Physics

James S. Walker

Chapter 12

Gravity - all with Video Answers

Educators

+ 1 more educators

Chapter Questions

Problem 1

System A has masses $m$ and $m$ separated by a distance $r ;$ system $B$ has masses $m$ and 2$m$ separated by a distance $2 r ;$ system $C$ has masses 2$m$ and 3$m$ separated by a distance $2 r,$ and system $D$ has masses 4 $\mathrm{m}$ and 5 $\mathrm{m}$ separated by a distance 3$r .$ Rank these systems in order of increasing gravitational force. Indicate ties where
appropriate.

Andy Chen

Andy Chen

Numerade Educator

Problem 2

A $6.3$ -kg bowling ball and a 7.1 -kg bowling ball rest on a rack 0.85 $\mathrm{m}$ apart. (a) What is the force of gravity exerted on each of the balls by the other ball? (b) At what separation is the force of gravity between the balls equal to $2.0 \times 10^{-9} \mathrm{N} ?$

Andy Chen

Numerade Educator

Problem 3

A communications satellite with a mass of 520 $\mathrm{kg}$ is in a circular orbit about the Earth. The radius of the orbit is $35,000 \mathrm{km}$ as measured from the center of the Earth. Calculate (a) the weight of the satellite on the surface of the Earth and (b) the gravitational force exerted on the orbiting satellite by the Earth.

Andy Chen

Numerade Educator

Problem 4

$\cdot$ The Attraction of Ceres Ceres, the largest asteroid known, has a mass of roughly $8.7 \times 10^{20} \mathrm{kg}$ . If Ceres passes within $14,000 \mathrm{km}$ of the spaceship in which you are traveling, what force does it exert on you? (Use an approximate value for your mass, and treat yourself and the asteroid as point objects.)

Andy Chen

Numerade Educator

Problem 5

In one hand you hold a 0.13 -kg apple, in the other hand a 0.22 -kg orange. The apple and orange are separated by 0.75 $\mathrm{m}$ . What is the magnitude of the force of gravity that (a) the orange exerts on the apple and (b) the apple exerts on the orange?

Andy Chen

Numerade Educator

Problem 6

Predict/Calculate A spaceship of mass $m$ travels from the Earth to the Moon along a line that passes through the center of the Earth and the center of the Moon. (a) At what distance from the center of the Earth is the force due to the Earth twice the magnitude of the force due to the Moon? (b) How does your answer to part (a) depend on the mass of the spaceship? Explain.

Andy Chen

Numerade Educator

Problem 7

At new moon, the Earth, Moon, and Sun are in a line, as indicated in FisuRE $12-33 .$ Find the direction and magnitude of the net gravitational force exerted on (a) the Earth, (b) the Moon, and
(c) the Sun.

Supratim Pal

Numerade Educator

Problem 8

When the Earth, Moon, and Sun form a right triangle, with the Moon located at the right angle, as shown in FIGURE $12 \cdot 34,$ the Moon is in its third-quarter phase. (The Earth is viewed here from above its North Pole.) Find the magnitude and direction of the net force exerted on the Moon. Give the direction relative to the line connecting the Moon and the Sun.

Andy Chen

Numerade Educator

Problem 9

Repeat the previous problem, this time finding the magnitude and direction of the net force acting on the Sun. Give the direction relative to the line connecting the Sun and the Moon.

Andy Chen

Numerade Educator

Problem 10

Predict/Calculate Three 7.25 -kg masses are at the corners of an equilateral triangle and located in space far from any other masses. (a) If the sides of the triangle are 0.610 $\mathrm{m}$ long, find the magnitude of the net force exerted on each of the three masses. (b) How does your answer to part (a) change if the sides of the triangle are doubled in length?

Andy Chen

Numerade Educator

Problem 11

Predict/Calculate Four masses are positioned at the corners of a rectangle, as indicated in FIGURE $12-35$ (a) Find the magnitude and direction ofthe net force acting on the 2.0 -kg mass. (b) How
do your answers to part (a) change (if at all) if all sides of the rectangle are doubled in length?

Andy Chen

Numerade Educator

Problem 12

Suppose that three astronomical objects $(1,2,$ and 3$)$ are observed to lie on a line, and that the distance from object 1 to object 3 is $D .$ Given that object 1 has four times the mass of object
3 and seven times the mass of object $2,$ find the distance between objects 1 and 2 for which the net force on object 2 is zero.

Penny Riley

Numerade Educator

Problem 13

Find the acceleration due to gravity on the surface of (a) Mercury and (b) Venus.

Andy Chen

Numerade Educator

Problem 14

At what altitude above the Earth's surface is the acceleration due to gravity equal to $g / 4 ?$

Andy Chen

Numerade Educator

Problem 15

Two 6.4 -kg bowling balls, each with a radius of $0.11 \mathrm{m},$ are in contact with one another. What is the gravitational attraction between the bowling balls?

Andy Chen

Numerade Educator

Problem 16

What is the acceleration due to Earth's gravity at a distance from the center of the Earth equal to the orbital radius of the Moon?

