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: An Escherichia coli bacterium rotates its flagellum at 1000 rev/min. What is the angular speed in rad/s?

$$\omega = \frac{1000 \times 2\pi}{60} \approx 104.7$$ 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 best describes the direction of the angular velocity vector for a rotating ceiling fan when viewed from below?

Accepted Answer

It points upward, perpendicular to the plane of rotation, following 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 hummingbird can flap its wings up to 70 beats per second, while an Andean condor flaps about one beat per second. What physical property of their wings primarily explains this difference?

Accepted Answer

The moment of inertia of the wings

Question 5

A wheel starts from rest and accelerates at a constant angular acceleration $$\alpha_z = 2\ \mathrm{$$rad/s^2$$}$$ for 5 seconds. What is its final angular velocity?

Accepted Answer

$$\omega_z = 10\ \mathrm{rad/s}$$

Question 6

Which formula correctly relates the linear speed $$v of a $$point at a distance $$r$$ from the axis of rotation to the angular speed $$\omega$$?

Accepted Answer

$$v = r\omega$$

Question 7

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 8

A baseball bat of mass $$M$$ has a moment of inertia $$I_{cm}$$ about an axis through its center of mass. If the axis is shifted a distance $$d$$ from the center of mass, what is the new moment of inertia according to the parallel-axis theorem?

Accepted Answer

$$I_p = I_{cm} + Md^2$

Question 9

A point on a rotating disk is 0.2 m from the axis and has an angular velocity of $$15\ \mathrm{rad/s}. $$What is its linear speed?

Accepted Answer

$$v = 3\ \mathrm{m/s}$$

Question 10

Which of the following statements about rotational kinetic energy is correct?

Accepted Answer

It is given by $$K = \frac{1}{2}I\$$omega^2$$ for a rigid body rotating about a fixed axis.

Question 11

A slender 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 12

A rotating wheel slows down with angular acceleration $$\alpha_z$$ opposite to its angular velocity $$\omega_z. $$What is happening to the wheel?

Accepted Answer

The wheel is slowing down.

Question 13

Why is it important to use radians rather than degrees when relating linear and angular quantities in equations such as $$s = r	heta$$?

Accepted Answer

Radians are dimensionless and ensure the equation is consistent with SI units.

Question 14

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

Accepted Answer

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

Question 15

An Escherichia coli bacterium rotates its flagellum at 1000 rev/min. What is the angular speed in rad/s?

Accepted Answer

$$\omega = \frac{1000 	imes 2\pi}{60} \approx 104.7$$ rad/s

Question 16

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 17

A gymnast arches her body so that her center of mass passes under the bar during a high jump. Why does this technique require less gravitational potential energy than straddling the bar?

Accepted Answer

Her center of mass does not need to rise as high, minimizing $$U_{grav} = Mg y_{cm}.$$

Question 18

The moment of inertia of a body with a continuous mass distribution is calculated by:

Accepted Answer

$$I = \int r^2$ dV$$

Question 19

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

Accepted Answer

$$I = MR^2$

Rotation of Rigid Bodies: Angular Kinematics, Kinetic Energy, and Moment of Inertia

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