Q: A wind turbine blade rotates with a constant angular acceleration. If the initial angular velocity is $$\$$omega_0$$ and the angular acceleration is $$\alpha$$, what is the angular displacement $$\theta$$ after time t$?

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

Question 1

A wind turbine blade rotates with a constant angular acceleration. If the initial angular velocity is $$\$$omega_0$$ and the angular acceleration is $$\alpha$$, what is the angular displacement $$	heta$$ after time t$?

Accepted Answer

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

Question 2

A solid cylinder (mass $$M$$, radius $$R) $$and a hollow cylinder (same mass and radius) are released from rest at the top of an incline. Which reaches the bottom first and why?

Accepted Answer

The solid cylinder, because it has a smaller moment of inertia ($$I = \frac{1}{2}MR^2$) and thus accelerates faster.

Question 3

A gyroscope is spinning with angular velocity $$\omega. If no $$external torque acts on it, what happens to its angular momentum?

Accepted Answer

It remains constant in both magnitude and direction.

Question 4

The linear velocity $$v of a $$point at a distance $$r$$ from the axis of rotation of a rigid body rotating with angular velocity $$\omega is $$given by:

Accepted Answer

$$v = r \omega$$

Question 5

A yo-yo (solid cylinder, mass $$M$$, radius $$R) is $$released from rest and falls a distance $$h as $$its string unwinds without slipping. What is the speed $$v_{cm} of $$its center of mass at the bottom?

Accepted Answer

$$v_{cm} = \sqrt{\frac{4gh}{3}}$$

Question 6

Which of the following best describes the total kinetic energy $$K_{tot} of a $$rolling object (without slipping) of mass $$M$$ and radius $$R$$?

Accepted Answer

$$K_{tot} = \frac{1}{2} M v_{cm}^2 + \frac{1}{2} I_{cm} \$$omega^2$$

Question 7

A winch drum (solid cylinder, mass $$50\ \mathrm{kg}$$, diameter $$0.12\ \mathrm{m}) is $$turned by a cable pulled with a force of $$9.0\ \mathrm{N}$$ over $$2.0\ \mathrm{m}. If $$the drum starts from rest, what is the final angular velocity $$\omega$$?

Accepted Answer

$$\omega = \sqrt{\frac{4 F d}{M R^2$}}$$

Question 8

Which equation correctly relates angular acceleration $$\alpha to $$the change in angular velocity $$\omega$$ over time $$t$$?

Accepted Answer

$$\alpha = \frac{\omega - \$$omega_0$$}{t}$$

Question 9

A solid sphere, a solid cylinder, and a hollow cylinder (all with the same mass and radius) roll down the same incline. Which reaches the bottom first?

Accepted Answer

The solid sphere, because it has the smallest moment of inertia relative to its mass and radius.

Question 10

If a wheel of radius $$R$$ rolls without slipping for one revolution, what is the distance traveled by its center of mass?

Accepted Answer

$$2 \pi R$$

Question 11

The moment of inertia of a thin-walled hollow cylinder of mass $$M$$ and radius $$R$$ about its central axis is:

Accepted Answer

$$I = M R^2$

Question 12

Which of the following statements about the work-energy theorem for rotation is correct?

Accepted Answer

The net work done by torques equals the change in rotational kinetic energy: $$W_{net} = \Delta K_{rot}$$

Question 13

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

Accepted Answer

$$\omega = \alpha t$$

Question 14

A point on the rim of a rotating disk moves with speed $$v. If $$the disk has angular velocity $$\omega$$ and radius $$R$$, what is the relationship between $$v$$ and $$\omega$$?

Accepted Answer

$$v = \omega R$$

Question 15

The kinetic energy of a rotating rigid body is given by:

Accepted Answer

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

Question 16

A rolling object’s total kinetic energy is the sum of which two components?

Accepted Answer

Translational kinetic energy and rotational kinetic energy

Question 17

If the angular velocity of a rotating object doubles, by what factor does its rotational kinetic energy change?

Accepted Answer

It increases by a factor of 4.

Question 18

Which of the following is NOT a correct equation for constant angular acceleration?

Accepted Answer

$$\omega = \$$omega_0$$ + \alpha t^2$

Rotational Motion: Concepts, Equations, and Applications

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