Q: A force $$\vec{F} is $$applied at a point with position vector $$\vec{r}$$ relative to a pivot. Which vector operation gives the torque $$\vec{\tau}$$?

$$\vec{\tau} = \vec{r} \times \vec{F}$$

Question 1

A rigid body is defined as an object whose size and shape do not change as it moves. Which of the following statements is TRUE regarding the motion of a rigid body?

Accepted Answer

All points on a rigid body move with the same angular velocity and angular acceleration.

Question 2

Which of the following equations correctly relates the tangential velocity $$v_t of a $$point at a distance $$r$$ from the axis of rotation to the angular velocity $$\omega$$?

Accepted Answer

$$v_t = r \omega$$

Question 3

A uniform rod of length $$L$$ and mass $$M$$ lies along the $$x-$$axis from $$x=0 to$$ $$x=L. $$What is the $$x-$$coordinate of its center of mass?

Accepted Answer

$$x_{CM} = L/2$$

Question 4

The moment of inertia $$I of a $$rigid body depends on which of the following factors?

Accepted Answer

The mass distribution, shape, and axis of rotation.

Question 5

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

Accepted Answer

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

Question 6

A solid disk of mass $$M$$ and radius $$R$$ rotates about its center. What is its moment of inertia?

Accepted Answer

$$I = \frac{1}{2} M R^2$

Question 7

The parallel-axis theorem is used to find the moment of inertia about an axis that is:

Accepted Answer

Parallel to the axis through the center of mass and a distance $$d$$ away.

Question 8

Which of the following correctly expresses the parallel-axis theorem?

Accepted Answer

$$I = I_{CM} + M d^2$

Question 9

A force $$\vec{F} is $$applied at a distance $$r$$ from a pivot, making an angle $$\phi$$ with the position vector $$\vec{r}. $$What is the magnitude of the torque $$	au$$ produced?

Accepted Answer

$$	au = r F \sin \phi$$

Question 10

A wheel is rotating with a constant angular acceleration $$\alpha. $$Which kinematic equation relates the final angular velocity $$\omega_f to $$the initial angular velocity $$\omega_i$$, angular acceleration $$\alpha$$, and time $$t$$?

Accepted Answer

$$\omega_f = \omega_i + \alpha t$$

Question 11

A disk is rotating with angular velocity $$\omega$$ and has moment of inertia $$I. $$What is its angular momentum $$L$$ about the axis of rotation?

Accepted Answer

$$L = I \omega$$

Question 12

Which of the following statements about the conservation of angular momentum is correct?

Accepted Answer

Angular momentum is conserved if there is no net external torque on the system.

Question 13

A uniform solid sphere and a uniform solid disk, both with the same mass $$M$$ and radius $$R$$, are released from rest at the top of an incline. Which will reach the bottom first, assuming no slipping, and why?

Accepted Answer

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

Question 14

A ladder leans against a frictionless wall at an angle $$	heta. $$What condition must be satisfied for the ladder to be in static equilibrium?

Accepted Answer

The net force and the net torque on the ladder must both be zero.

Question 15

A rolling object moves without slipping. What is the relationship between the velocity of the center of mass $$v_{CM}$$ and the angular velocity $$\omega$$?

Accepted Answer

$$v_{CM} = \omega R$$

Question 16

Which of the following best describes the direction of the angular velocity vector $$\vec{\omega}$$ for a rotating object?

Accepted Answer

It points along the axis of rotation, in the direction given by the right-hand rule.

Question 17

A force $$\vec{F} is $$applied at a point with position vector $$\vec{r}$$ relative to a pivot. Which vector operation gives the torque $$\vec{	au}$$?

Accepted Answer

$$\vec{	au} = \vec{r} 	imes \vec{F}$$

Question 18

A particle of mass $$m$$ moves in a circle of radius $$r$$ with angular velocity $$\omega. $$What is its angular momentum $$L$$ about the center?

Accepted Answer

$$L = m r^2$ \omega$$

Question 19

A disk of moment of inertia $$I is $$spinning with angular velocity $$\omega_i. A $$second identical disk is dropped onto it and sticks. What is the final angular velocity $$\omega_f of $$the combined system (assuming no external torques)?

Accepted Answer

$$\omega_f = \omega_i / 2$$

Rotation of a Rigid Body: Kinematics, Dynamics, and Energy

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