Q: A car's speedometer needle rotates about a fixed axis. If the arc length traced by the tip of the needle is 5 cm and the radius from the axis to the tip is 2 cm, what is the angular displacement in radians?

$$\theta = \frac{5}{2} = 2.5$$ radians

Q: A point on a rotating disk is 0.2 m from the axis and the disk rotates at an angular speed of $$10$$ rad/s. What is the linear speed of the point?

$$v = r\omega = 0.2 \times 10 = 2 m/s$$

Q: A wheel starts from rest and accelerates at a constant angular acceleration $$\alpha = 2$$ rad/s$$^2. $$What is its angular velocity after 5 seconds?

$$\omega = \$$omega_0$$ + \alpha t = 0 + 2 \times 5 = 10$$ rad/s

Question 1

A car's speedometer needle rotates about a fixed axis. If the arc length traced by the tip of the needle is 5 cm and the radius from the axis to the tip is 2 cm, what is the angular displacement in radians?

Accepted Answer

$$	heta = \frac{5}{2} = 2.5$$ radians

Question 2

Which of the following statements about angular velocity is TRUE?

Accepted Answer

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

Question 3

A 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 4

A Ferris wheel rotates counterclockwise. According to the right-hand rule, in which direction does its angular velocity vector point?

Accepted Answer

Upward, along the axis of rotation

Question 5

A hummingbird can flap its wings up to 70 times per second, while a condor flaps about once per second. Which statement best explains this difference?

Accepted Answer

The hummingbird has a smaller moment of inertia, allowing for faster wing movement.

Question 6

A point on a rotating disk is 0.2 m from the axis and the disk rotates at an angular speed of $$10$$ rad/s. What is the linear speed of the point?

Accepted Answer

$$v = r\omega = 0.2 	imes 10 = 2 m/s$$

Question 7

Which formula correctly relates the rotational kinetic energy of a rigid body to its moment of inertia and angular speed?

Accepted Answer

$$K = \frac{1}{2}I\$$omega^2$$

Question 8

A thin-walled hollow sphere of mass $$M$$ and radius $$R is $$used in a physics experiment. What is its moment of inertia about an axis through its center?

Accepted Answer

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

Question 9

A wheel starts from rest and accelerates at a constant angular acceleration $$\alpha = 2$$ rad/s$$^2. $$What is its angular velocity after 5 seconds?

Accepted Answer

$$\omega = \$$omega_0$$ + \alpha t = 0 + 2 	imes 5 = 10$$ rad/s

Question 10

A solid sphere and a thin-walled hollow sphere, both of mass $$M$$ and radius $$R$$, are released from rest at the top of an incline. Which will reach the bottom first, and why?

Accepted Answer

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

Question 11

A baseball bat of mass $$M$$ has a moment of inertia $$I_{cm}$$ about an axis through 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} + Md^2$

Question 12

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

Accepted Answer

Radians are dimensionless and ensure correct proportionality in equations like $$s = r	heta.$$

Question 13

A point on a rotating object has a tangential acceleration $$a_{tan}$$ and a centripetal acceleration $$a_{rad}. If $$the object rotates at angular speed $$\omega$$ and angular acceleration $$\alpha$$, what are the correct expressions for these accelerations at a distance $$r$$ from the axis?

Accepted Answer

$$a_{tan} = r\alpha$$ and $$a_{rad} = r\$$omega^2$$

Question 14

A rectangular plate of mass $$M$$, width $$a$$, and length $$b$$ rotates about an axis along its edge. What is its moment of inertia?

Accepted Answer

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

Question 15

A solid cylinder and a thin-walled hollow cylinder, both of mass $$M$$ and radius $$R$$, are spun about their central axes. Which has the greater moment of inertia and why?

Accepted Answer

The thin-walled hollow cylinder, because all its mass is farther from the axis.

Question 16

The rotational kinetic energy of a ceiling fan is doubled. If its moment of inertia remains constant, by what factor does its angular speed change?

Accepted Answer

The angular speed increases by a factor of $$\sqrt{2}.$$

Question 17

A satellite orbiting Earth allows geophysicists to measure the planet's moment of inertia. What does a higher moment of inertia indicate about Earth's mass distribution?

Accepted Answer

More mass is distributed farther from the center.

Question 18

An athlete arches her body so her center of mass passes under the bar during a high jump. What is the advantage of this technique?

Accepted Answer

It requires a smaller increase in gravitational potential energy.

Rotation of Rigid Bodies: Angular Kinematics and Moment of Inertia

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