Q: A solid sphere rolls smoothly down a ramp inclined at angle $$\theta. $$Which equation gives the acceleration of the center of mass of the sphere?

$$a_{com,x} = -\frac{g \sin \theta}{1 + \frac{I_{com}}{MR^2$}}$$

Q: Which equation expresses Newton's second law in angular form for a single particle?

$$\vec{\tau}_{net} = \frac{d\vec{\ell}}{dt}$$

Q: Which equation gives the angular momentum $$\vec{\ell} of a $$particle relative to a fixed point?

$$\vec{\ell} = \vec{r} \times \vec{p}$$

Question 1

A solid sphere rolls smoothly down a ramp inclined at angle $$	heta. $$Which equation gives the acceleration of the center of mass of the sphere?

Accepted Answer

$$a_{com,x} = -\frac{g \sin 	heta}{1 + \frac{I_{com}}{MR^2$}}$$

Question 2

A yo-yo of mass $$M$$ and radius $$R_0$$ descends its string. If its moment of inertia about the center of mass is $$I_{com}$$, what is the linear acceleration of its center of mass?

Accepted Answer

$$a_{com} = -\frac{g}{1 + \frac{I_{com}}{MR_0$^2}}$$

Question 3

A rolling object has two types of kinetic energy. Which equation correctly expresses the total kinetic energy?

Accepted Answer

$$K = \frac{1}{2} M v_{com}^2 + \frac{1}{2} I_{com} \$$omega^2$$

Question 4

A disk rolls up a frictionless incline and another identical disk rolls up a rough incline. Both start with the same speed. Which disk reaches a greater height?

Accepted Answer

The disk on the rough incline reaches a lesser height.

Question 5

A wheel rolls smoothly without slipping. What is the relationship between the center-of-mass speed $$v_{com}$$ and the angular speed $$\omega$$?

Accepted Answer

$$v_{com} = \omega R$$

Question 6

Which of the following best describes the direction of the torque vector produced by a force applied to a particle at position $$\vec{r}$$?

Accepted Answer

It is perpendicular to both $$\vec{r}$$ and $$\vec{F}$$, determined by the right-hand rule.

Question 7

A gyroscope spins rapidly and precesses about a vertical axis. Which equation gives the precession rate $$\Omega$$?

Accepted Answer

$$\Omega = \frac{Mgr}{I \omega}$$

Question 8

A student spins on a stool and pulls her arms in. What happens to her rotational inertia, angular momentum, and angular speed?

Accepted Answer

Rotational inertia decreases, angular momentum remains constant, angular speed increases.

Question 9

Which equation expresses Newton's second law in angular form for a single particle?

Accepted Answer

$$\vec{	au}_{net} = \frac{d\vec{\ell}}{dt}$$

Question 10

A disk, a hoop, and a solid sphere are spun by pulling a string with the same constant tangential force $$\vec{F}$$ for the same time. Which object has the greatest angular speed after time $$t$$?

Accepted Answer

The solid sphere.

Question 11

Which equation gives the angular momentum $$\vec{\ell} of a $$particle relative to a fixed point?

Accepted Answer

$$\vec{\ell} = \vec{r} 	imes \vec{p}$$

Question 12

If the net external torque acting on a system is zero, what happens to the angular momentum $$\vec{L} of $$the system?

Accepted Answer

$$\vec{L}$$ remains constant.

Question 13

A rhinoceros beetle rides the rim of a rotating disk and crawls toward the center. What happens to the rotational inertia, angular momentum, and angular speed of the beetle-disk system (relative to the central axis)?

Accepted Answer

Rotational inertia decreases, angular momentum remains constant, angular speed increases.

Question 14

Which equation expresses the total angular momentum $$\vec{L} of a $$system of $$n$$ particles?

Accepted Answer

$$\vec{L} = \sum_{j=1}^n \vec{\ell}_j$$

Question 15

A wheel rolls smoothly without slipping. What is the linear speed at the very top of the wheel relative to the ground?

Accepted Answer

$$2 v_{com}$$

Question 16

A yo-yo moves down its string, losing potential energy $$mgh. $$What types of kinetic energy does it gain?

Accepted Answer

Translational and rotational kinetic energy.

Question 17

Which of the following is true about the precession rate $$\Omega of a $$gyroscope?

Accepted Answer

It is independent of the gyroscope's mass.

Question 18

Which equation expresses the conservation of angular momentum for an isolated system?

Accepted Answer

$$\vec{L}_{initial} = \vec{L}_{final}$$

Question 19

A long jumper windmills her arms during flight. What is the purpose of this motion?

Accepted Answer

To transfer angular momentum and keep her upright.

Rolling, Torque, and Angular Momentum: Study Notes

Guida di studio - Domande di pratica

Study guide - Flashcard