University Physics with Modern Physics

Hugh D. Young

Chapter 13

Gravitation - all with Video Answers

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Chapter Questions

Problem 1

What is the ratio of the gravitational pull of the sun on the moon to that of the earth on the moon? (Assume the distance of the moon from the sun can be approximated by the distance of the earth from the sun.) Use the data in Appendix F. Is it more accurate to say that the moon orbits the earth, or that the moon orbits the sun?

David Gold

David Gold

Numerade Educator

Problem 2

In the Cavendish balance apparatus shown in Fig. 13.4, suppose that ${m_1}$ $=$ 1.10 kg, ${m_2}$ $=$ 25.0 kg, and the rod connecting the ${m_1}$ pairs is 30.0 cm long.
If, in each pair, ${m_1}$ and ${m_2}$ are 12.0 cm apart center to center, find (a) the net force and (b) the net torque (about the rotation axis) on the rotating part of the apparatus. (c) Does it seem that the torque
in part (b) would be enough to easily rotate the rod? Suggest some ways to improve the sensitivity of this experiment.

Salamat Ali

Numerade Educator

Problem 3

A couple of astronauts agree to rendezvous in space after hours. Their plan is to let gravity bring them together. One of them has a mass of 65 kg and the other a mass of 72 kg, and they start from rest 20.0 m apart. (a) Make a free-body diagram of each astronaut, and use it to find his or her initial acceleration. As a rough approximation, we can model the astronauts as uniform spheres. (b) If the astronauts' acceleration remained constant, how many days would they have to wait before reaching each other? (Careful! They $both$ have acceleration toward each other.) (c) Would their acceleration, in fact, remain constant? If not, would it increase or decrease? Why?

David Gold

Numerade Educator

Problem 4

Two uniform spheres, each with mass $M$ and radius $R$, touch each other. What is the magnitude of their gravitational force of attraction?

Jilin Wang

Boston University

Problem 5

Two uniform spheres, each of mass 0.260 kg, are fixed at points $A$ and $B$ ($\textbf{Fig. E13.5}$). Find the magnitude and direction of the initial acceleration of a uniform sphere with mass 0.010 kg if released from rest at point $P$ and acted on only by forces of gravitational attraction of the spheres at $A$ and $B$.
Figure e13.5 (FIGURE CAN'T COPY)

David Gold

Numerade Educator

Problem 6

Find the magnitude and direction of the net gravitational force on mass $A$ due to masses $B$ and $C$ in $\textbf{Fig. E13.6.}$ Each mass is 2.00 kg.
Figure e13.6 (FIGURE CAN'T COPY)

Jilin Wang

Boston University

Problem 7

A typical adult human has a mass of about 70 kg. (a) What force does a full moon exert on such a human when it is directly overhead with its center 378,000 km away? (b) Compare this force with the force exerted on the human by the earth.

David Gold

Numerade Educator

Problem 8

An 8.00-kg point mass and a 12.0-kg point mass are held in place 50.0 cm apart. A particle of mass $m$ is released from a point between the two masses 20.0 cm from the 8.00-kg mass along the line connecting the two fixed masses. Find the magnitude and direction of the acceleration of the particle.

Jilin Wang

Boston University

Problem 9

A particle of mass 3$m$ is located 1.00 m from a particle of mass $m$. (a) Where should you put a third mass $M$ so that the net gravitational force on $M$ due to the two masses is exactly zero? (b) Is the equilibrium of $M$ at this point stable or unstable (i) for points along the line connecting m and 3$m$, and (ii) for points along the line passing through $M$ and perpendicular to the line connecting $m$ and 3$m$?

David Gold

Numerade Educator

Problem 10

The point masses $m$ and 2$m$ lie along the x-axis, with $m$ at the origin and 2$m$ at $x$ $=$ $L$. A third point mass $M$ is moved along the $x$-axis. (a) At what point is the net gravitational force on $M$ due to the other two masses equal to zero? (b) Sketch the $x$-component of the net force on $M$ due to $m$ and 2$m$, taking quantities to the right as positive. Include the regions $x < 0$, $0 < x < L$, and $x > L$. Be especially careful to show the behavior of the graph on either side of $x = 0$ and $x = L$.

Salamat Ali

Numerade Educator

Problem 11

At what distance above the surface of the earth is the acceleration due to the earth's gravity 0.980 m/s$^2$ if the acceleration due to gravity at the surface has magnitude 9.80 m/s$^2$?

David Gold

Numerade Educator

Problem 12

The mass of Venus is 81.5% that of the earth, and its radius is 94.9% that of the earth. (a) Compute the acceleration due to gravity on the surface of Venus from these data. (b) If a rock weighs 75.0 N on earth, what would it weigh at the surface of Venus?

