Essential College Physics

Andrew F. Rex, Richard Wolfson

Chapter 9

Gravitation - all with Video Answers

Educators

Chapter Questions

Problem 1

Explain in your own words why gravitational acceleration $g$ is smaller at the equator than at the poles.

Shahab Ullah

Shahab Ullah

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Problem 2

A satellite orbits Earth in a circle with radius $R$. A second satellite orbits Mars in a circle with the same radius. Which (if either) has the longer orbital period?

Ajay Singhal

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Problem 3

A satellite is in an elliptical orbit around Earth. Does Earth do work on the satellite at any point in its orbit? Repeat for a circular orbit.

Ajay Singhal

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Problem 4

Earth's core is denser than its outer layers. How does this affect gravitational acceleration $g$ at the surface, compared to an Earthlike planet of the same total mass but uniform density?

Ajay Singhal

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Problem 5

Taking air resistance into account, would escape speed from Earth increase, decrease, or remain the same?

Ajay Singhal

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Problem 6

At what time of year is Earth traveling fastest in its orbit? When is it traveling slowest?

Ajay Singhal

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Problem 7

What's the advantage to launching satellites eastward rather then westward? Is it easier to put a satellite into orbit from Florida or Alaska?

Ajay Singhal

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Problem 8

How can you measure the mass of a planet with a moon? What if it has no moons?

Ajay Singhal

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Problem 9

Does escape speed depend on launch angle?

Narayan Hari

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Problem 10

A comet can have a total mechanical energy that's positive, negative, or zero. Describe its path in each case.

Ajay Singhal

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Problem 11

Compare the fuel required for a trip from Earth to Moon with that required for the return trip.

Ajay Singhal

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Problem 12

Why is it said that the Cavendish balance is used to "weigh the Earth"?

Ajay Singhal

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Problem 13

Why doesn't a rocket's escape speed depend on its mass?

Shahab Ullah

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Problem 14

Describe the subsequent motion of a rocket launched with speed twice the escape speed.

Ajay Singhal

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Problem 15

Does a planet's orbital period depend on the planet's mass?

Ajay Singhal

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Problem 16

Why does the Moon dominate Earth's tides, even though the Sun dominates Earth's orbital motion?

Ajay Singhal

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Problem 17

Mars has mass $6.42 \times 10^{23} \mathrm{~kg}$ and radius $3.37 \times 10^{6} \mathrm{~m} .$ What's the gravitational acceleration at the surface of Mars?
(a) $3.8 \mathrm{~m} / \mathrm{s}^{2}$,
(b) $4.9 \mathrm{~m} / \mathrm{s}^{2}$
(c) $6.2 \mathrm{~m} / \mathrm{s}^{2}$,
(d) $9.8 \mathrm{~m} / \mathrm{s}^{2}$

Narayan Hari

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Problem 18

What's the gravitational force between two lead balls with masses $25 \mathrm{~kg}$ and $60 \mathrm{~kg}$ separated by $0.50 \mathrm{~m}$ center to center?
(a) $5 \times$
$10^{-8} \mathrm{~N}$
(b) $1 \times 10^{-7} \mathrm{~N}$;
(c) $2 \times 10^{-7} \mathrm{~N}$
(d) $4 \times 10^{-7} \mathrm{~N}$

Narayan Hari

Numerade Educator

Problem 19

A $150-\mathrm{kg}$ satellite is in circular orbit with radius $7.1 \times 10^{6} \mathrm{~m}$ around another planet. If its orbital period is 4 hours, the planet's mass is
(a) $10^{27} \mathrm{~kg}$;
(b) $10^{26} \mathrm{~kg}$
(c) $10^{25} \mathrm{~kg}$;
(d) $10^{24} \mathrm{~kg}$.

Narayan Hari

Numerade Educator

Problem 20

A satellite is in a 1.5 -hour-period circular orbit around Earth. If the satellite moves to a new orbit with radius twice that of the original, the new period is
(a) 2.4 hours;
(b) 2.8 hours;
(c) 3.0 hours;
(d) 4.2 hours.