Andy Chen

Numerade Educator

Problem 17

Extrasolar Planet Gravity Kepler-62e is an exoplanet that orbits within the habitable zone around its parent star. The planet has a mass that is 3.57 times larger than Earth's and a radius that is 1.61 times larger than Earth's. Calculate the acceleration of gravity on the surface of Kepler-62e.

Andy Chen

Numerade Educator

Problem 18

Predict/Calculate At a certain distance from the center of the Earth, a 5.0 -kg object has a weight of 3.6 $\mathrm{N}$ . (a) Find this distance. (b) If the object is released at this location and allowed to fall toward the Earth, what is its initial acceleration? (c) If the object is now moved twice as far from the Earth, by what factor does its weight change? Explain. (d) By what factor does its initial acceleration change? Explain.

Andy Chen

Numerade Educator

Problem 19

The acceleration due to gravity on the Moon's surface is known to be about one-sixth the acceleration due to gravity on the Earth. Given that the radius of the Moon is roughly one-quarter that of
the Earth, find the mass of the Moon in terms of the mass of the Earth.

Andy Chen

Numerade Educator

Problem 20

Gravitational Tug of $\mathrm{War}$ At some point along the direct path from the center of the Earth to the center of the Moon, the gravitational force of attraction on a spacecraft from the Moon becomes greater than the force from the Earth. (a) How far from the center of the Earth does this occur? (b) At this location, how far is the spacecraft from the surface of the Moon? How far is it from the surface of the Earth?

Andy Chen

Numerade Educator

Problem 21

Predict/Calculate An Extraterrestrial volcano Several volcanoes have been observed erupting on the surface of Jupiter's closest Galilean moon, lo. Suppose that material ejected from one of these volcanoes reaches a height of 5.00 $\mathrm{km}$ after being projected straight upward with an initial speed of 134 $\mathrm{m} / \mathrm{s}$ . Given that the radius of Io is $1820 \mathrm{km},$ (a) outline a strategy that allows you to calculate the mass of lo. (b) Use your strategy to calculate Io's mass.

Andy Chen

Numerade Educator

Problem 22

Consider an asteroid with a radius of 19 $\mathrm{km}$ and a mass of $3.35 \times 10^{15} \mathrm{kg}$ . Assume the asteroid is roughly spherical. (a) What is the acceleration due to gravity on the surface of the asteroid? (b) Suppose the asteroid spins about an axis through its center, like
the Earth, with a rotational period $T .$ What is the smallest value $T$ can have before loose rocks on the asteroid's equator begin to fly off the surface?

Andy Chen

Numerade Educator

Problem 23

A satellite orbits the Earth in a circular orbit of radius $r .$ At some point its rocket engine is fired so that its speed increases rapidly by a small amount. As a result, do the (a) apogee distance and (b) perigee distance increase, decrease, or stay the same?

Andy Chen

Numerade Educator

Problem 24

Predict/Explain The Earth-Moon Distance Is Increasing Laser reflectors left on the surface of the Moon by the Apollo astronauts show that the average distance from the Earth to the Moon is increasing at the rate of 3.8 $\mathrm{cm}$ per year. (a) As a result, will the length of the month increase, decrease, or remain the same? (b) Choose the best explanation from among the following:
\begin{equation}
\begin{array}{l}{\text { 1. The greater the radius of an orbit, the greater the period, }} \\ {\text { which implies a longer month. }} \\ {\text { II. The length of the month will remain the same due to conservation }} \\ {\text { of angular momentum. }} \\ {\text { III. The speed of the Moon is greater with increasing radius; }} \\ {\text { therefore, the length of the month will decrease. }}\end{array}
\end{equation}

Andy Chen

Numerade Educator

Problem 25

Apollo Missions On Apollo missions to the Moon, the command module orbited at an altitude of 110 $\mathrm{km}$ above the lunar surface. How much time did it take for the command module to complete one orbit?

Andy Chen

Numerade Educator

Problem 26

Find the orbital speed of a satellite in a geosynchronous circular orbit $3.58 \times 10^{7}$ m above the surface of the Earth.

Andy Chen

Numerade Educator

Problem 27

An Extrasolar Planet In July of 1999 a planet was reported to be orbiting the Sun-like star lota Horologii with a period of 320 days. Find the radius of the planet's orbit, assuming that lota Horologii has the same mass as the Sun. (This planet is presumably similar to Jupiter, but it may have large, rocky moons that enjoy a relatively pleasant climate.)

Andy Chen

Numerade Educator

Problem 28

Phobos, one of the moons of Mars, orbits at a distance of 9378 $\mathrm{km}$ from the center of the red planet. What is the orbital period of Phobos?

