 3 questions
- [FRQ 3 – Experimental Design](/ap-physics-c-mechanics/frqs/experimental-design?unit=5): 5 questions
- [FRQ 4 – Qualitative/Quantitative Translation](/ap-physics-c-mechanics/frqs/qualitative-quantitative-translation?unit=5): 11 questions

## Practice Questions

### Question 1

A rigid body rotates with an angular position given by $$\theta(t) = 2t^3 - 4t$$, where $$\theta$$ is in radians and $$t$$ is in seconds. How does the angular acceleration $$\alpha$$ at time $$t = 4$$ s compare to the angular acceleration at time $$t = 2$$ s?

- (A) The angular acceleration at $$t = 4$$ s is two times the angular acceleration at $$t = 2$$ s
- (B) The angular acceleration at $$t = 4$$ s is four times the angular acceleration at $$t = 2$$ s
- (C) The angular acceleration at $$t = 4$$ s is half the angular acceleration at $$t = 2$$ s
- (D) The angular acceleration at $$t = 4$$ s is equal to the angular acceleration at $$t = 2$$ s

**Answer: (A)** Differentiating the position function twice yields $$\alpha(t) = 12t$$. Comparing values shows $$\alpha(4) = 48$$ rad/s² and $$\alpha(2) = 24$$ rad/s², resulting in a ratio of 2:1.

### Question 2

Two flywheels, A and B, start from rest and rotate with constant angular accelerations $$\alpha_A$$ and $$\alpha_B$$, respectively. If $$\alpha_B = 4\alpha_A$$, how does the time $$t_A$$ required for flywheel A to rotate through a displacement $$\Delta\theta$$ compare to the time $$t_B$$ required for flywheel B to rotate through the same displacement?

- (A) The time $$t_A$$ is two times the time $$t_B$$
- (B) The time $$t_A$$ is four times the time $$t_B$$
- (C) The time $$t_A$$ is half the time $$t_B$$
- (D) The time $$t_A$$ is equal to the time $$t_B$$

**Answer: (A)** Using $$\Delta\theta = \frac{1}{2}\alpha t^2$$, time is inversely proportional to the square root of acceleration ($$t \propto 1/\sqrt{\alpha}$$). Since $$\alpha_B = 4\alpha_A$$, $$t_B = t_A/2$$.

### Question 3

A turbine blade rotates with an angular velocity given by $$\omega(t) = k\sqrt{t}$$, where $$k$$ is a positive constant. How does the angular acceleration $$\alpha$$ at time $$t = T$$ compare to the angular acceleration at time $$t = 4T$$?

- (A) The acceleration at $$T$$ is eight times the acceleration at $$4T$$
- (B) The acceleration at $$T$$ is four times the acceleration at $$4T$$
- (C) The acceleration at $$T$$ is two times the acceleration at $$4T$$
- (D) The acceleration at $$T$$ is half the acceleration at $$4T$$

**Answer: (C)** Differentiating $$\omega(t)$$ gives $$\alpha(t) = \frac{k}{2\sqrt{t}}$$. Comparing values, $$\alpha(T) \propto 1$$ and $$\alpha(4T) \propto 1/2$$, so the ratio is 2:1.

### Question 4

A disk rotates with constant angular acceleration $$\alpha$$. It takes time $$t_1$$ to accelerate from rest to angular velocity $$\omega_f$$, and time $$t_2$$ to accelerate from $$\omega_f$$ to $$2\omega_f$$. How does $$t_2$$ compare to $$t_1$$?

- (A) The interval $$t_2$$ is equal to the interval $$t_1$$
- (B) The interval $$t_2$$ is twice the interval $$t_1$$
- (C) The interval $$t_2$$ is half the interval $$t_1$$
- (D) The interval $$t_2$$ is four times the interval $$t_1$$

**Answer: (A)** Using $$\Delta\omega = \alpha \Delta t$$, the change in velocity for both intervals is $$\omega_f$$ ($$\omega_f - 0$$ and $$2\omega_f - \omega_f$$). Since $$\alpha$$ is constant, the time intervals must be identical.

### Question 5

A system rotates with angular acceleration $$\alpha(t) = 3t^2$$. If the system starts from rest at $$t=0$$, how does the angular velocity $$\omega$$ at $$t=2$$ s compare to the angular velocity at $$t=1$$ s?

- (A) The velocity at $$t=2$$ s is eight times the velocity at $$t=1$$ s
- (B) The velocity at $$t=2$$ s is four times the velocity at $$t=1$$ s
- (C) The velocity at $$t=2$$ s is two times the velocity at $$t=1$$ s
- (D) The velocity at $$t=2$$ s is sixteen times the velocity at $$t=1$$ s

**Answer: (A)** Integrating acceleration $$\alpha(t) = 3t^2$$ gives $$\omega(t) = t^3$$. Comparing values, $$\omega(2) = 8$$ and $$\omega(1) = 1$$, resulting in a ratio of 8.

### Question 6

Two points, P and Q, are located on a rigid rotating disk. Point P is at distance $$R$$ from the axis, and Point Q is at distance $$2R$$. How does the angular acceleration $$\alpha_P$$ of Point P compare to the angular acceleration $$\alpha_Q$$ of Point Q?

- (A) The angular acceleration $$\alpha_P$$ is equal to $$\alpha_Q$$
- (B) The angular acceleration $$\alpha_P$$ is half of $$\alpha_Q$$
- (C) The angular acceleration $$\alpha_P$$ is twice $$\alpha_Q$$
- (D) The angular acceleration $$\alpha_P$$ is four times $$\alpha_Q$$

**Answer: (A)** For a rigid body, all points share the same angular quantities ($$\theta$$, $$\omega$$, $$\alpha$$) regardless of their distance from the axis of rotation.