Narayan Hari

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Problem 21

A satellite is in a $7000-\mathrm{km}$ -radius circular orbit around Earth. If the satellite moves to a new orbit with period twice that of the original, the new radius is
(a) $10,000 \mathrm{~km}$
(b) $11,100 \mathrm{~km}$;
(c) $14,000 \mathrm{~km}$
(d) $19,800 \mathrm{~km}$

Narayan Hari

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Problem 22

A rocket launched from Earth at $2400 \mathrm{~m} / \mathrm{s}$ reaches a maximum height of (a) $300 \mathrm{~km} ;$
(b) $600 \mathrm{~km} ;$ (c) $900 \mathrm{~km}$ (d) infinite height (escape).

Narayan Hari

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Problem 23

Ignoring air resistance, a ball dropped from rest at altitude $250 \mathrm{~km}$
strikes the ground with speed (a) $1500 \mathrm{~m} / \mathrm{s} ;$ (b) $2200 \mathrm{~m} / \mathrm{s}$; (c) $2800 \mathrm{~m} / \mathrm{s}$ (d) $3400 \mathrm{~m} / \mathrm{s}$

Narayan Hari

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Problem 24

The period of a satellite's orbit depends on (a) its mass; (b) the radius of the planet it's orbiting; (c) the planet's mass;
(d) all three factors (a), (b), and (c).

Narayan Hari

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Problem 25

A planet's closest approach to the Sun is its (a) aphelion: (b) perihelion; (c) minor axis; (d) eccentricity.

Ajay Singhal

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Problem 26

The total energy of a satellite in elliptical orbit (a) is zero; (b) is positive; (c) is negative; (d) cannot be determined without more information about the orbit.

Ajay Singhal

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Problem 27

What's the period of a satellite in circular orbit $4000 \mathrm{~km}$ above Earth's surface? (a) 1.5 hours; (b) 2.9 hours; (c) 5.2 hours; (d) 7.1 hours.

Narayan Hari

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Problem 28

Saturn's moon Titan is in a nearly circular orbit of radius $1.22 \times 10^{9} \mathrm{~m}$ and period 15.9 days. From these data determine the mass of Saturn and compare with the value in Appendix $\mathrm{E}$

Ajay Singhal

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Problem 28

Which of the following is true about ocean tides? (a) They're caused primarily by the Sun. (b) They're most extreme when the Moon is full or new. (c) They're most extreme close to the North Pole. (d) They aren't affected by the Sun.

Ajay Singhal

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Problem 29

Which of the following is true about ocean tides? (a) They're caused primarily by the Sun. (b) They're most extreme when the Moon is full or new. (c) They're most extreme close to the North Pole. (d) They aren't affected by the Sun.

Ajay Singhal

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Problem 30

Two identical balls, their centers $25.0 \mathrm{~cm}$ apart, experience a mutual gravitational force of $8.8 \times 10^{-6} \mathrm{~N}$. Find each ball's mass.

Narayan Hari

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Problem 31

Two stars with masses $2.3 \times 10^{30} \mathrm{~kg}$ and $6.8 \times 10^{30} \mathrm{~kg}$ form a binary system, with interstellar separation $8.8 \times 10^{11} \mathrm{~m}$. (a) Find the magnitude of the force between these stars. (b) Compare with the force between Earth and Sun.

Narayan Hari

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Problem 32

Compute the net force on Earth due to the Sun and Moon

Susan Hallstrom

Susan Hallstrom

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Problem 33

In a hydrogen atom, proton and electron are separated by $5.29 \times 10^{-11} \mathrm{~m} .$ What's the gravitational force between the two particles?

Narayan Hari

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Problem 34

Use the data in Appendix $\mathrm{E}$ to compute the gravitational acceleration $g$ at the surfaces of Mercury and Venus.

Narayan Hari

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Problem 35

Use the data in Appendix E to compute the gravitational acceleration $g$ at the surfaces of Saturn and Jupiter.

