Q: Which of the following is the correct expression for the torque $$\tau$$ produced by a force $$\vec{F}$$ applied at a position vector $$\vec{r}$$ from the axis of rotation?

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

Question 1

Which of the following best describes the condition for static equilibrium in a rigid body?

Accepted Answer

The net force and net torque on the body must both be zero.

Question 2

A uniform beam of length $$L$$ and mass $$m is $$supported horizontally at its ends. A mass $$M is $$hung at a distance $$d$$ from the left end. What is the upward force exerted by the left support?

Accepted Answer

$$\frac{m g}{2} + \frac{M g (L-d)}{L}$$

Question 3

Given a graph of angular velocity $$\omega$$ versus time $$t$$ for a rotating object, how can you determine the angular displacement $$	heta$$ over a time interval?

Accepted Answer

By calculating the area under the $$\omega vs$$. $$t$$ curve for that interval.

Question 4

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

Accepted Answer

$$\frac{1}{2} m R^2$ \omega$$

Question 5

A figure skater pulls in her arms while spinning, 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

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

Question 6

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

Accepted Answer

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

Question 7

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 a smaller moment of inertia.

Question 8

A wheel of radius $$R$$ rotates with constant angular acceleration $$\alpha. If it $$starts from rest, what is its angular velocity $$\omega$$ after time $$t$$?

Accepted Answer

$$\omega = \alpha t$$

Question 9

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 tangential speed $$v$$?

Accepted Answer

$$v = r \omega$$

Question 10

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

Accepted Answer

$$\alpha = \frac{F R}{I}$$

Question 11

A rotating object has a constant angular velocity $$\omega. $$What is its angular acceleration $$\alpha$$?

Accepted Answer

$$\alpha = 0$$

Question 12

A rod of length $$L$$ and mass $$m$$ rotates about one end. What is its moment of inertia?

Accepted Answer

$$I = \frac{1}{3} m L^2$

Question 13

A car's wheels lock and it skids to a stop. Which principle explains why the car's angular momentum is not conserved?

Accepted Answer

An external torque acts on the wheels due to friction with the road.

Question 14

A satellite orbits Earth in a circular path. If no external torque acts, what happens to its angular momentum?

Accepted Answer

It remains constant.

Question 15

A door of width $$w is $$pushed with force $$F at $$its edge, perpendicular to the surface. What is the torque about the hinges?

Accepted Answer

$$	au = F w$$

Question 16

A spinning neutron star contracts, reducing its radius by half. Assuming mass remains constant, by what factor does its angular velocity change?

Accepted Answer

It increases by a factor of 4.

Question 17

A rotating object slows down due to friction. Which of the following best describes the effect of friction?

Accepted Answer

Friction exerts an external torque, reducing angular velocity.

Question 18

A gymnast jumps and tucks her body to spin faster. Which law explains this change?

Accepted Answer

Conservation of angular momentum.

Question 19

A rotating platform has a moment of inertia $$I$$ and angular velocity $$\omega. If a $$person of mass $$m$$ stands at a distance $$r$$ from the center, what is the new moment of inertia?

Accepted Answer

$$I_{new} = I + m r^2$

Question 20

A bicycle wheel spins freely. If you touch the rim lightly, what happens to its angular velocity and why?

Accepted Answer

It decreases due to an external torque from friction.

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