### Question 7

A wheel rotates with an angular velocity $$\omega$$ related to its angular position $$\theta$$ by the equation $$\omega = c\theta$$, where $$c$$ is a constant. How does the angular acceleration $$\alpha$$ at position $$\theta_2 = 2$$ rad compare to the angular acceleration at $$\theta_1 = 1$$ rad?

- (A) The acceleration at $$\theta_2$$ is two times the acceleration at $$\theta_1$$
- (B) The acceleration at $$\theta_2$$ is four times the acceleration at $$\theta_1$$
- (C) The acceleration at $$\theta_2$$ is equal to the acceleration at $$\theta_1$$
- (D) The acceleration at $$\theta_2$$ is half the acceleration at $$\theta_1$$

**Answer: (A)** Using the chain rule, $$\alpha = \frac{d\omega}{dt} = \frac{d\omega}{d\theta}\omega$$. With $$\omega = c\theta$$, $$\alpha = (c)(c\theta) = c^2\theta$$. Thus, acceleration is directly proportional to $$\theta$$, so doubling $$\theta$$ doubles $$\alpha$$.

### Question 8

A fan blade slows down with an angular acceleration given by $$\alpha(t) = -k t$$. If the blade stops at time $$T$$, how does the magnitude of the angular displacement $$\Delta\theta$$ in the first half of the time interval ($$0$$ to $$T/2$$) compare to the displacement in the second half ($$T/2$$ to $$T$$)?

- (A) The displacement in the first half is greater than in the second half
- (B) The displacement in the first half is less than in the second half
- (C) The displacement in the first half is equal to that in the second half
- (D) The displacement in the first half is zero compared to the second half

**Answer: (A)** Since the blade is slowing down, its angular velocity is higher during the first interval than the second. Therefore, it covers more angular displacement in the first half of the time.

### Question 9

Two gears, A and B, are meshed. Gear A has radius $$R$$ and Gear B has radius $$3R$$. If Gear A accelerates with angular acceleration $$\alpha_A$$, how does the angular acceleration $$\alpha_B$$ of Gear B compare to $$\alpha_A$$?

- (A) The acceleration $$\alpha_B$$ is one-third of $$\alpha_A$$
- (B) The acceleration $$\alpha_B$$ is three times $$\alpha_A$$
- (C) The acceleration $$\alpha_B$$ is nine times $$\alpha_A$$
- (D) The acceleration $$\alpha_B$$ is equal to $$\alpha_A$$

**Answer: (A)** The tangential acceleration at the contact point must be equal: $$a_t = R\alpha_A = 3R\alpha_B$$. Solving for $$\alpha_B$$ yields $$\alpha_B = \alpha_A / 3$$.

### Question 10

An object rotates with angular position $$\theta(t) = A \cos(bt)$$. How does the magnitude of the angular acceleration $$\alpha$$ at $$t = 0$$ compare to the magnitude of the angular acceleration at $$t = \pi/(2b)$$?

- (A) The magnitude at $$t=0$$ is greater than at $$t=\pi/(2b)$$
- (B) The magnitude at $$t=0$$ is less than at $$t=\pi/(2b)$$
- (C) The magnitude at $$t=0$$ is equal to that at $$t=\pi/(2b)$$
- (D) The magnitude at $$t=0$$ is half that at $$t=\pi/(2b)$$

**Answer: (A)** Differentiating twice gives $$\alpha(t) = -Ab^2 \cos(bt)$$. At $$t=0$$, $$|\alpha| = Ab^2$$. At $$t=\pi/(2b)$$, the cosine term is 0, so $$|\alpha| = 0$$.

## Review the Unit

- [5.1 Rotation](/ap-physics-c-mechanics/unit-5/1-rotation/study-guide/0tVqvv29lj9DIxVt)
- [5.2 Connecting Linear and Rotational Motion](/ap-physics-c-mechanics/unit-5/2-connecting-linear-and-rotational-motion/study-guide/79Ym6NXzWOJH6ZWx)
- [5.3 Torque](/ap-physics-c-mechanics/unit-5/3-torque/study-guide/kQhoEJrKtYjpul5K)
- [5.4 Rotational Inertia](/ap-physics-c-mechanics/unit-5/4-rotational-inertia/study-guide/lSrDkHqB6EviD5CA)
- [5.5 Rotational Equilibrium and Newton's First Law in Rotational Form](/ap-physics-c-mechanics/unit-5/5-rotational-equilibrium-and-newtons-first-law-in-rotational-form/study-guide/hkkIAh5Csy8T6d0p)
- [5.6 Newton's Second Law in Rotational Form](/ap-physics-c-mechanics/unit-5/6-newtons-second-law-in-rotational-form/study-guide/VuXCF8dn0RYwBEaK)

## Other Unit Tests

- [Unit 1: Kinematics](/ap-physics-c-mechanics/unit-test/1)
- [Unit 2: Force and Translational Dynamics](/ap-physics-c-mechanics/unit-test/2)
- [Unit 3: Work, Energy, and Power](/ap-physics-c-mechanics/unit-test/3)
- [Unit 4: Linear Momentum](/ap-physics-c-mechanics/unit-test/4)
- [Unit 6: Energy and Momentum of Rotating Systems](/ap-physics-c-mechanics/unit-test/6)
- [Unit 7: Oscillations](/ap-physics-c-mechanics/unit-test/7)

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