Q: A force of $10\ \mathrm{N}$ is applied perpendicular to a wrench $0.3\ \mathrm{m}$ long. What is the torque produced?

$\tau = 3.0\ \mathrm{N\cdot m}$

Question 1

A uniform solid sphere and a uniform solid cylinder, both of mass $M$ and radius $R$, are released from rest at the top of an incline. Which object reaches the bottom first, and why?

Accepted Answer

The sphere, because it has a smaller $\frac{I}{MR^2}$ ratio.

Question 2

Which of the following correctly expresses the $x$-coordinate of the center of mass for a system of discrete particles?

Accepted Answer

$x_{CM} = \frac{\sum m_i x_i}{\sum m_i}$

Question 3

A high jumper's body can have its center of mass located outside the body during a jump. Which concept does this illustrate?

Accepted Answer

The center of mass can lie outside the physical object.

Question 4

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

Accepted Answer

$I = \frac{1}{3}ML^2$

Question 5

If a force $F$ is applied perpendicular to a lever arm of length $\ell$, what is the resulting torque?

Accepted Answer

$	au = F \cdot \ell$

Question 6

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

Accepted Answer

$v_{CM} = R\omega$

Question 7

Which of the following is the correct expression for the kinetic energy of a rolling object (without slipping)?

Accepted Answer

$KE = \frac{1}{2} M v_{CM}^2 + \frac{1}{2} I \omega^2$

Question 8

A disc of mass $M$ and radius $R$ is rolling down an incline without slipping. What is the total kinetic energy at the bottom?

Accepted Answer

$KE_{total} = \frac{1}{2} M v_{CM}^2 + \frac{1}{2} I \omega^2$

Question 9

Which statement about the center of mass (CM) of a uniform symmetric object is correct?

Accepted Answer

The CM is always at the geometric center.

Question 10

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

Accepted Answer

$I = MR^2$

Question 11

If the angular velocity of a rotating object changes from $\omega_1$ to $\omega_2$ in time $\Delta t$, what is the angular acceleration?

Accepted Answer

$\alpha = \frac{\Delta \omega}{\Delta t}$

Question 12

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

Accepted Answer

$I = \frac{1}{2}MR^2$

Question 13

Which of the following best describes the path of the center of mass (CM) of a system under a net external force?

Accepted Answer

It follows the same path as a point particle under the same force.

Question 14

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

Accepted Answer

$I = \frac{1}{12}ML^2$

Question 15

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

Accepted Answer

$I = \frac{2}{5}MR^2$

Question 16

Which formula correctly relates arc length $s$ to radius $r$ and angular displacement $	heta$?

Accepted Answer

$s = r	heta$

Question 17

A force of $10\ \mathrm{N}$ is applied perpendicular to a wrench $0.3\ \mathrm{m}$ long. What is the torque produced?

Accepted Answer

$	au = 3.0\ \mathrm{N\cdot m}$

Question 18

If a wheel is rolling without slipping, which equation relates its angular velocity $\omega$ and the velocity of its center of mass $v_{CM}$?

Accepted Answer

$v_{CM} = R\omega$

Question 19

Which of the following is the rotational analog of Newton's Second Law?

Accepted Answer

$	au_{net} = I\alpha$

Center of Mass, Rotational Motion, and Rolling Dynamics

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