Q: A car is rounding a curve of radius $$30\ \mathrm{m} at a $$constant speed. If the coefficient of static friction between the tires and the road is $$0.8$$, what is the maximum speed at which the car can make the turn without sliding?

$$v_{\text{max}} = \sqrt{0.8 \times 9.8 \times 30} = 15.3\ \mathrm{m/s}$$

Q: A 0.5 kg ball is swung in a horizontal circle of radius $$2\ \mathrm{m} at a $$speed of $$4\ \mathrm{m/s}. $$What is the tension in the string?

$$T = \frac{mv^2$}{r} = \frac{0.5 \times 4^2$}{2} = 4\ \mathrm{N}$$

Q: A satellite orbits Earth at a radius $$r$$ with speed $$v. If $$the radius of the orbit is doubled, what happens to the orbital speed?

It decreases by a factor of $$\sqrt{2}$$: $$v_{\text{new}} = \frac{v}{\sqrt{2}}$$

Q: Two asteroids attract each other with a gravitational force $$F. If $$the distance between them is tripled, what is the new force?

$$F_{\text{new}} = \frac{F}{9}$$

Q: A 1500 kg car goes over a hill of radius $$60\ \mathrm{m} at$$ $$20\ \mathrm{m/s}. $$What is the normal force on the car at the top of the hill?

$$n = mg - \frac{mv^2$}{r} = 1500 \times 9.8 - \frac{1500 \times 20^2$}{60} = 14,700 - 10,000 = 4,700\ \mathrm{N}$$

Q: A 0.0030 kg sample is spun in an ultracentrifuge with a centripetal acceleration of $$2.45 \times 10^6$\ \mathrm{m/s^2$}. $$What is its apparent weight?

$$w_{\text{app}} = ma = 0.0030 \times 2.45 \times 10^6$ = 7.4\ \mathrm{N}$$

Question 1

A car is rounding a curve of radius $$30\ \mathrm{m} at a $$constant speed. If the coefficient of static friction between the tires and the road is $$0.8$$, what is the maximum speed at which the car can make the turn without sliding?

Accepted Answer

$$v_{	ext{max}} = \sqrt{0.8 	imes 9.8 	imes 30} = 15.3\ \mathrm{m/s}$$

Question 2

A 0.5 kg ball is swung in a horizontal circle of radius $$2\ \mathrm{m} at a $$speed of $$4\ \mathrm{m/s}. $$What is the tension in the string?

Accepted Answer

$$T = \frac{mv^2$}{r} = \frac{0.5 	imes 4^2$}{2} = 4\ \mathrm{N}$$

Question 3

A satellite orbits Earth at a radius $$r$$ with speed $$v. If $$the radius of the orbit is doubled, what happens to the orbital speed?

Accepted Answer

It decreases by a factor of $$\sqrt{2}$$: $$v_{	ext{new}} = \frac{v}{\sqrt{2}}$$

Question 4

A 70-kg astronaut is in a spacecraft orbiting Earth. The spacecraft and astronaut are both in free fall. What is the astronaut’s apparent weight?

Accepted Answer

Zero, because both are in free fall and there is no normal force.

Question 5

A coin sits on a turntable rotating at constant speed. What force provides the centripetal acceleration?

Accepted Answer

Friction between the coin and the turntable.

Question 6

A roller coaster car of mass $$m$$ goes over the top of a circular hill of radius $$R at $$speed $$v. $$What is the normal force on the car at the top?

Accepted Answer

$$n = mg - \frac{mv^2$}{R}$$

Question 7

Two asteroids attract each other with a gravitational force $$F. If $$the distance between them is tripled, what is the new force?

Accepted Answer

$$F_{	ext{new}} = \frac{F}{9}$$

Question 8

A 1500 kg car goes over a hill of radius $$60\ \mathrm{m} at$$ $$20\ \mathrm{m/s}. $$What is the normal force on the car at the top of the hill?

Accepted Answer

$$n = mg - \frac{mv^2$}{r} = 1500 	imes 9.8 - \frac{1500 	imes 20^2$}{60} = 14,700 - 10,000 = 4,700\ \mathrm{N}$$

Question 9

A communications satellite is in a geostationary orbit. What is true about its period?

Accepted Answer

It is equal to the period of Earth’s rotation: $$T = 24\ \mathrm{hours}$$

Question 10

A ball is shot through a horizontal tube and exits into open space. What path does it follow after leaving the tube?

Accepted Answer

It moves in a straight line tangent to the exit point.

Question 11

A 0.0030 kg sample is spun in an ultracentrifuge with a centripetal acceleration of $$2.45 	imes 10^6$\ \mathrm{m/s^2$}. $$What is its apparent weight?

Accepted Answer

$$w_{	ext{app}} = ma = 0.0030 	imes 2.45 	imes 10^6$ = 7.4\ \mathrm{N}$$

Question 12

A planet has twice the mass and twice the radius of Earth. If $$g_{	ext{Earth}} = 9.8\ \mathrm{m/s^2$}$$, what is the surface gravity on this planet?

Accepted Answer

$$g = \frac{GM}{R^2$} = \frac{2GM_{	ext{Earth}}}{(2R_{	ext{Earth}})^2$} = \frac{2}{4}g_{	ext{Earth}} = 4.9\ \mathrm{m/s^2$}$$

Question 13

A satellite is moved from a low Earth orbit to a much higher orbit. Which of the following increases?

Accepted Answer

The period of the satellite.

Question 14

A 65 kg student sits 0.60 m from another 65 kg student. What is the gravitational force between them?

Accepted Answer

$$F = \frac{Gm_1m_2}{r^2$} = \frac{6.67 	imes 10$$^{-11}$$ 	imes 65 	imes 65}{0.60^2$} = 7.8 	imes 10$$^{-7}$$\ \mathrm{N}$$

Question 15

A ball at the end of a string is swung in a horizontal circle at constant speed. What is the direction of the acceleration?

Accepted Answer

Toward the center of the circle.

Question 16

A car is turning a corner on a level road. What force provides the centripetal acceleration?

Accepted Answer

Static friction between the tires and the road.

Chapter 6: Circular Motion, Orbits, and Gravity – Study Notes

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