Q: Which of the following is the correct equation for angular displacement $$\theta$$ under constant angular acceleration $$\alpha$$, starting from initial angular velocity $$\$$omega_0$$?

$$\theta = \$$omega_0$$ t + \frac{1}{2} \alpha t^2$

Q: A disk rotates from rest with a constant angular acceleration $$\alpha. $$After time $$t$$, what is its angular displacement $$\theta$$?

$$\theta = \frac{1}{2} \alpha t^2$

Question 1

A point on the edge of a rotating record and a point near the center both complete one revolution in the same amount of time. Which statement best describes their linear speeds?

Accepted Answer

The point on the edge moves faster because it travels a larger circumference in the same time.

Question 2

Which equation correctly relates the arc length $$s to $$the radius $$r$$ and the angle $$	heta in $$radians?

Accepted Answer

$$s = r 	heta$$

Question 3

A rigid body rotates counterclockwise around a fixed axis. If the angular velocity is positive, which direction is considered positive according to the standard convention?

Accepted Answer

Counterclockwise

Question 4

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

Accepted Answer

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

Question 5

A solid disk and a hollow ring, both of equal mass and radius, are released from rest at the top of an incline. Which will reach the bottom first, assuming no slipping?

Accepted Answer

The solid disk, because it has less rotational inertia.

Question 6

If a yo-yo of mass $$m = 0.10\ \mathrm{kg}$$ descends a height $$h = 0.40\ \mathrm{m}$$, what is the speed of its center of mass just before it reaches the bottom, neglecting air resistance? (Use $$g = 9.8\ \mathrm{m/s^2$})$$

Accepted Answer

$$v = \sqrt{2gh/(1 + I/(mR^2$))}$$

Question 7

Which of the following best describes the moment of inertia $$I$$ for a point mass $$m at a $$distance $$r$$ from the axis of rotation?

Accepted Answer

$$I = m r^2$

Question 8

A wheel rotates with a constant angular acceleration $$\alpha. $$Which equation gives its angular velocity $$\omega$$ after time $$t if $$its initial angular velocity is $$\$$omega_0$$?

Accepted Answer

$$\omega = \$$omega_0$$ + \alpha t$$

Question 9

Which of the following is the correct relationship between linear velocity $$v$$ and angular velocity $$\omega$$ for a point at radius $$r$$?

Accepted Answer

$$v = r \omega$$

Question 10

A rolling object moves without slipping. What is the relationship between its center-of-mass speed $$v_{\mathrm{cm}}$$ and its angular speed $$\omega$$?

Accepted Answer

$$v_{\mathrm{cm}} = R \omega$$

Question 11

Which of the following is the correct equation for angular displacement $$	heta$$ under constant angular acceleration $$\alpha$$, starting from initial angular velocity $$\$$omega_0$$?

Accepted Answer

$$	heta = \$$omega_0$$ t + \frac{1}{2} \alpha t^2$

Question 12

A rotating object has both translational and rotational kinetic energy. Which equation gives the total kinetic energy?

Accepted Answer

$$K_{\mathrm{total}} = \frac{1}{2} M v^2$ + \frac{1}{2} I \$$omega^2$$

Question 13

If a bicycle wheel spins at $$120$$ revolutions per minute (rpm), what is its angular velocity in radians per second?

Accepted Answer

$$\omega = 12.57\ \mathrm{rad/s}$$

Question 14

Which of the following best describes the difference between a rigid body and a deformable body?

Accepted Answer

A rigid body does not change shape during motion, while a deformable body can change shape.

Question 15

A disk rotates from rest with a constant angular acceleration $$\alpha. $$After time $$t$$, what is its angular displacement $$	heta$$?

Accepted Answer

$$	heta = \frac{1}{2} \alpha t^2$

Question 16

Which of the following is the correct expression for the gravitational potential energy of an extended-size object?

Accepted Answer

$$U_{\mathrm{grav}} = M g y_{\mathrm{cm}}$$

Question 17

A car tire has a radius of $$0.3\ \mathrm{m}$$ and rotates at $$10\ \mathrm{rad/s}. $$What is the linear speed of a point on its edge?

Accepted Answer

$$v = 3.0\ \mathrm{m/s}$$

Question 18

Which of the following is a correct statement about angular acceleration $$\alpha$$?

Accepted Answer

Angular acceleration is the rate of change of angular velocity.

Chapter 9: Rotational Motion – Study Notes

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