Narayan Hari

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Problem 36

A spherical asteroid with radius of $9.50 \mathrm{~km}$ has uniform density $3500 \mathrm{~kg} / \mathrm{m}^{3}$. Find the gravitational acceleration at the asteroid's surface

Ajay Singhal

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Problem 37

Deimos is one of Mars's two small moons. It orbits Mars in a nearly circular orbit of radius $23,500 \mathrm{~km}$ with a period of 1.26 days. From these data determine the mass of Mars and compare with the value in Appendix E

Ajay Singhal

Numerade Educator

Problem 39

In a typical Cavendish experiment, two small lead balls on a rod are attracted to two large lead balls, as shown in Figure $9.7 .$ The small and large balls are, respectively, $1.0 \mathrm{~cm}$ and $3.4 \mathrm{~cm}$ in diameter, and the density of lead is $11,350 \mathrm{~kg} / \mathrm{m}^{3}$. (a) Find the attractive force between a small ball and a large one when their sur faces are $0.50 \mathrm{~cm}$ apart. (b) The rod connecting the small balls is $15 \mathrm{~cm}$ long. Find the torque on the apparatus under these conditions.

Ajay Singhal

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Problem 40

Earth's orbit is approximately a circle of radius $1.50 \times 10^{11} \mathrm{~m}$
(a) Find Earth's centripetal acceleration.
(b) Use your answer to compute the gravitational force exerted on Earth by the Sun.
(c) Show that your answer to part (b) agrees with the force computed directly from Newton's law of gravitation.

Shoukat Ali

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Problem 41

If At what height above Earth's surface is the gravitational acceleration reduced from its sea-level value by (a) $0.1 \%$,
(b) $1.0 \%$ and (c) $10 \% ?$

Ajay Singhal

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Problem 42

In ure P9.42. Find the net gravitational force on the ball at the origin.

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Problem 43

Find the eccentricity of an ellipse whose major axis is twice its minor axis.

Narayan Hari

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Problem 44

Draw ellipses with the following eccentricities: 0.01,0.1,0.5,0.9

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Problem 45

Use data for Mars in Appendix $\mathrm{E}$ to determine the distance of the Sun (at one focus of the ellipse) from the orbit's geometric center.

Narayan Hari

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Problem 46

Draw to scale on the same diagram the elliptical orbits of Earth and Mars.

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Problem 47

Compute the constant $C=a^{3} / T^{2}$ for the orbits of Mercury, Venus, and Jupiter.

Narayan Hari

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Problem 48

In Find the perihelion and aphelion distances for Pluto. Compare with Neptune's average distance from the Sun.

Shoukat Ali

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Problem 49

In Satellites $A$ and $B$ are in circular orbits around Earth, with $A$ twice as far as $\mathrm{B}$ from Earth's center. How do their orbital periods compare?

Ajay Singhal

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Problem 50

Use conservation of angular momentum to estimate the difference in Earth's orbital speed from perihelion to aphelion.

Narayan Hari

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Problem 51

In A satellite is in circular orbit at a height $R_{\mathrm{E}}$ above Earth's surface. (a) Find its orbital period. (b) What height is required for a circular orbit with a period double that found in part (a)?

Shoukat Ali

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Problem 52

In The Moon orbits Earth in a nearly circular orbit of radius $3.84 \times 10^{8} \mathrm{~m}$ and period 27.3 days. Compute the constant $C=a^{3} / T^{2}$ for this orbit, and use the result to find the period of a satellite in circular orbit $500 \mathrm{~km}$ above Earth's surface.

Shoukat Ali

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Problem 53

A Hohmann ellipse is the trajectory for a spacecraft going from one planet to another that requires the least energy. This ellipse has its perihelion at one planet's orbit and its aphelion at the other's (Figure P9.53). Find the travel time from Earth to
Jupiter along such a tra jectory, and compare your answer with the orbital periods of those planets.

Shoukat Ali

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Problem 54

Find the potential energy of (a) the Earth-Sun system and (b) the Earth-Moon system.

Ajay Singhal

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Problem 55

Compare the potential energy of (a) the Sun-Jupiter system and (b) the Sun-Saturn system.