Andy Chen

Numerade Educator

Problem 29

Predict/Calculate An Asteroid with its 0 wn Moon The asteroid 243 Ida has its own small moon, Dactyl. (See the photo on p. 394 . Another such system, asteroid 624 Hektor, has a mass of $7.9 \times 10^{18} \mathrm{kg},$ with its moon orbiting at a radius of 623.5 $\mathrm{km}$ and
a period of 2.965 days. (a) Given that the orbital radius of Dactyl is $108 \mathrm{km},$ and its period is 1.54 days, is the mass of 243 Ida greater than, less than, or equal to the mass of 624 Hektor? (b) Calculate the mass of 243 Ida.

Andy Chen

Numerade Educator

Problem 30

GPS Satellites GPS (Global Positioning System) satellites orbit at an altitude of $2.0 \times 10^{7} \mathrm{m} .$ Find (a) the orbital period, and (b) the orbital speed of such a satellite.

Andy Chen

Numerade Educator

Problem 31

Predict/Calculate Two satellites orbit the Earth, with satellite 1 at a higher altitude than satellite 2 . (a) Which satellite has the greater orbital speed? Explain. (b) Calculate the orbital speed of a satellite that orbits at an altitude of one Earth radius above the surface of the Earth. (c) Calculate the orbital speed of a satellite that orbits at an altitude of two Earth radii above the surface of the Earth.

Andy Chen

Numerade Educator

Problem 32

Predict/Calculate Satellite A has a mass of 1000 $\mathrm{kg}$ and satellite $\mathrm{B}$ has a mass of 2000 $\mathrm{kg}$ . (a) When each satellite orbits at one Earth radius above the surface of the Earth, will the period of satellite $\mathrm{B}$ be longer, shorter, or the same as the period of satellite A? (b) Calculate the orbital periods of satellites A and B.

Andy Chen

Numerade Educator

Problem 33

Predict/Calculate The Martian moon Deimos has an orbital period that is greater than the other Martian moon, Phobos. Both moons have approximately circular orbits. (a) Is Deimos closer to or farther from Mars than Phobos? Explain. (b) Calculate the distance from the center of Mars to Deimos given that its orbital period is $1.10 \times 10^{5} \mathrm{s}$ .

Andy Chen

Numerade Educator

Problem 34

$\because$ Predict/Calculatel(a) Calculate the orbital period of a satellite that orbits at two Earth radii above the surface of the Earth. (b) If the Earth suddenly became more massive while the satellite remains in circular orbit at the same altitude, will the orbital period increase, decrease, or remain the same? (c) Calculate the orbital period of the satellite if the Earth's mass were to double to $1.194 \times 10^{25} \mathrm{kg}$ and everything else were to remain the same.

Andy Chen

Numerade Educator

Problem 35

Binary Stars Alpha Centauri $\mathrm{A}$ and Alpha Centauri $\mathrm{B}$ are binary stars with a separation of $3.45 \times 10^{12} \mathrm{m}$ and an orbital period of $2.52 \times 10^{9}$ . Assuming the two stars are equally massive (which is approximately the case), determine their mass.

Andy Chen

Numerade Educator

Problem 36

Sputnik The first artificial satellite to orbit the Earth was Sputnik I, launched October $4,1957$ The mass of Sputnik I was 83.5 $\mathrm{kg}$ , and its distances from the center of the Earth at apogee and perigee were 7330 $\mathrm{km}$ and $6610 \mathrm{km},$ respectively. Find the difference in gravitational potential energy for Sputnik I as it moved from apogeeto perigee.

Andy Chen

Numerade Educator

Problem 37

How much gravitational potential energy is required to lift a $9380-$ kg Progress spacecraft to the altitude of the International Space Station, 422 $\mathrm{km}$ above the surface of the Earth?

Andy Chen

Numerade Educator

Problem 38

Predict/Explain (a) Is the amount of energy required to get a spacecraft from the Earth to the Moon greater than, less than, or equal to the energy required to get the same spacecraft from the Moon to the Earth? (b) Choose the best explanation from among the following:
\begin{equation}
\begin{array}{l}{\text { 1. The escape speed of the Moon is less than that of the Earth; }} \\ {\text { therefore, less energy is required to leave the Moon. }} \\ {\text { II. The situation is symmetric, and hence the same amount of }} \\ {\text { energy is required to travel in either direction. }} \\ {\text { III. It takes more energy to go from the Moon to the Earth because }} \\ {\text { the Moon is orbiting the Earth. }}\end{array}
\end{equation}

Andy Chen

Numerade Educator

Problem 39

Predict/Calculate Consider the four masses shown in Figure $12-35 .$ (a) Find the total gravitational potential energy of this system. (b) How does your answer to part (a) change if all the masses in the system are doubled? (c) How does your answer to part (a) change if, instead, all the sides of the rectangle are halved in length?

Andy Chen

Numerade Educator

Problem 40

Calculate the gravitational potential energy of a 9.50 -kg mass (a) on the surface of the Earth and $(b)$ at an altitude of 325 $\mathrm{km}$ . (c) Take the difference between the results for parts (b) and (a), and compare with $m g h,$ where $h=325 \mathrm{km} .$

Andy Chen

Numerade Educator

Problem 41

Three bowling balls form an equilateral triangle, as shown in FIGURE $12-36 .$ Each ball has a radius of 10.8 $\mathrm{cm}$ and a mass of 7.26 $\mathrm{kg} .$ What is the total gravitational potential energy of this system?

