Q: A car has a track width $$t$$ and center of mass height $$h. $$What is the critical angle $$\theta_c at $$which it will tip over?

$$\theta_c = \tan$$^{-1}$$ \left( \frac{t}{2h} \right)$$

Q: Which of the following is the correct vector expression for torque?

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

Question 1

A rigid body is rotating about a fixed axis. Which statement is TRUE regarding the angular velocity $$\omega$$ and angular acceleration $$\alpha of $$points on the rigid body?

Accepted Answer

Both $$\omega$$ and $$\alpha$$ are the same for all points on the rigid body.

Question 2

A wheel rotates counterclockwise with a positive angular velocity $$\omega. If $$the wheel is speeding up, what is the sign of the angular acceleration $$\alpha$$?

Accepted Answer

$$\alpha \u003e 0$$

Question 3

For a particle on a rotating wheel, which equation correctly describes the tangential velocity $$v_t$$?

Accepted Answer

$$v_t = \omega r$$

Question 4

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

Accepted Answer

$$x_{CM} = \frac{L}{2}$$

Question 5

The moment of inertia $$I of a $$rigid body depends on:

Accepted Answer

The mass, shape, and the axis of rotation

Question 6

Which formula gives the rotational kinetic energy of a rigid body?

Accepted Answer

$$K = \frac{1}{2} I \$$omega^2$$

Question 7

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

Accepted Answer

$$I = \frac{2}{5} M R^2$

Question 8

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

Accepted Answer

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

Question 9

The parallel-axis theorem states that the moment of inertia about an axis a distance $$d$$ from the center of mass is:

Accepted Answer

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

Question 10

A disk of mass $$M$$ and radius $$R is $$rolling without slipping. What is the relationship between the velocity of its center of mass $$v_{CM}$$ and its angular velocity $$\omega$$?

Accepted Answer

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

Question 11

A car has a track width $$t$$ and center of mass height $$h. $$What is the critical angle $$	heta_c at $$which it will tip over?

Accepted Answer

$$	heta_c = 	an$$^{-1}$$ \left( \frac{t}{2h} ight)$$

Question 12

If a rigid body is in static equilibrium, which of the following must be TRUE?

Accepted Answer

Net force and net torque are both zero.

Question 13

A rope passes over a pulley with radius $$R$$ and rotates with angular velocity $$\omega. $$What is the velocity of the rope?

Accepted Answer

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

Question 14

The angular momentum $$L of a $$particle moving in a circle of radius $$r$$ with mass $$m$$ and angular velocity $$\omega is$$:

Accepted Answer

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

Question 15

Which equation expresses Newton's second law for rotational motion?

Accepted Answer

$$	au = I \alpha$$

Question 16

A rigid body rotates about a fixed axis. If no net external torque acts on it, what happens to its angular momentum?

Accepted Answer

Angular momentum remains constant.

Question 17

A disk of mass $$M$$ and radius $$R is $$rolling without slipping. What is its total kinetic energy?

Accepted Answer

$$K_{tot} = \frac{1}{2} I_{CM} \$$omega^2$$ + \frac{1}{2} M R^2$ \$$omega^2$$

Question 18

Which of the following is the correct vector expression for torque?

Accepted Answer

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

Question 19

A 2.0 kg solid disk (radius 0.1 m) rotates at 200 rpm. A 1.0 kg circular loop (same radius) is dropped onto it and friction causes them to rotate together. What is the final angular velocity $$\omega_f of $$the system? (Disk: $$I_{disk} = \frac{1}{2} M R^2$, Loop: I$$_{loop} = M $$R^2)$$

Accepted Answer

$$\omega_f = \frac{I_{disk} \omega_{i}}{I_{disk} + I_{loop}}$$

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

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