Q: A car's speedometer needle rotates about a fixed axis. If the needle sweeps an arc length of 0.15 m on a circle of radius 0.05 m, what is the angular displacement in radians?

$$\theta = \dfrac{0.15}{0.05} = 3$$ radians

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

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

Q: 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?

$$\omega_z = \omega_{z0} + \alpha_z t = 0 + 2 \times 5 = 10\ \mathrm{rad/s}$$

Q: Why is it important to use radians rather than degrees when relating linear and angular quantities?

Because equations like $$s = r\theta$$ are only valid when $$\theta is in $$radians.

Question 1

A car's speedometer needle rotates about a fixed axis. If the needle sweeps an arc length of 0.15 m on a circle of radius 0.05 m, what is the angular displacement in radians?

Accepted Answer

$$	heta = \dfrac{0.15}{0.05} = 3$$ radians

Question 2

Which of the following best describes the direction of the angular velocity vector for a ceiling fan rotating counterclockwise when viewed from below?

Accepted Answer

It points upward, along the axis 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 = \dfrac{1}{2}MR^2$

Question 4

A hummingbird can flap its wings up to 70 beats per second, while an Andean condor flaps its wings about once per second. Which physical property of their wings most directly explains this difference?

Accepted Answer

Moment of inertia

Question 5

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

Accepted Answer

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

Question 6

Which formula gives the rotational kinetic energy of a rigid body?

Accepted Answer

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

Question 7

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

Accepted Answer

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

Question 8

A baseball bat of mass $$M$$ has a moment of inertia $$I_{cm}$$ about its center of mass. If the rotation axis is moved a distance $$d$$ from the center of mass, what is the new moment of inertia?

Accepted Answer

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

Question 9

Which of the following is the correct SI unit for moment of inertia?

Accepted Answer

$$\mathrm{kg\ m^2$}$$

Question 10

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

$$\omega_z = \omega_{z0} + \alpha_z t = 0 + 2 	imes 5 = 10\ \mathrm{rad/s}$$

Question 11

A solid sphere and a thin-walled hollow sphere of equal 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 ($$I = \dfrac{2}{5}MR^2$) and thus accelerates faster.

Question 12

Which equation correctly relates the tangential acceleration $$a_{tan} of a $$point at distance $$r$$ from the axis to the angular acceleration $$\alpha$$?

Accepted Answer

$$a_{tan} = r\alpha$$

Question 13

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

Accepted Answer

Because equations like $$s = r	heta$$ are only valid when $$	heta is in $$radians.

Question 14

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: $$U_{grav} = Mg y_{cm}$$

Question 15

A hollow cylinder has inner radius $$R_1$$ and outer radius $$R_2. $$What is its moment of inertia about its central axis?

Accepted Answer

$$I = \dfrac{1}{2}M(R_1$^2 + R_2$^2)$$

Question 16

Which integral gives the moment of inertia for a continuous mass distribution?

Accepted Answer

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

Question 17

If a rotating object is speeding up, what is the relationship between the directions of its angular velocity $$\vec{\omega}$$ and angular acceleration $$\vec{\alpha}$$ vectors?

Accepted Answer

They point in the same direction.

Question 18

Escherichia coli bacteria swim by rotating their flagella at angular speeds up to 100 rad/s. If a flagellum accelerates from 20 rad/s to 100 rad/s in 2 seconds, what is its average angular acceleration?

Accepted Answer

$$\alpha_{avg} = \dfrac{100 - 20}{2} = 40\ \mathrm{$$rad/s^2$$}$$

Question 19

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}ML^2$

Rotation of Rigid Bodies – Study Notes

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