Question 1

A solid sphere of mass $$m$$ and radius $$R$$ rolls without slipping down an inclined plane of height $$h. $$What is the final speed $$v of $$the sphere at the bottom of the incline?

Accepted Answer

$$v = \sqrt{\frac{10}{7}gh}$$

Question 2

Which of the following correctly expresses the moment of inertia for a thin hoop of mass $$m$$ and radius $$R$$ about its central axis?

Accepted Answer

$$I = mR^2$

Question 3

A solid cylinder and a solid sphere, both with the same mass and radius, are released from rest at the top of an incline. Which one reaches the bottom first and why?

Accepted Answer

The solid sphere, because it has a smaller moment of inertia.

Question 4

What is the moment of inertia of a solid disk of mass $$m$$ and radius $$R$$ about its central axis?

Accepted Answer

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

Question 5

A thin rod of length $$L$$ and mass $$m$$ rotates about an axis through one end perpendicular to its length. What is its moment of inertia?

Accepted Answer

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

Question 6

A hollow cylinder and a solid cylinder, both with mass $$m$$ and radius $$R$$, are rolling without slipping. Which has the greater rotational kinetic energy for the same linear speed $$v$$?

Accepted Answer

The hollow cylinder, because its moment of inertia is larger.

Question 7

If a solid sphere and a thin hoop, both with mass $$m$$ and radius $$R$$, are spun with the same angular velocity $$\omega$$, which has the greater rotational kinetic energy?

Accepted Answer

The thin hoop, because its moment of inertia is larger.

Question 8

A uniform solid sphere of mass $$m$$ and radius $$R$$ rotates about a diameter. What is its moment of inertia?

Accepted Answer

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

Question 9

A solid disk and a thin hoop, both with mass $$m$$ and radius $$R$$, are rotating with the same angular velocity $$\omega. $$What is the ratio of their rotational kinetic energies $$K_{hoop}/K_{disk}$$?

Accepted Answer

$$2$$

Question 10

A uniform 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 11

A solid sphere of mass $$m$$ and radius $$R is $$rolling without slipping at speed $$v. $$What is its total kinetic energy?

Accepted Answer

$$K = \frac{7}{10}mv^2$

Question 12

A thin spherical shell of mass $$m$$ and radius $$R$$ rotates about a diameter. What is its moment of inertia?

Accepted Answer

$$I = \frac{2}{3}mR^2$

Question 13

A uniform solid cylinder of mass $$m$$ and radius $$R$$ rotates about its central axis. What is its moment of inertia?

Accepted Answer

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

Question 14

A solid sphere, a solid cylinder, and a thin hoop, all with the same mass and radius, are released from rest at the top of an incline. Which one reaches the bottom last?

Accepted Answer

The thin hoop

Question 15

A uniform rod of length $$L$$ and mass $$m$$ rotates about an axis perpendicular to the rod and passing through its center. What is the correct formula for its moment of inertia?

Accepted Answer

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

Question 16

A solid sphere of mass $$m$$ and radius $$R is $$rolling without slipping. What fraction of its total kinetic energy is rotational?

Accepted Answer

$$\frac{2}{7}$$

Rotational Motion: Moments of Inertia and Exam Preparation

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