Q: A 10 kg solid sphere of radius 0.4 m is rotated about an axis through its center. What is its moment of inertia?

$$\frac{2}{5} \times 10 \times (0.4)^2$

Q: A rigid body rotates with constant angular acceleration $$\alpha. If $$its initial angular velocity is $$\$$omega_0$$, what is its angular displacement $$\theta$$ after time t$?

$$\theta = \$$omega_0$$ t + \frac{1}{2} \alpha t^2$

Question 1

A solid disk of mass $$M$$ and radius $$R$$ rotates about its central axis with angular velocity $$\omega. $$What is its rotational kinetic energy?

Accepted Answer

$$\frac{1}{2} \left(\frac{1}{2} M R^2$\right) \$$omega^2$$

Question 2

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

Accepted Answer

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

Question 3

A uniform thin 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

$$\frac{1}{3} M L^2$

Question 4

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

Accepted Answer

$$F \cdot r$$

Question 5

A solid sphere and a 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, assuming no slipping?

Accepted Answer

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

Question 6

If a net torque $$	au is $$applied to a rigid body with moment of inertia $$I$$, what is the resulting angular acceleration $$\alpha$$?

Accepted Answer

$$\alpha = 	au / I$$

Question 7

A disk of mass $$M$$ and radius $$R is $$spinning with angular velocity $$\omega. If $$the mass is redistributed so that it is concentrated at the rim (like a hoop), what happens to the moment of inertia?

Accepted Answer

It increases.

Question 8

A bar of length $$L$$ and mass $$M is $$pivoted at one end and held horizontally. What is the magnitude of the torque due to gravity about the pivot?

Accepted Answer

$$M g \frac{L}{2}$$

Question 9

A wheel of radius $$R$$ rolls without slipping on a horizontal surface. What is the relationship between the linear speed $$v of $$the center and the angular speed $$\omega$$?

Accepted Answer

$$v = R \omega$$

Question 10

A uniform solid cylinder of mass $$M$$ and radius $$R is $$released from rest at the top of a ramp of height $$h. $$What is its speed at the bottom, assuming it rolls without slipping?

Accepted Answer

$$\sqrt{\frac{4gh}{3}}$$

Question 11

Which of the following best describes static equilibrium for a rigid body?

Accepted Answer

Both the sum of the forces and the sum of the torques are zero.

Question 12

A force $$F is $$applied at an angle $$	heta to a $$lever arm of length $$r. $$What is the general formula for the torque $$	au$$?

Accepted Answer

$$	au = r F \sin 	heta$$

Question 13

A uniform disk of mass $$M$$ and radius $$R is $$mounted on a frictionless axle. A string is wrapped around the rim and pulled with a constant force $$F$$ tangentially. What is the angular acceleration $$\alpha of $$the disk?

Accepted Answer

$$\alpha = \frac{2F}{M R^2$}$$

Question 14

A 10 kg solid sphere of radius 0.4 m is rotated about an axis through its center. What is its moment of inertia?

Accepted Answer

$$\frac{2}{5} 	imes 10 	imes (0.4)^2$

Question 15

A uniform rod of length $$L$$ and mass $$M is $$rotated about an axis perpendicular to the rod and passing through a point a distance $$D$$ from its center. What theorem allows you to calculate its moment of inertia about this new axis?

Accepted Answer

Parallel Axis Theorem

Question 16

A pulley of mass $$M$$ and radius $$R is $$used in a system where a mass $$m$$ hangs from a string wrapped around the pulley. If the pulley is a solid disk, what is its moment of inertia?

Accepted Answer

$$\frac{1}{2} M R^2$

Question 17

A child pushes a merry-go-round at its edge with a force $$F$$ perpendicular to the radius $$R. If $$the merry-go-round has moment of inertia $$I$$, what is the angular acceleration?

Accepted Answer

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

Question 18

A door is easier to open if you push at the edge farthest from the hinges. This is because:

Accepted Answer

Torque increases with distance from the axis of rotation.

Question 19

A rigid body rotates with constant angular acceleration $$\alpha. If $$its initial angular velocity is $$\$$omega_0$$, what is its angular displacement $$	heta$$ after time t$?

Accepted Answer

$$	heta = \$$omega_0$$ t + \frac{1}{2} \alpha t^2$

Rotational Motion, Moment of Inertia, and Torque: Study Notes

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