Jilin Wang

Boston University

Problem 13

Titania, the largest moon of the planet Uranus, has $\frac{1}{8}$ the radius of the earth and $\frac{1}{1700}$ the mass of the earth. (a) What is the acceleration due to gravity at the surface of Titania? (b) What is the average density of Titania? (This is less than the density of rock, which is one piece of evidence that Titania is made primarily of ice.)

David Gold

Numerade Educator

Problem 14

Rhea, one of Saturn's moons, has a radius of 764 km and an acceleration due to gravity of 0.265 m/s$^2$ at its surface. Calculate its mass and average density.

Jilin Wang

Boston University

Problem 15

Calculate the earth's gravity force on a 75-kg astronaut who is repairing the Hubble Space Telescope 600 km above the earth's surface, and then compare this value with his weight at the earth's surface. In view of your result, explain why it is said that astronauts are weightless when they orbit the earth in a satellite such as a space shuttle. Is it because the gravitational pull of the earth is negligibly small?

David Gold

Numerade Educator

Problem 16

Jupiter's moon Io has active volcanoes (in fact, it is the most volcanically active body in the solar system) that eject material as high as 500 km (or even higher) above the surface. Io has a mass of 8.93 $\times$ 10$^{22}$ kg and a radius of 1821 km. For this calculation, ignore any variation in gravity over the 500-km range of the debris. How high would this material go on earth if it were ejected with the same speed as on Io?

Jilin Wang

Boston University

Problem 17

Use the results of Example 13.5 (Section 13.3) to calculate the escape speed for a spacecraft (a) from the surface of Mars and (b) from the surface of Jupiter. Use the data in Appendix F. (c) Why is the escape speed for a spacecraft independent of the spacecraft's mass?

David Gold

Numerade Educator

Problem 18

Ten days after it was launched toward Mars in December 1998, the $Mars$ $Climate$ $Orbiter$ spacecraft (mass 629 kg) was 2.87 $\times$ 10$^6$ km from the earth and traveling at 1.20 $\times$ 10$^4$ km/h relative to the earth. At this time, what were (a) the spacecraft's kinetic energy relative to the earth and (b) the potential energy of the earth$-$spacecraft system?

Jilin Wang

Boston University

Problem 19

A planet orbiting a distant star has radius 3.24 $\times$ 10$^6$ m. The escape speed for an object launched from this planet's surface is 7.65 $\times$ 10$^3$ m/s. What is the acceleration due to gravity at the surface of the planet?

David Gold

Numerade Educator

Problem 20

On July 15, 2004, NASA launched the $Aura$ spacecraft to study the earth's climate and atmosphere. This satellite was injected into an orbit 705 km above the earth's surface. Assume a circular orbit. (a) How many hours does it take this satellite to make one orbit? (b) How fast (in km/s) is the $Aura$ spacecraft moving?

Jilin Wang

Boston University

Problem 21

For a satellite to be in a circular orbit 890 km above the surface of the earth, (a) what orbital speed must it be given, and (b) what is the period of the orbit (in hours)?

David Gold

Numerade Educator

Problem 22

On July 15, 2004, NASA launched the $Aura$ spacecraft to study the earth's climate and atmosphere. This satellite was injected into an orbit 705 km above the earth's surface. Assume a circular orbit. (a) How many hours does it take this satellite to make one orbit? (b) How fast (in km/s) is the $Aura$ spacecraft moving?

Jilin Wang

Boston University

Problem 23

Two satellites are in circular orbits around a planet that has radius 9.00 $\times$ 10$^6$m. One satellite has mass 68.0 kg, orbital radius 7.00 $\times$ 10$^7$m, and orbital speed 4800 m/s. The second satellite has mass 84.0 kg and orbital radius 3.00 $\times$ 10$^7$m. What is the orbital speed of this second satellite?

David Gold

Numerade Educator

Problem 24

In its orbit each day, the International Space Station makes 15.65 revolutions around the earth. Assuming a circular orbit, how high is this satellite above the surface of the earth?

Jilin Wang

Boston University

Problem 25

Deimos, a moon of Mars, is about 12 km in diameter with mass 1.5 $\times$ 10$^{15}$ kg. Suppose you are stranded alone on Deimos and want to play a one-person game of baseball. You would be the pitcher, and you would be the batter! (a) With what speed would you have to throw a baseball so that it would go into a circular orbit just above the surface and return to you so you could hit it? Do you think you could actually throw it at this speed? (b) How long (in hours) after throwing the ball should you be ready to hit it? Would this be an action-packed baseball game?

David Gold

Numerade Educator

Problem 26

Planet Vulcan. Suppose that a planet were discovered between the sun and Mercury, with a circular orbit of radius equal to $\frac{2}{3}$ of the average orbit radius of Mercury. What would be the orbital period of such a planet? (Such a planet was once postulated, in part to explain the precession of Mercury's orbit. It was even given the name Vulcan, although we now have no evidence that it actually exists. Mercury's precession has been explained by general relativity.)

