Q: A force $F$ is applied at a distance $r$ from the axis at an angle $\theta$. Which component of the force produces torque?

The perpendicular component: $F_\perp = F \sin \theta$

Q: If a system is both translating and rotating, which laws must be applied?

Both $\sum F = m a$ and $\sum \tau = I \alpha$

Question 1

Which equation correctly expresses the torque $	au$ produced by a force $F$ applied at a distance $r$ from the axis, at an angle $	heta$ from the lever arm?

Accepted Answer

$	au = F r \sin 	heta$

Question 2

A solid cylinder of mass $M$ and radius $R$ rotates about its center. What is its moment of inertia?

Accepted Answer

$I = \frac{1}{2} M R^2$

Question 3

You need to hold a door against a zombie pressing with a force of $30\ \mathrm{N}$ at an angle of $30^\circ$ from normal at a radius of $0.4\ \mathrm{m}$. What minimum force must you apply at the edge of the door ($0.8\ \mathrm{m}$) at normal incidence to keep the door closed?

Accepted Answer

$13.0\ \mathrm{N}$

Question 4

A see-saw of length $1.0\ \mathrm{m}$ and mass $10\ \mathrm{kg}$ is at an angle of $30^\circ$ with respect to the ground. The pivot is $0.6\ \mathrm{m}$ from one end. What mass should be added to the raised end to barely start rotation?

Accepted Answer

$2.5\ \mathrm{kg}$

Question 5

Which of the following is the correct form of Newton's Second Law for rotation?

Accepted Answer

$\sum 	au = I \alpha$

Question 6

A merry-go-round (solid cylinder, $M = 200\ \mathrm{kg}$, $R = 1\ \mathrm{m}$) is started by a $200\ \mathrm{N}$ force at the edge. What is the angular acceleration produced?

Accepted Answer

$2\ \mathrm{rad/s^2}$

Question 7

How long does it take for the merry-go-round to reach an angular speed of $10\ \mathrm{rad/s}$, starting from rest with the angular acceleration found above?

Accepted Answer

$5\ \mathrm{s}$

Question 8

A $2\ \mathrm{kg}$ bucket is attached to a massless string wrapped around a $4\ \mathrm{cm}$ diameter cylinder. If the bucket falls $2\ \mathrm{m}$, how does the moment of inertia of the cylinder affect the time to reach the bottom?

Accepted Answer

It increases the time, because rotational inertia resists acceleration.

Question 9

Which object has the largest moment of inertia for the same mass and radius?

Accepted Answer

A thin hoop about its center: $I = M R^2$

Question 10

If a force is applied tangentially to a rotating object, which variable does NOT affect the torque produced?

Accepted Answer

The mass of the object

Question 11

A long uniform rod of length $L$ and mass $M$ rotates about its center. What is its moment of inertia?

Accepted Answer

$I = \frac{1}{12} M L^2$

Question 12

Which step should you take first when solving a rotational dynamics problem?

Accepted Answer

Draw a diagram and label the forces

Question 13

A force $F$ is applied at a distance $r$ from the axis at an angle $	heta$. Which component of the force produces torque?

Accepted Answer

The perpendicular component: $F_\perp = F \sin 	heta$

Question 14

If the net torque on a system is zero, what can be said about its angular acceleration?

Accepted Answer

Angular acceleration is zero: $\alpha = 0$

Question 15

A uniform sphere of mass $M$ and radius $R$ rotates about its center. What is its moment of inertia?

Accepted Answer

$I = \frac{2}{5} M R^2$

Question 16

Which of the following best describes the difference between translational and rotational dynamics?

Accepted Answer

Translational dynamics uses $\sum F = m a$, rotational uses $\sum 	au = I \alpha$

Question 17

A hollow cylinder of mass $M$ and radius $R$ rotates about its center. What is its moment of inertia?

Accepted Answer

$I = M R^2$

Question 18

If a system is both translating and rotating, which laws must be applied?

Accepted Answer

Both $\sum F = m a$ and $\sum 	au = I \alpha$

Question 19

A rectangular plate of mass $M$, length $l$, and width $w$ rotates about its center. What is its moment of inertia?

Accepted Answer

$I = \frac{1}{12} M (l^2 + w^2)$

Rotation II: Torque, Rotational Dynamics, and Problem Solving

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