Ajay Singhal

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Problem 56

(a) What's the gravitational potential energy of the electron and proton in an atom, $5.29 \times 10^{-11} \mathrm{~m}$ apart? (b) How does this compare with the atomic binding energy of about $2 \times 10^{-18} \mathrm{~J} ?$ (c) Comment on the influence of gravity in the atom.

Shoukat Ali

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Problem 57

Ignoring air resistance, find the speed of a $1-\mathrm{kg}$ ball when it strikes the ground after being dropped from rest at heights of
(a) $10 \mathrm{~km}$;
(b) $1000 \mathrm{~km}$;
(c) $10^{7} \mathrm{~m}$.

Shoukat Ali

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Problem 58

Find the maximum height of a rocket shot straight up from Earth's surface at $2200 \mathrm{~m} / \mathrm{s}$

Ajay Singhal

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Problem 59

What's the change in potential energy of the Earth-Sun system from perihelion to aphelion? Use your answer and conservation of energy to find the corresponding difference in Earth's orbital speed.

Shoukat Ali

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Problem 60

Halley's comet reaches a maximum speed of $54.6 \mathrm{~km} / \mathrm{s}$ at perihelion, $8.84 \times 10^{10} \mathrm{~m}$ from the Sun. Find its speed at aphelion, $5.27 \times 10^{12} \mathrm{~m}$ from the Sun.

Narayan Hari

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Problem 61

What's the total mechanical energy of a $500-\mathrm{kg}$ satellite in circular orbit $1500 \mathrm{~km}$ above Earth's surface?

Narayan Hari

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Problem 62

What's the total mechanical energy of a $500-\mathrm{kg}$ satellite in circular orbit $1500 \mathrm{~km}$ above Earth's surface?

Shoukat Ali

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Problem 63

With what speed does a rock strike the Moon when it's dropped from rest from a great distance above the Moon's surface?

Shoukat Ali

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Problem 64

One proposal for dealing with radioactive waste is to shoot it into the Sun. Suppose a waste canister were dropped from rest, starting in the vicinity of Earth's orbit. At what speed would it hit the Sun?

Shoukat Ali

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Problem 65

What's the orbital radius of a satellite in a 48 -hour-period circular Earth orbit?

Narayan Hari

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Problem 66

If Find the period of a satellite with a circular orbit $150 \mathrm{~km}$ above Earth's surface

Narayan Hari

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Problem 67

For elliptical Earth orbits, the near and far points are called the perigee and apogee. Find the period of a satellite with perigee and apogee $200 \mathrm{~km}$ and $1600 \mathrm{~km}$ above Earth's surface, respectively.

Shoukat Ali

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Problem 67

Find the energy needed to place a $1.0-\mathrm{kg}$ satellite, initially at rest on Earth's surface, into geosynchronous orbit.

Ajay Singhal

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Problem 68

Mars has mass $6.42 \times 10^{23} \mathrm{~kg}$ and radius $3.37 \times 10^{6} \mathrm{~m} .$ Find the escape speed from Mars's surface.

Narayan Hari

Numerade Educator

Problem 69

Determine the escape speed from (a) Jupiter's moon Callisto, with mass $1.07 \times 10^{23} \mathrm{~kg}$ and radius $2.40 \times 10^{6} \mathrm{~m},$ and $(\mathrm{b})$ a neutron star with the Sun's mass and a radius of $6.0 \mathrm{~km}$.

Ajay Singhal

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Problem 70

A satellite is in geosynchronous orbit. (a) What is the satellite's speed? (b) What additional speed is required for the satellite to escape Earth's gravity completely?

Shoukat Ali

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Problem 71

The Moon's rotation period is 27.3 days. What's the height of a "lunosynchronous" satellite that orbits the Moon, appearing fixed in the sky relative to a lunar observer? Compare your answer to the Moon's radius.

Shoukat Ali

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Problem 72

(a) Calculate the orbital period of Jupiter's moon Io, which is in a nearly circular orbit of radius $4.22 \times 10^{5} \mathrm{~km}$. (b) What's Io's orbital speed?