Andy Chen

Numerade Educator

Problem 42

Two 0.59 -kg basketballs, each with a radius of $12 \mathrm{cm},$ are just touching. How much energy is required to change the separation between the centers of the basketballs to (a) 1.0 $\mathrm{m}$ and (b) 10.0 $\mathrm{m}$ ? (Ignore any other gravitational interactions.)

Andy Chen

Numerade Educator

Problem 43

Find the minimum kinetic energy needed for a $32,000-\mathrm{kg}$ rocket to escape (a) the Moon or (b) the Earth.

Andy Chen

Numerade Educator

Problem 44

Predict/Explain Suppose the Earth were to suddenly shrink to half its current diameter, with its mass remaining constant. (a) Would the escape speed of the Earth increase, decrease, or stay the same? (b) Choose the best explanation from among the following:
\begin{equation}
\begin{array}{l}{\text { 1. Since the radius of the Earth would be smaller, the escape }} \\ {\text { speed would also be smaller. }} \\ {\text { II. The Earth would have the same amount of mass, and hence }} \\ {\text { its escape speed would be unchanged. }} \\ {\text { III. The force of gravity would be much stronger on the surface of }} \\ {\text { the compressed Earth, leading to a greater escape speed. }}\end{array}
\end{equation}

Andy Chen

Numerade Educator

Problem 45

Is the energy required to launch rocket vertically to a height $h$ greater than, less than, or equal to the energy required to put the same rocket into orbit at the height $h$ ? Explain.

Andy Chen

Numerade Educator

Problem 46

Suppose one of the Global Positioning System satellites has a speed of 4.46 $\mathrm{km} / \mathrm{s}$ at perigee and a speed of 3.64 $\mathrm{km} / \mathrm{s}$ at apogee. If the distance from the center of the Earth to the satellite at perigee is $2.00 \times 10^{4} \mathrm{km},$ what is the corresponding distance at apogee?

Andy Chen

Numerade Educator

Problem 47

Meteorites from Mars Several meteorites found in Antarctica are believed to have come from Mars, including the famous ALH84001 meteorite that was once thought to contain fossils of ancient life
on Mars. Meteorites from Mars are thought to get to Earth by being blasted off the Martian surface when a large object (such as an asteroid or a comet) crashes into the planet. What speed must a rock have to escape Mars?

Andy Chen

Numerade Educator

Problem 48

What is the launch speed of a projectile that rises vertically above the Earth to an altitude equal to one Earth radius before coming to rest momentarily?

Andy Chen

Numerade Educator

Problem 49

A projectile launched vertically from the surface of the Moon rises to an altitude of 425 $\mathrm{km} .$ What was the projectile's initial speed?

Andy Chen

Numerade Educator

Problem 50

(a) How much gravitational potential energy must a 3130 -kg satellite acquire in order to attain a geoynchronous orbit? (b) How much kinetic energy must it gain? Note that because of the rotation of the Earth on its axis, the satellite had a velocity of 463 $\mathrm{m} / \mathrm{s}$ relative to the center of the Earth just before launch.

Andy Chen

Numerade Educator

Problem 51

Predict/Calculate Halley's Comet Halley's comet, which passes around the Sun every 76 years, has an elliptical orbit. When closest to the Sun (perihelion) it is at a distance of $8.823 \times 10^{10} \mathrm{m}$ and moves with a speed of 54.6 $\mathrm{km} / \mathrm{s}$ . The greatest distance between Halley's comet and the Sun (aphelion) is $6.152 \times 10^{12} \mathrm{m}$ . (a) Is the
speed of Halley's comet greater than or less than 54.6 $\mathrm{km} / \mathrm{s}$ when it
is at aphelion? Explain. (b) Calculate its speed at aphelion.

Andy Chen

Numerade Educator

Problem 52

The End of the Lunar Module On Apollo Moon missions, the lunar module would blast off from the Moon's surface and dock with the command module in lunar orbit. After docking, the lunar module
would be jettisoned and allowed to crash back onto the lunar surface. Seismometers placed on the Moon's surface by the astronauts would then pick up the resulting seismic waves. Find the impact
speed of the lunar module, given that is jettisoned from an orbit 110 km above the lunar surface moving with a speed of 1630 $\mathrm{m} / \mathrm{s}$ .

Andy Chen

Numerade Educator

Problem 53

If a projectile is launched vertically from the Earth with a speed equal to the escape speed, how high above the Earth's surface is it when its speed is one-third the escape speed?

Andy Chen

Numerade Educator

Problem 54

Suppose a planet is discovered orbiting a distant star. If the mass of the planet is 10 times the mass of the Earth, and its radius is one-tenth the Earth's radius, how does the escape speed of this planet
compare with that of the Earth?

