Q: A ball of mass $$m$$ and velocity $$v$$ collides with a stationary sphere and bounces off with the same speed but reversed direction. What is the change in angular momentum of the ball about the sphere's center?

$$\Delta L = 2 m v R \sin \theta$$

Question 1

A 3 kg mass is pushed into a hole to a depth $$D = 30 cm$$, compressing a spring with spring constant $$k = 500 N/m. If a $$constant frictional force of $$f = 10 N $$acts as the mass moves up, what is the speed of the mass when it reaches a height $$h = 20 cm $$above the bottom? (Assume the mass starts from rest at the bottom.)

Accepted Answer

$$v = \sqrt{\frac{k(D^2$ - h^2$)}{m} - \frac{2fh}{m}}$$

Question 2

A ball of mass $$m is $$thrown at a stationary sphere of mass $$M$$ and radius $$R$$ with velocity $$v at an $$angle $$	heta$$ from the center. What is the initial angular momentum of the ball about the sphere's center?

Accepted Answer

$$L = m v R \sin 	heta$$

Question 3

A uniform rod of mass $$M$$ and length $$L is $$pivoted at its center. Masses $$m_1$$ and $$m_2$$ are attached to its ends. What is the angular acceleration $$\alpha of $$the rod immediately after release?

Accepted Answer

$$\alpha = \frac{g(m_2$ - m_1$)L}{2I}$$, where $$I = \frac{1}{12}ML^2$ + m_1$(\frac{L}{2})^2$ + m_2$(\frac{L}{2})^2$

Question 4

A rod of length $$l = 10 m is $$hanging from an axle. A small mass $$m = 2 kg is $$attached at a distance $$r = 2 m $$from the axle. A string pulls horizontally at the end of the rod with tension $$T = 200 N at an $$angle $$\phi = 30^\circ$$ from vertical. What is the mass $$M of $$the rod if it is stationary?

Accepted Answer

$$M = \frac{2 T l \sin \phi - 2 m g r}{g l^2$}$$

Question 5

A car makes a quarter-circle turn of radius $$R = 30 m on a $$road with coefficient of static friction $$\mu_s = 0.8. $$What is the shortest time $$t$$ the car could make the turn without sliding?

Accepted Answer

$$t = \frac{\pi R}{2 \sqrt{\mu_s g R}}$$

Question 6

The Moon orbits the Earth in a circular path with period $$T = 27$$ days. Given the mass of the Earth $$M_E$$ and radius $$R_E$$, what is the distance $$r$$ from the center of the Earth to the Moon?

Accepted Answer

$$r = \left( \frac{G M_E T^2$}{4 \pi^2$} \$$right)^{1/3}$$

Question 7

A spaceship hovers at a distance $$r_0$$ from the center of the Moon. An astronaut drops a ball out the window. What is the speed $$v_f of $$the ball when it hits the Moon's surface (radius $$R_m$$)?

Accepted Answer

$$v_f = \sqrt{2 G M_m \left( \frac{1}{R_m} - \frac{1}{r_0$} \right)}$$

Question 8

Which of the following correctly expresses the moment of inertia $$I$$ for a solid sphere of mass $$M$$ and radius $$R$$ about an axis through its center?

Accepted Answer

$$I = \frac{2}{5} M R^2$

Question 9

A rod of length $$L$$ and mass $$M$$ rotates about an axis perpendicular to its length and passing through one end. What is its moment of inertia?

Accepted Answer

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

Question 10

A ball of mass $$m$$ and velocity $$v$$ collides with a stationary sphere and bounces off with the same speed but reversed direction. What is the change in angular momentum of the ball about the sphere's center?

Accepted Answer

$$\Delta L = 2 m v R \sin 	heta$$

Question 11

Which equation gives the gravitational force between two masses $$m_1$$ and $$m_2$$ separated by distance $$r$$?

Accepted Answer

$$F = G \frac{m_1$ m_2$}{r^2$}$$

Question 12

A mass $$m is $$attached to a spring with constant $$k$$ and compressed a distance $$x. $$What is the elastic potential energy stored in the spring?

Accepted Answer

$$U = \frac{1}{2} k x^2$

Question 13

A uniform disk of mass $$M$$ and radius $$R$$ rotates about its center. What is its moment of inertia?

Accepted Answer

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

Question 14

A car of mass $$m$$ moves in a circle of radius $$R at $$speed $$v. $$What is the minimum coefficient of static friction $$\mu_s$$ required to prevent sliding?

Accepted Answer

$$\mu_s = \frac{v^2$}{g R}$$

Question 15

A rod of length $$L$$ and mass $$M is $$suspended horizontally from its center. What is its moment of inertia about the center?

Accepted Answer

$$I = \frac{1}{12} M L^2$

Question 16

A mass $$m is $$dropped from a height $$h$$ above the ground. What is its speed $$v$$ just before hitting the ground (neglecting air resistance)?

Accepted Answer

$$v = \sqrt{2 g h}$$

Question 17

A solid sphere of mass $$M$$ and radius $$R is $$spinning with angular velocity $$\omega. $$What is its rotational kinetic energy?

Accepted Answer

$$K = \frac{1}{2} I \$$omega^2$$ = \frac{2}{5} M R^2$ \$$omega^2$$

Physics Study Guide: Rotational Dynamics, Energy Conservation, Angular Momentum, Static Equilibrium, Circular Motion, and Gravity

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