Q: After a perfectly inelastic collision, two identical rovers (each mass $$m) $$move together at $$5\ \mathrm{m/s}. To $$what vertical height $$h$$ can they rise on Mars ($$g = 3.7\ \mathrm{m/s^2$}) if $$all kinetic energy is converted to gravitational potential energy?

$$h = \dfrac{25}{2 \times 3.7} = 3.4\ \mathrm{m}$$

Question 1

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

Accepted Answer

$$v = \sqrt{\dfrac{7gh}{5}}$$

Question 2

Why do astronauts in low orbit around Jupiter feel weightless, even though the gravitational force is stronger than on Earth?

Accepted Answer

Because they are in free fall, experiencing no normal force.

Question 3

A car is driving at $$30\ \mathrm{m/s}$$ around a circular turn with a radius of $$100\ \mathrm{m}. If $$the coefficient of static friction is $$0.9$$, will the car make the turn without sliding? (Take $$g = 9.8\ \mathrm{m/s^2$})$$

Accepted Answer

No, because the required centripetal force exceeds the maximum frictional force.

Question 4

A hollow sphere and a solid sphere of equal mass and radius roll down the same incline from rest. Which reaches the bottom first and why?

Accepted Answer

The solid sphere, because it has a smaller moment of inertia and thus more translational kinetic energy.

Question 5

If two adult humans can push a car with almost as much force as a car engine, why can't they accelerate the car to a much higher speed?

Accepted Answer

Because their maximum power output is much lower than that of a car engine.

Question 6

A tree falls to the ground. Why do taller trees fall more slowly than shorter trees?

Accepted Answer

Taller trees are longer, so they have more rotational inertia.

Question 7

A solid marble of mass $$0.1\ \mathrm{kg}$$ rolls down a $$7\ \mathrm{m}$$ high slope. How much rotational kinetic energy does it have at the bottom?

Accepted Answer

$$K_r = 2\ \mathrm{J}$$

Question 8

A plane's center of mass is ahead of its main wings. Why must the tail provide a downward force during level flight?

Accepted Answer

To balance the upward lift from the wings and maintain zero net torque.

Question 9

A rover of mass $$m$$ travels at $$10\ \mathrm{m/s} on $$Mars and collides with an identical stationary rover. After sticking together, what is their velocity?

Accepted Answer

$$5\ \mathrm{m/s}$$

Question 10

Given Mars' mass $$6.4 	imes 10$$^{23}$$\ \mathrm{kg}$$ and radius $$3.4 	imes 10^6$\ \mathrm{m}$$, what is the gravitational acceleration $$g on $$Mars?

Accepted Answer

$$3.7\ \mathrm{m/s^2$}$$

Question 11

A solid sphere and a hollow sphere of the same mass and radius roll down the same incline. Which has more rotational kinetic energy at the bottom and why?

Accepted Answer

The hollow sphere, because more of its energy is in rotation due to a larger moment of inertia.

Question 12

A car of mass $$m$$ makes a turn of radius $$r at $$speed $$v. $$What is the minimum coefficient of static friction $$\mu_s$$ required to prevent slipping?

Accepted Answer

$$\mu_s = \dfrac{v^2$}{rg}$$

Question 13

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

Accepted Answer

$$I = \dfrac{1}{3}ML^2$

Question 14

A solid sphere of mass $$m$$ and radius $$R$$ rolls without slipping. What fraction of its total kinetic energy is rotational?

Accepted Answer

$$\dfrac{2}{7}$$

Question 15

A car of mass $$m is $$pushed by two adults with a constant force $$F$$ but cannot reach high speeds. Which equation best explains this limitation?

Accepted Answer

$$P = Fv$$

Question 16

A marble of mass $$m$$ rolls down a $$7\ \mathrm{m}$$ high slope. If it is a hollow sphere, how does its speed at the bottom compare to a solid sphere?

Accepted Answer

It is less, because the hollow sphere has a larger moment of inertia.

Question 17

A rover of mass $$m on $$Mars ($$g = 3.7\ \mathrm{m/s^2$}) $$moves at $$5\ \mathrm{m/s}$$ after a collision. What is its kinetic energy?

Accepted Answer

$$K = 25m\ \mathrm{J}$$

Question 18

After a perfectly inelastic collision, two identical rovers (each mass $$m) $$move together at $$5\ \mathrm{m/s}. To $$what vertical height $$h$$ can they rise on Mars ($$g = 3.7\ \mathrm{m/s^2$}) if $$all kinetic energy is converted to gravitational potential energy?

Accepted Answer

$$h = \dfrac{25}{2 	imes 3.7} = 3.4\ \mathrm{m}$$

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

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