Q: A conical pendulum of length $$r$$ makes an angle $$\theta$$ with the vertical. What is the expression for $$g in $$terms of $$r$$, $$\omega$$, and $$\theta$$?

$$g = r \$$omega^2$$ \tan \theta$$

Q: A ladder of mass $$m$$ leans against a wall with negligible friction. What is the minimum angle $$\theta$$ for static equilibrium, given the coefficient of friction $$\mu at $$the floor?

$$\tan \theta = \frac{1}{2\mu}$$

Q: Which condition must be satisfied for an object to be in static equilibrium?

$$\vec{F}_{net} = 0$$ and $$\vec{\tau}_{net} = 0$$

Question 1

A car of mass $$m$$ starts from rest at height $$h$$ and rolls down a frictionless hill. What is its speed at the bottom of the hill?

Accepted Answer

$$v = \sqrt{2gh}$$

Question 2

Two cars collide inelastically at the bottom of a hill: car 1 has mass $$m$$ and car 2 has mass $$2m. If $$car 1 rolls down from height $$h$$ and collides with car 2 (initially at rest), what fraction of the initial kinetic energy remains after the collision?

Accepted Answer

$$\frac{1}{3}$$

Question 3

A car of mass $$3m$$ travels up a hill after an inelastic collision at the bottom. If the initial height was $$h$$, what is the maximum height $$h'$$ the car can reach after the collision?

Accepted Answer

$$h' = \frac{h}{9}$$

Question 4

A 2,000 kg car with a maximum engine power of 100 kW attempts to drive up a $$20^\circ$$ incline. What is the maximum speed it can maintain up the hill?

Accepted Answer

$$v_{max} = 15\ \mathrm{m/s}$$

Question 5

Which equation correctly relates the period $$T of a $$wheel rolling without slipping to its angular velocity $$\omega$$?

Accepted Answer

$$T = \frac{2\pi}{\omega}$$

Question 6

A car is turning on a flat road. What is the maximum speed it can maintain without skidding, given the coefficient of static friction $$\mu_s$$ and gravitational acceleration $$g$$?

Accepted Answer

$$v_{max} = \sqrt{\mu_s g r}$$

Question 7

A conical pendulum of length $$r$$ makes an angle $$	heta$$ with the vertical. What is the expression for $$g in $$terms of $$r$$, $$\omega$$, and $$	heta$$?

Accepted Answer

$$g = r \$$omega^2$$ 	an 	heta$$

Question 8

What is the escape velocity from the surface of the Earth, given mass $$M$$ and radius $$r_E$$?

Accepted Answer

$$v_{escape} = \sqrt{\frac{2GM}{r_E}}$$

Question 9

A ladder of mass $$m$$ leans against a wall with negligible friction. What is the minimum angle $$	heta$$ for static equilibrium, given the coefficient of friction $$\mu at $$the floor?

Accepted Answer

$$	an 	heta = \frac{1}{2\mu}$$

Question 10

Which of the following is the correct statement of the parallel-axis theorem for moment of inertia?

Accepted Answer

$$I = I_{cm} + Md^2$

Question 11

A disk of mass $$M$$ and radius $$R is $$released from rest with its center of mass at a horizontal pivot. What is its angular acceleration $$\alpha$$ immediately after release?

Accepted Answer

$$\alpha = \frac{2g \sin \phi}{3R}$$

Question 12

For a solid sphere rolling down a hill of height $$h$$ without slipping, what is its speed at the bottom?

Accepted Answer

$$v = \sqrt{\frac{10gh}{7}}$$

Question 13

What is the ratio $$\beta of $$rotational to translational kinetic energy for a solid sphere rolling without slipping?

Accepted Answer

$$\beta = \frac{2}{5}$$

Question 14

A marble of radius $$R$$ must complete a vertical loop of radius $$R. $$What is the minimum height $$h$$ from which it must start to just make it around the loop?

Accepted Answer

$$h = \frac{27}{10}R$$

Question 15

Which of the following best explains why astronauts feel weightless in orbit?

Accepted Answer

The sensation of weightlessness is due to the lack of normal force, not the absence of gravity.

Question 16

What is the gravitational potential energy $$U(r) of a $$mass $$m at a $$distance $$r$$ from the center of the Earth (mass $$M$$)?

Accepted Answer

$$U(r) = -\frac{GMm}{r}$$

Question 17

A weight $$m is $$attached to a string and rotates in a circle of radius $$r on a $$frictionless table, connected to a hanging mass $$M. $$What is the required angular speed $$\omega$$ for the hanging mass to remain stationary?

Accepted Answer

$$\omega = \sqrt{\frac{Mg}{mr}}$$

Question 18

Which condition must be satisfied for an object to be in static equilibrium?

Accepted Answer

$$\vec{F}_{net} = 0$$ and $$\vec{	au}_{net} = 0$$

Question 19

For a falling ruler of length $$L$$, what is the angular acceleration $$\alpha at $$the instant it is released from a small angle $$\phi$$?

Accepted Answer

$$\alpha = \frac{3g \sin \phi}{L}$$

Physics 211: Momentum, Energy, Rotation, and Gravity – Midterm 3 Review Study Notes

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