Q: For a conical pendulum of length $$L$$ making an angle $$\theta$$ with the vertical, what is the relationship between the gravitational acceleration $$g$$, the radius $$r of $$the circle, and the angular velocity $$\omega$$?

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

Q: A uniform ladder of mass $$m$$ and length $$L$$ leans against a frictionless wall, making an angle $$\theta$$ with the ground. What is the minimum coefficient of static friction $$\mu_s at $$the ground required to prevent slipping?

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

Q: A uniform rod of mass $$m$$ and length $$L is $$pivoted at one end and released from rest at an angle $$\theta$$ above the horizontal. What is the angular acceleration $$\alpha$$ just after release?

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

Question 1

A car of mass $$m$$ starts from rest at the top of a hill of height $$h. It $$rolls down without friction and collides inelastically with a stationary car of mass $$2m at $$the bottom. The two stick together and move up a second hill. What is the maximum height $$h'$$ they reach on the second hill in terms of $$h$$?

Accepted Answer

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

Question 2

Which of the following statements is true for an inelastic collision between two objects where they stick together?

Accepted Answer

Momentum is conserved, but kinetic energy is not.

Question 3

A 2000 kg car with a maximum engine power of 100,000 W attempts to drive up a $$20^\circ$$ incline. What is the maximum speed it can maintain up the hill? (Use $$g = 9.8\ \mathrm{m/s^2$}$$ and $$\sin 20^\circ \approx 0.342)$$

Accepted Answer

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

Question 4

A car of mass $$m is $$turning in a circle of radius $$r on a $$flat road. If the coefficient of static friction is $$\mu_s$$, what is the maximum speed $$v_{max}$$ the car can have without skidding?

Accepted Answer

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

Question 5

A small mass $$m is $$attached to a string and whirled in a horizontal circle of radius $$r on a $$frictionless table. The string passes through a hole in the center and is attached to a hanging mass $$M. $$What angular velocity $$\omega is $$needed so that $$M$$ remains stationary?

Accepted Answer

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

Question 6

For a conical pendulum of length $$L$$ making an angle $$	heta$$ with the vertical, what is the relationship between the gravitational acceleration $$g$$, the radius $$r of $$the circle, and the angular velocity $$\omega$$?

Accepted Answer

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

Question 7

A uniform ladder of mass $$m$$ and length $$L$$ leans against a frictionless wall, making an angle $$	heta$$ with the ground. What is the minimum coefficient of static friction $$\mu_s at $$the ground required to prevent slipping?

Accepted Answer

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

Question 8

A solid disk of mass $$M$$ and radius $$R$$ rotates about an axis through its center. What is its moment of inertia?

Accepted Answer

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

Question 9

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

Accepted Answer

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

Question 10

A solid sphere of mass $$M$$ and radius $$R$$ rolls without slipping down a hill of height $$h. $$What is its speed at the bottom?

Accepted Answer

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

Question 11

A marble (solid sphere) of mass $$m$$ and radius $$R$$ starts from rest at a height $$h$$ and rolls without slipping down a track into a vertical loop of radius $$R. $$What is the minimum height $$h_{min}$$ required for the marble to complete the loop?

Accepted Answer

$$h_{min} = \frac{27}{10}R$$

Question 12

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

Accepted Answer

The net force and the net torque on the body must both be zero.

Question 13

A satellite orbits the Earth at a height $$h$$ above the surface. If the Earth's radius is $$R_E$$, what is the satellite's orbital speed $$v$$?

Accepted Answer

$$v = \sqrt{\frac{GM_E}{R_E + h}}$$

Question 14

What is the escape velocity from the surface of a planet of mass $$M$$ and radius $$R$$?

Accepted Answer

$$v_{esc} = \sqrt{\frac{2GM}{R}}$$

Question 15

A uniform rod of mass $$m$$ and length $$L is $$pivoted at one end and released from rest at an angle $$	heta$$ above the horizontal. What is the angular acceleration $$\alpha$$ just after release?

Accepted Answer

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

Question 16

A rolling object has both translational and rotational kinetic energy. For a solid cylinder of mass $$m$$ and radius $$R$$ rolling without slipping at speed $$v$$, what is its total kinetic energy?

Accepted Answer

$$K = \frac{1}{2}mv^2$ + \frac{1}{4}mv^2$ = \frac{3}{4}mv^2$

Question 17

A car of mass $$m is $$moving at speed $$v$$ and collides inelastically with a stationary car of mass $$2m. $$What fraction of the initial kinetic energy remains after the collision?

Accepted Answer

$$\frac{1}{6}$$

Question 18

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

Accepted Answer

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

Question 19

A block slides down a frictionless hill of height $$h$$ and reaches the bottom with speed $$v. $$What is the relationship between $$v$$ and $$h$$?

Accepted Answer

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

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

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