Narayan Hari

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Problem 73

A binary system comprises two stars of equal mass $M$ separated by distance $d$, orbiting their common center of mass. (a) Find the period of their orbital motion. (b) Compare your result with the period of a small planet $(m<M)$ orbiting an isolated star of the same mass $M$ in a circular orbit with the same radius $d$.

Shoukat Ali

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Problem 74

I I Suppose a satellite with a 24 -hour period over Earth's equator has a noncircular, elliptical orbit. (a) Explain why this satellite doesn't stay above the same point. (b) What is the maximum possible eccentricity of its orbital ellipse?

Shoukat Ali

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Problem 75

In Find (a) the kinetic energy, (b) the potential energy, and (c) the total mechanical energy of the Moon in its orbit.

Ajay Singhal

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Problem 77

I mechanical energy $-4.0 \times 10^{10} \mathrm{~J} .$ Find the satellite's (a) kinetic energy, (b) mass, and (c) speed.

Ajay Singhal

Numerade Educator

Problem 78

The Moon has no atmosphere, making extremely low orbits possible. Find the period of a circular orbit just above the Moon's surface $\left(M_{\text {Moon }}=7.35 \times 10^{22} \mathrm{~kg}, R_{\text {Moon }}=1740 \mathrm{~km}\right)$

Narayan Hari

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Problem 79

In 1968 , Frank Borman, Jim Lovell, and Bill Anders became the first humans to orbit the Moon. Their orbit was nearly circular, ranging from 59.7 to 60.7 miles above the lunar surface $(1$ mile $=1.609 \mathrm{~km}) .$ Find the orbital period, and compare it with the results of the preceding problem.

Shoukat Ali

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Problem 80

(a) To what radius would Earth have to be shrunk, with no loss of mass, in order for escape speed from its surface to double?
(b) What would be Earth's average density under those conditions?

Ajay Singhal

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Problem 81

Find the gravitational acceleration $20 \mathrm{~km}$ from the center of a $9.9 \times 10^{30}-\mathrm{kg}$ black hole. (This is far enough for Newtonian physics to be applicable.)

Narayan Hari

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Problem 82

Our solar system is 25,000 light years from the center of the Milky Way galaxy. (a) If our solar system orbits the galactic center in a circular orbit with speed $230 \mathrm{~km} / \mathrm{s},$ what's its orbital period? (b) What's the approximate mass of the galactic center assuming it's essentially spherical?

Shoukat Ali

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Problem 83

The mean center-to-center Earth-Moon distance is $3.84 \times 10^{8} \mathrm{~m} .$ Evaluate the Moon's tidal effect by calculating the gravitational acceleration of a water droplet on the side of the Earth (a) closest to and (b) farthest from the Moon. (c) Calculate the difference between these two accelerations -which is a measure of the Moon's tidal effect.

Shoukat Ali

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Problem 84

Repeat the preceding problem for the Earth-Sun tidal interaction. You should find a greater solar acceleration in (a) and (b), but a smaller difference in (c), showing why the Sun's tidal effect is smaller than the Moon's despite the Sun's greater gravitational force on Earth.

Shoukat Ali

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Problem 85

Tidal forces can break apart objects that orbit too close to a planet, producing rings like those of Saturn. Consider an asteroid $500 \mathrm{~km}$ in radius, 75 Mm from Saturn's center. What's the difference in gravitational acceleration between the two sides of the asteroid?

Shoukat Ali

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Problem 86

A small, spherical asteroid has radius $3.50 \mathrm{~km}$ and density $4830 \mathrm{~kg} / \mathrm{m}^{3} .$ (a) What's the gravitational acceleration at its surface? (b) What's the escape speed from the surface?

