Q: Given a net torque $$\tau_{\text{net}}$$ applied to a rigid body with moment of inertia $$I$$, what is the resulting angular acceleration $$\alpha$$?

$$\alpha = \tau_{\text{net}} / I$$

Q: A baseball bat (modeled as a uniform rod) of length $$L is $$pivoted at one end and allowed to fall from rest at an angle $$\theta$$ from the vertical. What is the torque due to gravity about the pivot when the bat makes an angle $$\theta$$ with the vertical?

$$\tau = mg\frac{L}{2} \sin\theta$$

Q: Which of the following is the correct analogy between linear and rotational dynamics?

Force $$F is to $$mass $$m as $$torque $$\tau is to $$moment of inertia $$I.$$

Question 1

Which of the following best describes the physical meaning of the moment of inertia $$I$$ for a rotating object?

Accepted Answer

It is the rotational equivalent of mass, quantifying how difficult it is to change the object's angular velocity.

Question 2

A solid cylinder (mass $$M$$, radius $$R) $$and a cylindrical hoop (same mass $$M$$ and radius $$R) $$are both released from rest at the top of an incline. Which will reach the bottom first, and why?

Accepted Answer

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

Question 3

Given a net torque $$	au_{	ext{net}}$$ applied to a rigid body with moment of inertia $$I$$, what is the resulting angular acceleration $$\alpha$$?

Accepted Answer

$$\alpha = 	au_{	ext{net}} / I$$

Question 4

A uniform thin rod of length $$L$$ and mass $$M$$ rotates about its center. What is its moment of inertia?

Accepted Answer

$$I = \frac{1}{12}ML^2$

Question 5

A force $$F is $$applied tangentially at a distance $$r$$ from the axis of rotation of a disk. What is the torque $$	au$$ produced by this force?

Accepted Answer

$$	au = Fr$$

Question 6

A merry-go-round has children sitting at different distances from the center. Why is it harder to spin when children sit farther from the center?

Accepted Answer

Because the moment of inertia increases as mass is distributed farther from the axis.

Question 7

A bucket of mass $$m is $$attached to a rope wound around a solid cylinder of mass $$M$$ and radius $$R. If $$the bucket is released and falls, which equation correctly relates the linear acceleration $$a of $$the bucket to the angular acceleration $$\alpha of $$the cylinder (assuming no slipping)?

Accepted Answer

$$a = \alpha R$$

Question 8

A baseball bat (modeled as a uniform rod) of length $$L is $$pivoted at one end and allowed to fall from rest at an angle $$	heta$$ from the vertical. What is the torque due to gravity about the pivot when the bat makes an angle $$	heta$$ with the vertical?

Accepted Answer

$$	au = mg\frac{L}{2} \sin	heta$$

Question 9

A flywheel (solid cylinder) of mass $$540\ \mathrm{kg}$$ and radius $$0.30\ \mathrm{m} is $$spun up by a constant torque of $$2.8\ \mathrm{N\cdot m}. $$How long does it take to reach an angular speed of $$1360\ \mathrm{rad/s}$$ from rest? (Use $$I = \frac{1}{2}MR^2$.)

Accepted Answer

$$\Delta t = 11,800\ \mathrm{s}$$

Question 10

Which of the following statements about the moment of inertia is correct?

Accepted Answer

It depends on both the mass and how the mass is distributed relative to the axis of rotation.

Question 11

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

Accepted Answer

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

Question 12

A solid sphere and a spherical shell have the same mass $$M$$ and radius $$R. $$Which has the larger moment of inertia about its diameter?

Accepted Answer

The spherical shell, because $$I = \frac{2}{3}MR^2$.

Question 13

A plane slab of mass $$M$$ and side length $$a$$ rotates about its center. What is its moment of inertia?

Accepted Answer

$$I = \frac{1}{6}Ma^2$

Question 14

Which of the following is the correct analogy between linear and rotational dynamics?

Accepted Answer

Force $$F is to $$mass $$m as $$torque $$	au is to $$moment of inertia $$I.$$

Question 15

A bucket of mass $$m is $$attached to a rope wound around a cylinder of mass $$M$$ and radius $$R. If $$the bucket is released, which of the following best describes the effect of the cylinder's mass on the bucket's acceleration?

Accepted Answer

The acceleration is reduced compared to free fall because some energy goes into spinning the cylinder.

Question 16

A force $$F is $$applied at a distance $$r$$ from the axis of rotation, perpendicular to the lever arm. If the moment of inertia is $$I$$, what is the angular acceleration $$\alpha$$?

Accepted Answer

$$\alpha = \frac{Fr}{I}$$

Question 17

A uniform rod of length $$L$$ and mass $$M is $$pivoted at one end. What is the angular acceleration $$\alpha$$ when released from rest at an angle $$	heta$$ from the vertical? (Use $$I = \frac{1}{3}ML^2$.)

Accepted Answer

$$\alpha = \frac{3g\sin	heta}{2L}$$

Question 18

If a pulley of radius $$R$$ turns without the rope slipping, what is the relationship between the linear speed $$v of $$the rope and the angular speed $$\omega of $$the pulley?

Accepted Answer

$$v = \omega R$$

Rotational Dynamics and Moment of Inertia: Study Notes

스터디 가이드 - 연습 문제

학습 가이드 - 플래시카드