Q: A disk rotates from an initial angular position $$\theta_i = 0 to a $$final angular position $$\theta_f = \frac{\pi}{2}$$ radians in 2 seconds. What is its average angular velocity $$\omega_{av}$$?

$$\omega_{av} = \frac{\Delta \theta}{\Delta t} = \frac{\frac{\pi}{2}}{2} = \frac{\pi}{4} \, \text{rad/s}$$

Q: A force of 100 N is applied perpendicular to a wrench at a distance of 0.3 m from the pivot. What is the torque produced?

$$\tau = 30 \, \text{N} \cdot \text{m}$$

Question 1

A 3,000,000-kg boulder, such as the Balanced Rock in Arches National Park, has remained in stable equilibrium for millennia. Which statement best explains why the rock remains balanced?

Accepted Answer

The center of gravity of the rock is located directly above its base of support, resulting in zero net torque.

Question 2

Which of the following best defines the angular displacement $$\Delta 	heta of a $$rotating rigid object?

Accepted Answer

The angle through which the object rotates, given by $$\Delta 	heta = 	heta_f - 	heta_i.$$

Question 3

A disk rotates from an initial angular position $$	heta_i = 0 to a $$final angular position $$	heta_f = \frac{\pi}{2}$$ radians in 2 seconds. What is its average angular velocity $$\omega_{av}$$?

Accepted Answer

$$\omega_{av} = \frac{\Delta 	heta}{\Delta t} = \frac{\frac{\pi}{2}}{2} = \frac{\pi}{4} \, 	ext{rad/s}$$

Question 4

Which equation correctly relates the net torque $$\sum 	au on a $$rigid body to its angular acceleration $$\alpha$$ and moment of inertia $$I$$?

Accepted Answer

$$\sum 	au = I \alpha$$

Question 5

A force $$F is $$applied at an angle $$	heta to a $$lever arm of length $$r. $$What is the magnitude of the torque $$	au$$ produced by this force about the pivot point?

Accepted Answer

$$	au = r F \sin 	heta$$

Question 6

A wrench is used to loosen a bolt. If the force applied is doubled and the distance from the axis of rotation is halved, what happens to the torque?

Accepted Answer

The torque remains the same.

Question 7

Which of the following objects has the greatest moment of inertia about its central axis, assuming all have the same mass $$M$$ and radius $$R$$?

Accepted Answer

A hoop or thin cylindrical shell: $$I = M R^2$

Question 8

A uniform, solid ball of mass $$M$$ and radius $$R$$ rolls without slipping down a ramp of height $$h. $$What is its speed $$v at $$the bottom of the ramp?

Accepted Answer

$$v = \sqrt{\frac{10}{7}gh}$$

Question 9

A system consists of two blocks of masses $$m_1$$ and $$m_2$$ connected by a string over a pulley of mass $$M$$ and radius $$R. $$The pulley is a hollow cylinder. If $$m_2$$ \u003e $$m_1$$, which of the following correctly expresses the speed $$v of $$the blocks after $$m_2$$ descends a distance $$h ($$neglecting friction)?

Accepted Answer

$$v = \sqrt{\frac{2gh(m_2$ - m_1$)}{m_1$ + m_2$ + M}}$$

Question 10

Which of the following statements about the center of mass is correct?

Accepted Answer

The center of mass is the point where the total mass of the system can be considered to be concentrated for analyzing translational motion.

Question 11

A force is applied to a door at a distance $$r$$ from the hinges, making an angle $$	heta$$ with the door. Which component of the force is responsible for producing torque?

Accepted Answer

$$F \sin 	heta$$

Question 12

If the net torque acting on a rigid object is zero, which of the following must be true?

Accepted Answer

The object's angular velocity remains constant.

Question 13

A solid cylinder and a solid sphere, both with the same mass and radius, roll down an incline without slipping. Which reaches the bottom first and why?

Accepted Answer

The solid sphere, because it has a smaller moment of inertia relative to its mass.

Question 14

Which of the following is the correct expression for the rotational kinetic energy of a rigid body?

Accepted Answer

$$KE_{rot} = \frac{1}{2} I \$$omega^2$$

Question 15

A force of 100 N is applied perpendicular to a wrench at a distance of 0.3 m from the pivot. What is the torque produced?

Accepted Answer

$$	au = 30 \, 	ext{N} \cdot 	ext{m}$$

Question 16

Which of the following best describes the lever arm in the context of torque?

Accepted Answer

The perpendicular distance from the axis of rotation to the line of action of the force.

Question 17

A uniform rod of length $$L$$ and mass $$M$$ rotates about an axis through its center. What is its moment of inertia?

Accepted Answer

$$I = \frac{1}{12} M L^2$

Question 18

A disk is rotating with constant angular acceleration $$\alpha. If $$its initial angular velocity is $$\omega_i$$ and it rotates through an angle $$	heta$$, what is its final angular velocity $$\omega_f$$?

Accepted Answer

$$\omega_f^2$ = \omega_i^2$ + 2 \alpha 	heta$$

Question 19

Which of the following is NOT a correct analogy between linear and rotational motion?

Accepted Answer

Displacement $$x is to $$angular displacement $$	heta as $$mass $$m is to $$angular velocity $$\omega.$$

Chapter 7: Rotational Motion – Structured Study Notes

Study Guide - Practice Questions

Study Guide - Flashcards