Ajay Singhal

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Problem 87

Rank in increasing order the gravitational acceleration at the surface of planets that have the given masses and radii, where $M_{\mathrm{E}}$ and $R_{\mathrm{E}}$ are the mass and radius of Earth. Assume spherically symmetric planets. (a) $M=M_{\mathrm{E}}, R=R_{\mathrm{E}}$; (b) $M=2 M_{\mathrm{E}}, R=2 R_{\mathrm{E}}$
(c) $M=0.5 M_{\mathrm{E}}, \quad R=0.5 R_{\mathrm{E}}$;
(d) $M=1.8 M_{\mathrm{E}}, \quad R=1.5 R_{\mathrm{E}}$
(e) $M=0.75 M_{\mathrm{E}}, R=0.90 R_{\mathrm{E}}$

Ajay Singhal

Numerade Educator

Problem 88

An Earth satellite's elliptical orbit ranges from $230 \mathrm{~km}$ to $890 \mathrm{~km}$ altitude. At the high point, it's moving at $7.23 \mathrm{~km} / \mathrm{s} .$ How fast is it moving at the low point?

Donald Albin

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Problem 89

BI0 In High jump. The winning high jump in the 2004 Summer Olympics was $2.34 \mathrm{~m}$. Assuming the same takeoff conditions, how high could the winner jump on Mars? Repeat for the long jump, for which the winning jump was $8.59 \mathrm{~m}$

Ajay Singhal

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Problem 90

You're tour director for a lunar trip, and want to award your passengers with certificates commemorating their crossing the point where the Moon's gravity becomes stronger than Earth's. How far from Earth should you award the certificate? Express

Shoukat Ali

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Problem 91

Mercury's orbital speed varies from $38.8 \mathrm{~km} / \mathrm{s}$ at aphelion to $59.0 \mathrm{~km} / \mathrm{s}$ at perihelion.
(a) If aphelion is $6.99 \times 10^{10} \mathrm{~m}$ from the Sun's center, how far is perihelion? (b) Use your answer to compute Mercury's orbital eccentricity, and compare with Appendix E

Ajay Singhal

Numerade Educator

Problem 92

Voyager 1 and Voyager $2,$ launched in $1977,$ are the first human-made objects to leave our solar system. Find the minimum speed with which such an object must leave Earth's orbit to escape the Sun's gravity.

Narayan Hari

Numerade Educator

Problem 93

In a spacecraft is in circular Earth orbit at $5500 \mathrm{~km}$ altitude. By how much will its altitude decrease if it moves to a new circular orbit where (a) its orbital speed is $10 \%$ higher or (b) its orbital period is $10 \%$ lower?

Shoukat Ali

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Problem 94

A $50-\mathrm{kg}$ satellite is placed in circular orbit, $750 \mathrm{~m}$ above the surface of the asteroid in Problem $86 .$ What's its orbital period? Why is the answer so different from the periods of Earth-orbiting satellites?

Shoukat Ali

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Problem 95

Tidal effects cause the Moon's orbital period to increase at about $35 \mathrm{~ms}$ per century. Assuming a circular orbit, to what rate of change in the Earth-Moon distance does this correspond?

Shoukat Ali

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Problem 96

Two satellites are in geosynchronous orbit but diametrically opposite positions (Figure GP9.96). Into how much lower a circular orbit should one spacecraft descend if it's to catch up with the other after 10 complete orbits? Neglect rocket firing times and the time spent moving from one circular orbit to another.

Dading Chen

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Problem 97

Suppose you can jump $55 \mathrm{~cm}$ high from a standing start on Earth. Find the radius of the largest spherical asteroid you could escape by jumping, assuming a uniform asteroid density of $4000 \mathrm{~kg} / \mathrm{m}^{3}$

Shoukat Ali

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Problem 98

(a) Find the change in potential energy of the Earth-Sun system between Earth's aphelion and perihelion.
(b) Find the corresponding change in Earth's orbital kinetic energy.

Shoukat Ali

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Problem 99

An object collapses to a black hole when escape speed at its surface becomes the speed of light. At what radius does that occur for an object with (a) the Sun's mass and (b) the mass of a galaxy, approximately $10^{11}$ solar masses. (You can use Newtonian gravitation for this calculation.)

Shoukat Ali

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