Andy Chen

Numerade Educator

Problem 55

A projectile is launched vertically from the surface of the Moon with an initial speed of 1010 $\mathrm{m} / \mathrm{s}$ . At what altitude is the projectile's speed one-half its initial value?

Andy Chen

Numerade Educator

Problem 56

To what radius would the Sun have to be contracted for its escape speed to equal the speed of light?

Andy Chen

Numerade Educator

Problem 57

Predict/Calculate Two baseballs, each with a mass of 0.148 $\mathrm{kg}$ , are separated by a distance of 395 $\mathrm{m}$ in outer space, far from any other objects. (a) If the balls are released from rest, what speed do they have when their separation has decreased to 145 $\mathrm{m}$ ? (b) Suppose the mass of the balls is doubled. Would the speed found in part (a) increase, decrease, or stay the same? Explain.

Andy Chen

Numerade Educator

Problem 58

On Earth, a person can jump vertically and rise to a height $h .$ What is the radius of the largest spherical asteroid from which this person could escape by jumping straight upward? Assume that
each cubic meter of the asteroid has a mass of 3500 kg.

Andy Chen

Numerade Educator

Problem 59

The magnitude of the tidal force exerted on a linear object of mass $m$ and length $L$ is approximately 2$G m M L / r^{3}$ . In this expression, $M$ is the mass of the body causing the tidal force and $r$ is the distance from the center of $m$ to the center of $M .$ Suppose you are
1 million miles $\left(1.6 \times 10^{9} \mathrm{m}\right)$ away from a black hole whose mass is $1.99 \times 10^{36} \mathrm{kg}$ (one million times that of the Sun). (a) Estimate the tidal force exerted on your body $(L=1.8 \mathrm{m})$ by the black hole. (b) At what distance will the tidal force be approximately 10 times greater than your weight?

Andy Chen

Numerade Educator

Problem 60

The magnitude of the tidal force between the International Space Station (ISS) and a nearby astronaut on a spacewalk is approximately 2$G m M a / r^{3} .$ In this expression, $M$ is the mass of the Earth, $r=6.79 \times 10^{6} \mathrm{m}$ is the distance from the center of the Earth to the orbit of the ISS, $m=125 \mathrm{kg}$ is the mass of the astronaut, and $a=10 \mathrm{m}$ is the distance from the astronaut to the center of mass of the ISS. (a) Calculate the magnitude of the tidal force for this astronaut. This force tends to separate the astronaut from the ISS if the astronaut is located along the line that connects the center of the Earth with the center of mass of the ISS. (b) Calculate the force of gravitational attraction between the astronaut and the ISS if they are 10 $\mathrm{m}$ apart and the ISS has a mass of $420,000 \mathrm{kg}$ . (c) Is the ISS orbit inside or outside the Roche limit?

Andy Chen

Numerade Educator

Problem 61

A dumbbell has a mass $m$ on either end of a rod of length 2a. The center of the dumbbell is a distance $r$ from the center of the Earth, and the dumbbell is aligned radially. If $r \gg a,$ show that the difference in the gravitational force exerted on the two masses by the Earth is approximately 4$G m M_{\mathrm{E}} a / r^{3} .$ (Note: The difference in force causes a tension in the rod connecting the masses. We refer to this as a tidal force.) IHint: Use the fact that $1 /(r-a)^{2}-1 /(r+a)^{2} \sim 4 a / r^{3}$ for $r \gg a . ]$

Andy Chen

Numerade Educator

Problem 62

Referring to the previous problem, suppose the rod connecting the two masses $m$ is removed. In this case, the only force between the two masses is their mutual gravitational attraction. In addition, suppose the masses are spheres of radius $a$ and mass $m=\frac{4}{3} \pi a^{3} \rho$ that touch each other. (The Greek letter $\rho$ stands for the density of the masses.)(a) Write an expression for the gravitational force between the masses $m$ . (b) Find the distance from the center of the Earth, $r,$ for which the gravitational force found in part (a) is equal to the tidal force found in the previous problem. This distance is known as the Roche limit. (c) Calculate the Roche limit for Saturn, assuming $\rho=3330 \mathrm{kg} / \mathrm{m}^{3} .$ The famous rings of Saturn are within the Roche limit for that planet. Thus, the innumerable small objects, composed mostly of ice, that make up the rings will never coalesce to form a moon.)

Andy Chen

Numerade Educator

Problem 63

You weigh yourself on a scale inside an airplane flying due east above the equator. If the airplane now turns around and heads due west with the same speed, will the reading on the scale increase, decrease, or stay the same? Explain.

Andy Chen

Numerade Educator

Problem 64

Rank objects $A, B,$ and $C$ in FlGURE $12-37$ in order of increasing net gravitational force experienced by the object. Indicate ties where appropriate.

Andy Chen

Numerade Educator

Problem 65

Referring to Figure $12-37$ , rank objects $A, B$ , and $C$ in order of increasing initial acceleration
each would experience if it alone were allowed to move. Indicate ties where appropriate.

