Q: A child pushes tangentially on a merry-go-round, causing it to rotate. Which of the following expressions gives the work done by the torque $$\tau as $$the merry-go-round rotates from $$\$$theta_1$$ to $$\$$theta_2$$?

$$W = \int_{\$$theta_1$$}^{\$$theta_2$$} \tau \, d\theta$$

Q: A rigid body rotates about a moving axis. What is the total kinetic energy $$K of $$the body?

$$K = \frac{1}{2} M v_{\text{cm}}^2 + \frac{1}{2} I_{\text{cm}} \$$omega^2$$

Q: A wheel is rolling with slipping on a road. Which of the following is true?

$$v_{\text{cm}} \neq R \omega$$

Question 1

Which of the following best describes the torque $$	au$$ produced by a force $$\vec{F}$$ applied at a distance $$r$$ from the axis of rotation?

Accepted Answer

$$	au = rF \sin 	heta$$

Question 2

A mechanic uses a wrench to loosen a bolt. Which action will maximize the torque applied to the bolt?

Accepted Answer

Applying the force perpendicular to the wrench at the farthest point from the axis

Question 3

Which equation correctly expresses the rotational analog of Newton's second law for a rigid body?

Accepted Answer

$$\sum 	au = I \alpha$$

Question 4

A solid disk of mass $$M$$ and radius $$R$$ rolls without slipping down an incline. What is the condition for rolling without slipping?

Accepted Answer

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

Question 5

A child pushes tangentially on a merry-go-round, causing it to rotate. Which of the following expressions gives the work done by the torque $$	au as $$the merry-go-round rotates from $$\$$theta_1$$ to $$\$$theta_2$$?

Accepted Answer

$$W = \int_{\$$theta_1$$}^{\$$theta_2$$} 	au \, d	heta$$

Question 6

A helicopter's main rotor is spinning at a constant rate. Which statement is true about the work and power involved?

Accepted Answer

The engine does positive work, air resistance does negative work, and the net work is zero.

Question 7

A rigid body rotates about a moving axis. What is the total kinetic energy $$K of $$the body?

Accepted Answer

$$K = \frac{1}{2} M v_{	ext{cm}}^2 + \frac{1}{2} I_{	ext{cm}} \$$omega^2$$

Question 8

Which of the following statements about angular momentum $$\vec{L} is $$correct for a rigid body rotating about a symmetry axis?

Accepted Answer

$$\vec{L} is $$parallel to $$\vec{\omega}$$ and given by $$\vec{L} = I \vec{\omega}.$$

Question 9

A figure skater pulls in her arms during a spin, decreasing her moment of inertia. If no external torque acts, what happens to her angular velocity?

Accepted Answer

It increases to conserve angular momentum.

Question 10

A cat falls and twists its body so that it lands on its feet. What happens to the total angular momentum of the cat during the fall?

Accepted Answer

It remains zero throughout the fall.

Question 11

A gyroscope's axis of rotation changes direction while spinning. What is this motion called?

Accepted Answer

Precession

Question 12

If a flywheel is initially not spinning and is released, what is its initial angular momentum?

Accepted Answer

Zero

Question 13

Which of the following best describes the effect of rolling friction on a rolling object?

Accepted Answer

It causes mechanical energy to be lost if the object or surface deforms.

Question 14

A wheel is rolling with slipping on a road. Which of the following is true?

Accepted Answer

$$v_{	ext{cm}} 
eq R \omega$$

Question 15

A rotating flywheel is subjected to a torque perpendicular to its angular momentum. What is the result?

Accepted Answer

The direction of angular momentum changes, but its magnitude remains constant.

Question 16

Which of the following is the correct vector expression for torque $$\vec{	au}$$ produced by a force $$\vec{F}$$ applied at position vector $$\vec{r}$$ relative to the axis?

Accepted Answer

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

Question 17

A uniform rod of length $$L$$ and mass $$M is $$pivoted at one end and held horizontally. What is the magnitude of the torque due to gravity about the pivot?

Accepted Answer

$$	au = MgL/2$$

Question 18

A disk rotates with constant angular acceleration $$\alpha. If $$its initial angular velocity is $$\$$omega_0$$, what is its angular velocity $$\omega$$ after time t$?

Accepted Answer

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

Question 19

A solid sphere and a hollow cylinder of equal mass and radius roll without slipping down an incline. Which reaches the bottom first and why?

Accepted Answer

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

Dynamics of Rotational Motion – Study Notes

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