Q: Which equation correctly relates angular acceleration $$\alpha to $$net torque $$\tau_{net}$$ and moment of inertia $$I$$ for a rotating object?

$$\alpha = \frac{\tau_{net}}{I}$$

Q: A person is lying on their back with one leg pointing up, modeled as a uniform thin rod of mass $$12\ \mathrm{kg}$$ and length $$0.85\ \mathrm{m}. $$What is the moment of inertia of the leg about the hip joint?

$$I = \frac{1}{3} \times 12 \times (0.85)^2$

Question 1

A potter shapes clay on a spinning wheel, as shown in the image. If the wheel has an angular speed $$\omega$$ and the clay is at a distance $$r$$ from the center, what is the linear speed $$v of $$the clay?

Accepted Answer

$$v = \omega r$$

Question 2

Simone Biles performs a rotation on a balance beam. If her moment of inertia about the axis of rotation increases, which of the following will occur if the net torque remains constant?

Accepted Answer

Her angular acceleration will decrease.

Question 3

Which equation correctly relates angular acceleration $$\alpha to $$net torque $$	au_{net}$$ and moment of inertia $$I$$ for a rotating object?

Accepted Answer

$$\alpha = \frac{	au_{net}}{I}$$

Question 4

A gymnast spins faster when she pulls her arms in. This is because:

Accepted Answer

Her moment of inertia decreases, so her angular velocity increases.

Question 5

A uniform thin rod of mass $$m$$ and length $$L$$ rotates about one end. What is its moment of inertia?

Accepted Answer

$$I = \frac{1}{3} m L^2$

Question 6

A force $$F is $$applied tangentially at a distance $$r$$ from the axis of rotation of a particle of mass $$m. $$What is the torque $$	au$$ produced?

Accepted Answer

$$	au = F r$$

Question 7

A person is lying on their back with one leg pointing up, modeled as a uniform thin rod of mass $$12\ \mathrm{kg}$$ and length $$0.85\ \mathrm{m}. $$What is the moment of inertia of the leg about the hip joint?

Accepted Answer

$$I = \frac{1}{3} 	imes 12 	imes (0.85)^2$

Question 8

If a net torque is applied to a rigid body, which of the following best describes the resulting motion?

Accepted Answer

The body will experience angular acceleration.

Question 9

Which formula gives the rotational kinetic energy $$K_{rot} of a $$rotating object?

Accepted Answer

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

Question 10

A rolling object has both translational and rotational kinetic energy. Which equation correctly expresses its total kinetic energy?

Accepted Answer

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

Question 11

If the angular acceleration $$\alpha$$ and angular velocity $$\omega of a $$rotating object have opposite signs, what happens to the angular speed?

Accepted Answer

The angular speed decreases.

Question 12

A tightrope walker carries a long pole to help maintain balance. Why does holding the pole help?

Accepted Answer

It increases the moment of inertia, making it harder to rotate and easier to balance.

Question 13

The parallel axis theorem relates the moment of inertia $$I$$ about an axis to the moment of inertia $$I_{cm}$$ about the center of mass. Which equation is correct?

Accepted Answer

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

Question 14

A force of magnitude $$F is $$applied perpendicular to a massless rigid rod at a distance $$r$$ from the pivot. What is the resulting angular acceleration $$\alpha if $$the moment of inertia is $$I$$?

Accepted Answer

$$\alpha = \frac{F r}{I}$$

Question 15

Three identical balls are arranged in different shapes and connected by rigid rods with negligible inertia. Which configuration will have the greatest moment of inertia about a given axis?

Accepted Answer

The configuration with the balls farthest from the axis.

Question 16

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} m L^2$

Question 17

A gymnast rotates with angular velocity $$\omega$$ and moment of inertia $$I. If $$she changes her body position to decrease $$I$$ and no external torque acts, what happens to $$\omega$$?

Accepted Answer

$$\omega$$ increases to conserve angular momentum.

Question 18

A rolling object moves down a hill. Why does it reach the bottom slower than if it were sliding without rolling?

Accepted Answer

Some of its potential energy is converted into rotational kinetic energy.

Question 19

A leg modeled as a uniform rod of length $$0.85\ \mathrm{m}$$ rotates about the hip joint with angular acceleration $$7\ \mathrm{$$rad/s^2$$}. $$What is the tangential acceleration at the foot?

Accepted Answer

$$a_t = 0.85 	imes 7$$

Rotational Dynamics: Concepts, Formulas, and Applications

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