Vishal Gupta

Numerade Educator

Problem 66

The Crash of skylab Skylab, the largest spacecraft ever to fall back to the Earth, met its fiery end on July $11,1979,$ after flying directly over Everett, WA, on its last orbit. On the CBS Evening News the night before the crash, anchorman Walter Cronkite, in his rich baritone voice, made the following statement: "NASA says there is a little chance that Skylab will land in a populated area." After the commercial, he immediately corrected himself by saying, "I meant to say 'there is little chance' Skylab will hit a populated area." In fact, it landed primarily in the Indian Ocean off the west coast of Australia, though several pieces were recovered near the town of Esperance, Australia, which later sent the U.S. State Department a $\$ 400$ bill for littering. The cause of Skylab's crash was the friction it experienced in the upper reaches of the Earth's atmosphere. As the radius of Skylab's orbit decreased, did its speed increase, decrease, or stay the same? Explain.

Andy Chen

Numerade Educator

Problem 67

Consider a system consisting of three masses on the $x$ axis. Mass $m_{1}=1.00 \mathrm{kg}$ is at $x=1.00 \mathrm{m}$ ; mass $m_{2}=2.00 \mathrm{kg}$ is at $x=2.00 \mathrm{m}$ ; and mass $m_{3}=3.00 \mathrm{kg}$ is at $x=3.00 \mathrm{m} .$ What is the total gravitational potential energy of this system?

Andy Chen

Numerade Educator

Problem 68

An astronaut exploring a distant solar system lands on an unnamed planet with a radius of 3520 $\mathrm{km} .$ When the astronaut jumps upward with an initial speed of 2.91 $\mathrm{m} / \mathrm{s}$ , she rises to height of 0.525 $\mathrm{m} .$ What is the mass of the planet?

Prabhat Tyagi

Numerade Educator

Problem 69

Predict/Calculate When the Moon is in its third-quarter phase, the Earth, Moon, and Sun form a right triangle, as shown in Figure $12-34$ . Calculate the magnitude of the force exerted on the Moon by (a) the Earth and (b) the Sun. (c) Does it make more sense to think of the Moon as orbiting the Sun, with a small effect due to the Earth, or as orbiting the Earth, with a small effect due
to the Sun?

Andy Chen

Numerade Educator

Problem 70

An equilateral triangle 10.0 $\mathrm{m}$ on a side has a 1.00 -kg mass at one corner, a 2.00 -kg mass at another corner, and a 3.00 -kg mass at the third corner (FIGURE $12-38$ ). Find the magnitude and direction of the net force acting on the 2.00 -kg mass.

Narayan Hari

Numerade Educator

Problem 71

Suppose that each of the three masses in Figure $12-38$ is replaced by a mass of 5.95 $\mathrm{kg}$ and radius 0.0714 $\mathrm{m} .$ If the masses are released from rest, what speed will they have when they collide at the center of the triangle? Ignore gravitational effects from
any other objects.

Andy Chen

Numerade Educator

Problem 72

A Near Miss! In the early morning hours of June $14,2002,$ the Earth had a remarkably close encounter with an asteroid the size of a small city. The previously unknown asteroid, now designated
2002 $\mathrm{MN}$ , remained undetected until three days after it had passed the Earth. At its closest approach, the asteroid was $73,600$ miles from the center of the Earth-about a third of the distance to the Moon. (a) Find the speed of the asteroid at closest approach, assuming its speed at infinite distance to be zero and considering only its interaction with the Earth. (b) Observations indicate the asteroid to have a diameter of about 0.730 $\mathrm{km}$ . Estimate the kinetic
energy of the asteroid at closest approach, assuming it has an average density of 3.33 $\mathrm{g} / \mathrm{m}^{3} .$ (For comparison, a 1 -megaton nuclear weapon releases about $4.2 \times 10^{15} \mathrm{J}$ of energy.)

Andy Chen

Numerade Educator

Problem 73

Predict/Calculate Suppose a planet is discovered that has the total mass as the Earth, but half its radius. (a) Is the acceleration due to gravity on this planet more than, less than, or the same as the acceleration due to gravity on the Earth? Explain. (b) Calculate the acceleration due to gravity on this planet.

Andy Chen

Numerade Educator

Problem 74

Show that the speed of a satellite in a circular orbit a height $h$ above the surface of the Earth is
$$v=\sqrt{\frac{G M_{\mathrm{E}}}{R_{\mathrm{E}}+h}}$$

Andy Chen

Numerade Educator

Problem 75

Walking into 0 rbit A spherical asteroid of average density would have a mass of $8.7 \times 10^{13} \mathrm{kg}$ if its radius were 2.0 $\mathrm{km}$ . (a) If you and your spacesuit had a mass of 125 $\mathrm{kg}$ , how much would you weigh when standing on the surface of this asteroid? (b) If you could walk on the surface of this asteroid, what minimum speed would you need to launch yourself into an orbit just above the surface of the asteroid?

Andy Chen

Numerade Educator

Problem 76

In a binary star system, two stars orbit about their common center of mass, as shown in FIGURE $12-39 .$ If $r_{2}=2 r_{1},$ what is the ratio of the masses $m_{2} / m_{1}$ of the two stars?

