Question 1

A car's speedometer needle rotates about a fixed axis. If the needle moves through an arc length $$s$$ and the radius of rotation is $$r$$, what is the angular displacement $$	heta in $$radians?

Accepted Answer

$$	heta = \dfrac{s}{r}$$

Question 2

Which of the following best describes the moment of inertia $$I of a $$rigid body about a given axis?

Accepted Answer

$$I is $$the sum of each particle's mass times the square of its distance from the axis: $$I = \sum m_i r_i^2$

Question 3

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

Accepted Answer

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

Question 4

A Ferris wheel rotates at a constant angular velocity $$\omega. $$What is the linear speed $$v of a $$seat located a distance $$r$$ from the axis of rotation?

Accepted Answer

$$v = r \omega$$

Question 5

Which of the following statements about angular velocity is correct?

Accepted Answer

Angular velocity is a vector whose direction is given by the right-hand rule.

Question 6

A thin rectangular plate of mass $$M$$ and side length $$a$$ rotates about an axis along one edge. What is its moment of inertia?

Accepted Answer

$$I = \dfrac{1}{3} M a^2$

Question 7

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

Accepted Answer

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

Question 8

If a bird flaps its wings with a small moment of inertia, what effect does this have on its ability to move its wings rapidly?

Accepted Answer

It allows the bird to move its wings rapidly because less torque is needed for the same angular acceleration.

Question 9

A baseball bat of mass $$m$$ has a moment of inertia $$I_{cm}$$ about its center of mass. What is its moment of inertia $$I_P$$ about a parallel axis a distance $$d$$ away?

Accepted Answer

$$I_P = I_{cm} + m d^2$

Question 10

A rotating rigid body has an angular acceleration $$\alpha$$ and angular velocity $$\omega. $$What is the tangential acceleration $$a_t of a $$point at distance $$r$$ from the axis?

Accepted Answer

$$a_t = r \alpha$$

Question 11

Which of the following is the correct expression for the rotational kinetic energy $$K_{rot} of a $$rigid body?

Accepted Answer

$$K_{rot} = \dfrac{1}{2} I \$$omega^2$$

Question 12

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

Accepted Answer

$$I = \dfrac{2}{3} M R^2$

Question 13

Why is it important to use radians rather than degrees when relating linear and angular quantities in rotational motion equations?

Accepted Answer

Because only radians ensure the correct proportionality between arc length and radius in equations like $$s = r 	heta.$$

Question 14

A point on a rotating disk is at a distance $$r$$ from the axis. If the disk rotates with angular velocity $$\omega$$, what is the centripetal acceleration $$a_c of $$the point?

Accepted Answer

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

Question 15

The gravitational potential energy of an extended object is calculated as if all its mass were concentrated at which point?

Accepted Answer

At its center of mass.

Question 16

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

Accepted Answer

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

Question 17

Escherichia coli bacteria swim by rotating their flagella at angular speeds up to 100 rad/s. If a flagellum accelerates from rest to 100 rad/s in 0.5 seconds, what is its average angular acceleration?

Accepted Answer

$$\alpha = 200 \ \mathrm{$$rad/s^2$$}$$

Question 18

Which of the following integrals correctly expresses the moment of inertia $$I$$ for a continuous mass distribution?

Accepted Answer

$$I = \int r^2$ \rho \, dv$$

Question 19

If two equal-mass cylinders are locked close to the center of a rotating shaft, what happens to the moment of inertia and the rotational kinetic energy for a given angular velocity?

Accepted Answer

The moment of inertia decreases, so for the same angular velocity, the rotational kinetic energy decreases.

Chapter 09

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