Q: A bicycle wheel is spinning freely. If friction at the axle exerts a constant torque $$\tau$$ opposite to the direction of rotation, what is the angular velocity after time $$t if $$the initial angular velocity is $$\$$omega_0$$?

$$\omega = \$$omega_0$$ - \frac{\tau}{I} t$$

Question 1

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 point?

Accepted Answer

$$Mg \frac{L}{2}$$

Question 2

A disk of mass $$M$$ and radius $$R$$ rotates about its center with angular velocity $$\omega. $$What is its angular momentum?

Accepted Answer

$$\frac{1}{2}MR^2$ \omega$$

Question 3

A gymnast stands on a rotating platform with arms extended. When she pulls her arms in, her moment of inertia decreases. What happens to her angular velocity, assuming no external torque acts?

Accepted Answer

It increases

Question 4

A car of mass $$m$$ moves in a circle of radius $$r at $$constant speed $$v. $$What is the magnitude of the centripetal force acting on the car?

Accepted Answer

$$\frac{mv^2$}{r}$$

Question 5

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

Accepted Answer

The solid sphere, because it has less rotational inertia

Question 6

A force $$F is $$applied tangentially to a wheel of radius $$R$$ and moment of inertia $$I. $$What is the angular acceleration $$\alpha of $$the wheel?

Accepted Answer

$$\alpha = \frac{FR}{I}$$

Question 7

A figure skater spins with arms extended and then pulls them in, reducing her moment of inertia from $$I_1 to$$ $$I_2. If $$her initial angular velocity is $$\$$omega_1$$, what is her final angular velocity $$\$$omega_2$$?

Accepted Answer

$$\$$omega_2$$ = \$$omega_1$$ \frac{I_1$}{I_2$}$$

Question 8

A uniform beam of length $$L$$ and mass $$M is $$supported at its center. A mass $$m is $$hung at a distance $$d$$ from the center. What is the magnitude of the force required at the opposite end to keep the beam in static equilibrium?

Accepted Answer

$$\frac{md}{L/2}$$

Question 9

A point mass $$m$$ moves in a circle of radius $$r$$ with constant angular velocity $$\omega. $$What is its linear velocity $$v$$?

Accepted Answer

$$v = r \omega$$

Question 10

A rotating object has a 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 11

A child sits on a merry-go-round at a distance $$r$$ from the center, which rotates at angular velocity $$\omega. $$What is the child’s centripetal acceleration?

Accepted Answer

$$a_c = r \$$omega^2$$

Question 12

A wheel of radius $$R$$ rolls without slipping. If its center moves with velocity $$v$$, what is the angular velocity $$\omega of $$the wheel?

Accepted Answer

$$\omega = \frac{v}{R}$$

Question 13

A seesaw is balanced with a child of mass $$m_1$$ sitting at distance $$d_1$$ from the pivot and another child of mass $$m_2 at $$distance $$d_2 on $$the other side. What is the condition for equilibrium?

Accepted Answer

$$m_1$$ $$d_1$$ = $$m_2$$ $$d_2$$

Question 14

A rotating platform has a moment of inertia $$I$$ and is spinning at angular velocity $$\omega. If a $$person of mass $$m$$ steps onto the edge at radius $$R$$, what is the new angular velocity $$\omega'$$?

Accepted Answer

$$\omega' = \frac{I}{I + mR^2$} \omega$$

Question 15

A ball is thrown horizontally from a cliff and lands at a distance $$d$$ from the base. Neglecting air resistance, which statement is true about its horizontal velocity during flight?

Accepted Answer

It remains constant

Question 16

A torque $$	au is $$applied to a rigid body with moment of inertia $$I. $$What is the resulting angular acceleration $$\alpha$$?

Accepted Answer

$$\alpha = \frac{	au}{I}$$

Question 17

A bicycle wheel is spinning freely. If friction at the axle exerts a constant torque $$	au$$ opposite to the direction of rotation, what is the angular velocity after time $$t if $$the initial angular velocity is $$\$$omega_0$$?

Accepted Answer

$$\omega = \$$omega_0$$ - \frac{	au}{I} t$$

Question 18

A satellite orbits Earth in a circular path. Which force provides the centripetal acceleration?

Accepted Answer

Gravitational force

Question 19

A uniform rod of length $$L is $$pivoted at one end and released from rest in a vertical position. What is its angular acceleration immediately after release?

Accepted Answer

$$\alpha = \frac{3g}{2L}$$

Equilibrium, Rotational Kinematics, and Dynamics: Study Guide

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