Andy Chen

Numerade Educator

Problem 77

Find the orbital period of the binary star system described in the previous problem.

Andy Chen

Numerade Educator

Problem 78

Exploring Mars In the not-too-distant future astronauts will to Mars to carry out scientific explorations. As part of their mission, it is likely that a "geosynchronous" satellite will be placed above a given point on the Martian equator to facilitate communications. At what altitude above the surface of Mars should such a satellite orbit? (Note: The Martian "day" is 24.6229 hours. Other relevant information can be found in Appendix C.)

Andy Chen

Numerade Educator

Problem 79

Comet wild 2 In $2004,$ a NASA spacecraft named Stardust flew within 147 miles of Comet wild 2 (pronounced "Vilt $2^{\prime \prime}$ , zooming by it at 6200 $\mathrm{m} / \mathrm{s}$ , about six times the speed of a rifle bullet. Photos taken by Stardust show that the comet is roughly spherical, as shown in FilGuRE $12-40,$ with a radius of 2.7 $\mathrm{km}$ . It has also been determined that the acceleration due to gravity on the surface of Wild 2 is 0.000010$g .$ What is the minimum speed needed for an object to escape from the surface of Wild 2$?$

Andy Chen

Numerade Educator

Problem 80

Predict/Calculate (a) If you want to launch a probe that orbits the Sun with a period of 2.00 years, what should be its orbital radius? Give your answer in astronomical units. The astronomical
unit AU is defined as the mean distance from the Sun to the Earth $\left(1 \mathrm{AU}=1.50 \times 10^{11} \mathrm{m}\right) .$ (b) Is the orbital speed of this probe greater than, less than, or equal to the orbital speed of the Earth? Explain. (c) What is the orbital speed of this probe?

Andy Chen

Numerade Educator

Problem 81

Predict/Calculate A satellite is placed in Earth orbit at an altitude of $23,300 \mathrm{mi},$ which is 1100 miles higher than the altitude of a geosynchronous satellite. (a) Is the period of this satellite greater than or less than 24 hours? (b) As viewed from the surface of the Earth, does the satellite move eastward or westward? Explain. (c) Find the orbital period of this satellite.

Andy Chen

Numerade Educator

Problem 82

Show that the force of gravity between the Moon and the Sun is always greater than the force of gravity between the Moon and the Earth.

Andy Chen

Numerade Educator

Problem 83

The astronomical unit AU is defined as the mean distance from the Sun to the Earth $\left(1 \mathrm{AU}=1.50 \times 10^{11} \mathrm{m}\right) .$ Apply Kepler's third law (Equation $12-7 )$ to the solar system, and show that it can be written as
$$T=C r^{3 / 2}$$
In this expression, the period $T$ is measured in years, the distance $r$ is measured in astronomical units, and the constant $C$ has a magnitude that you must determine.

Andy Chen

Numerade Educator

Problem 84

(a) Find the kinetic energy of a $1940-$ kg satellite in a circular orbit about the Earth, given that the radius of the orbit is $12,600$ miles. (b) How much energy is required to move this satellite to a circular orbit with a radius of $25,200$ miles?

Andy Chen

Numerade Educator

Problem 85

Predict/Calculate Space Station Orbit The International Space Station $\left(m=4.5 \times 10^{5} \mathrm{kg}\right)$ orbits at an altitude of 415 $\mathrm{km}$ above the Earth's surface. (a) Does the orbital speed of the station depend on its mass? Explain. (b) Find the speed of the station in its orbit.
(c) How much time does it take for the station to complete one orbit of the Earth?

Andy Chen

Numerade Educator

Problem 86

Approaching the ISS A Russian Soyuz module, with three astronauts and a full load of cargo, has a mass of 7500 kg. The International Space Station (ISS) has a mass of $420,000$ kg. When
Soyuz docks at the ISS, the two centers of mass are separated by 9.5 (a) Find the average of the gravitational force of attraction between the two spacecraft when Soyuz is docked and when it is
110 $\mathrm{m}$ from the ISS. (b) If Soyuz approached the ISS from rest a distance of $110 \mathrm{m},$ a thruster would have to counteract the average force of gravitational attraction between it and the ISS. If this thruster has an exhaust velocity of $590 \mathrm{m} / \mathrm{s},$ at what average rate must it burn fuel (in $\mathrm{kg} / \mathrm{s}$ ) to counteract the pull of the ISS? (See Section $9-8$ for a discussion of thrust. $(\mathrm{c})$ If the approach requires
$330 \mathrm{s},$ how much fuel will the thruster burn in order to counteract
the gravitational attraction?

Andy Chen

Numerade Educator

Problem 87

Predict/Calculate Consider an object of mass $m$ orbiting the Earth at a radius $r .$ (a) Find the speed of the object. (b) Show that the total mechanical energy of this object is equal to $(-1)$ times
its kinetic energy. (c) Does the result of part (b) apply to an object orbiting the Sun? Explain.

Andy Chen

Numerade Educator

Problem 88

The Earth's orbit around the Sun is slightly elliptical (see Conceptual Example $12-7$ ). At Earth's closest approach to the Sun (perihelion) the orbital radius is $1.471 \times 10^{11} \mathrm{m},$ and at its farthest distance (aphelion) the orbital radius is $1.521 \times 10^{11} \mathrm{m.Given}$ that the mass of the Earth is $5.972 \times 10^{24} \mathrm{kg}$ , and that of the Sun is $1.989 \times 10^{30} \mathrm{kg},($ a) find the difference in gravitational potential energy between when the Earth is at its aphelion and perihelion radii. (b) If the orbital speed of the Earth is $29,290 \mathrm{m} / \mathrm{s}$ at aphelion, what is its orbital speed at perihelion?

Andy Chen

Numerade Educator

Problem 89

Three identical stars, at the vertices of an equilateral triangle, orbit about their common center of mass (FIGURE $12-41 ) .$ Find the period of this orbital motion in terms of the orbital radius, $R,$ and the mass of each star, $M .$

Andy Chen

Numerade Educator

Problem 90

Find an expression for the kinetic energy of a satellite of mass $m$ in an orbit of radius $r$ about a planet of mass $M .$

Andy Chen

Numerade Educator

Problem 91

What is the minimum kinetic energy an impact must give to a 3.0 - -diameter rock for the rock to beable to escape from the Earth?
$$\begin{array}{l}{\text { A. } 5.3 \times 10^{3} \mathrm{J} \quad \text { B. } 1.3 \times 10^{6} \mathrm{J}} \\ {\text { C. } 2.7 \times 10^{12} \mathrm{J} \quad \text { D. } 2.1 \times 10^{13} \mathrm{J}}\end{array}$$

Andy Chen

Numerade Educator

Problem 92

What is the speed a rock needs to be given at the surface of the Earth in order for it to have a residual speed of 2.5 $\mathrm{km} / \mathrm{s} ?$
$$\begin{array}{ll}{\text { A. } 8.7 \mathrm{km} / \mathrm{s}} & {\text { B. } 10.9 \mathrm{km} / \mathrm{s}} \\ {\mathrm{C.} 11.5 \mathrm{km} / \mathrm{s}} & {\text { D. } 13.7 \mathrm{km} / \mathrm{s}}\end{array}$$

Andy Chen

Numerade Educator

Problem 93

Would increasing the ejection speed from 12 $\mathrm{km} / \mathrm{s}$ to 13 $\mathrm{km} / \mathrm{s}$ change the residual speed by more than, less than, or the same
amount as increasing the ejection speed from 15 $\mathrm{km} / \mathrm{s}$ to 16 $\mathrm{km} / \mathrm{s} ?$

Andy Chen

Numerade Educator

Problem 94

Consider a similar plot for rocks ejected from Mars. Where would this plot intercept the $x$ axis?
\begin{equation}
\begin{array}{l}{\text { A. The plot for Mars would intercept the } x \text { axis at } 5.0 \mathrm{km} / \mathrm{s} \text { . }} \\ {\text { B. The plot for Mars would intercept the } x \text { axis at } 16.2 \mathrm{km} / \mathrm{s} \text { . }} \\ {\text { C. The plot for Mars would intercept the } x \text { axis at } 11.2 \mathrm{km} / \mathrm{s} \text { . }} \\ {\text { D. The plot for Mars would not intercept the } x \text { axis. }}\end{array}
\end{equation}

Andy Chen

Numerade Educator

Problem 95

REFERRING TO EXAMPLE 12-8 Find the orbital radius that corresponds to a "year" of 150 days.

Andy Chen

Numerade Educator

Problem 96

Predict/Calculate REFERRING TO EXAMPLE $12-8$ Suppose the mass of the Sun is suddenly doubled, but the Earth's orbital radius remains the same. (a) Would the length of an Earth year increase, decrease, or stay the same? (b) Find the length of a year for the case of a Sun
with twice the mass. (c) Suppose the Sun retains its present mass, but the mass of the Earth is doubled instead. Would the length of the year increase, decrease, or stay the same?

Andy Chen

Numerade Educator

Problem 97

Predict/Calculate REFERRING TO EXAMPLE $12-17$ (a) If the mass of the Earth were doubled, would the escape speed of a rocket increase, decrease, or stay the same? (b) Calculate the escape speed of a rocket for the case of an Earth with twice its present mass. (c) If the mass of the Earth retains its present value, but the mass of the rocket is doubled, does the escape speed increase, decrease, or stay the same?

Andy Chen

Numerade Educator

Problem 98

Predict/Calculate REFERRING TO EXAMPLE $12-17$ Suppose the Earth is suddenly shrunk to half its present radius without losing any of its mass.(a) Would the escape speed of a rocket increase, decrease, or stay the same? (b) Find the escape speed for an Earth with half its present radius.

Andy Chen

Numerade Educator