Jilin Wang

Boston University

Problem 27

The star Rho$^1$ Cancri is 57 light-years from the earth and has a mass 0.85 times that of our sun. A planet has been detected in a circular orbit around Rho$^1$ Cancri with an orbital radius equal to 0.11 times the radius of the earth's orbit around the sun. What are (a) the orbital speed and (b) the orbital period of the planet of Rho$^1$ Cancri?

David Gold

Numerade Educator

Problem 28

In March 2006, two small satellites were discovered orbiting Pluto, one at a distance of 48,000 km and the other at 64,000 km. Pluto already was known to have a large satellite Charon, orbiting at 19,600 km with an orbital period of 6.39 days. Assuming that the satellites do not affect each other, find the orbital
periods of the two small satellites $without$ using the mass of Pluto.

Jilin Wang

Boston University

Problem 29

The dwarf planet Pluto has an elliptical orbit with a semimajor axis of 5.91 $\times$ 10$^{12}$ m and eccentricity 0.249. (a) Calculate Pluto's orbital period. Express your answer in seconds and in earth
years. (b) During Pluto's orbit around the sun, what are its closest and farthest distances from the sun?

David Gold

Numerade Educator

Problem 30

In 2004 astronomers reported the discovery of a large Jupiter-sized planet orbiting very close to the star HD 179949 (hence the term "hot Jupiter"). The orbit was just $\frac{1}{9}$
the distance of Mercury from our sun, and it takes the planet only 3.09 days to make one orbit (assumed to be circular). (a) What is the mass of the star? Express your answer in kilograms and as a multiple of our sun's mass. (b) How fast (in km/s) is this planet moving?

Jilin Wang

Boston University

Problem 31

On October 15, 2001, a planet was discovered orbiting around the star HD 68988. Its orbital distance was measured to be 10.5 million kilometers from the center of the star, and its orbital period was estimated at 6.3 days. What is the mass of HD 68988? Express your answer in kilograms and in terms of our sun's mass. (Consult Appendix F.)

David Gold

Numerade Educator

Problem 32

A uniform, spherical, 1000.0-kg shell has a radius of 5.00 m. (a) Find the gravitational force this shell exerts on a 2.00-kg point mass placed at the following distances from the center of the shell: (i) 5.01 m, (ii) 4.99 m, (iii) 2.72 m. (b) Sketch a qualitative graph of the magnitude of the gravitational force this sphere exerts on a point mass $m$ as a function of the distance $r$ of $m$ from the center of the sphere. Include the region from $r = 0$ to $r$ $\rightarrow$ $\infty$.

Jilin Wang

Boston University

Problem 33

A uniform, solid, 1000.0-kg sphere has a radius of 5.00 m. (a) Find the gravitational force this sphere exerts on a 2.00-kg point mass placed at the following distances from the center of the sphere: (i) 5.01 m, (ii) 2.50 m. (b) Sketch a qualitative graph of the magnitude of the gravitational force this sphere exerts
on a point mass $m$ as a function of the distance $r$ of $m$ from the center of the sphere. Include the region from $r = 0$ to $r$ $\rightarrow$ $\infty$.

David Gold

Numerade Educator

Problem 34

A thin, uniform rod has length $L$ and mass $M$. A small uniform sphere of mass $m$ is placed a distance $x$ from one end of the rod, along the axis of the rod ($\textbf{Fig. E13.34}$). (a) Calculate the gravitational potential energy of the rod$-$sphere system. Take the potential energy to be zero when the rod and sphere are infinitely far apart. Show that your answer reduces to the expected result when $x$ is much larger than $L$. ($Hint:$ Use the power series expansion for $\mathrm{l}$n(1 $+$ $x$) given in Appendix B.) (b) Use ${F_x}$ $= -$$dU$/$dx$ to find the magnitude and direction of the gravitational force exerted on the sphere by the rod (see Section 7.4). Show that your answer reduces to the expected result when $x$ is much larger than $L$.
Figure e13.34 (FIGURE CAN'T COPY)

Jilin Wang

Boston University

Problem 35

Consider the ringshaped body of $\textbf{Fig. E13.35.}$ A particle with mass $m$ is placed a distance $x$ from the center of the ring, along the line through the center of the ring and perpendicular to its plane. (a) Calculate the gravitational potential energy $U$ of this system. Take the potential energy to be zero when the two objects are far apart. (b) Show that your answer to part (a) reduces to the expected result when $x$ is much larger than the radius a of the ring. (c) Use ${F_x} = -dU/dx$ to find the magnitude and direction of the force on the particle (see Section 7.4). (d) Show that your answer to part (c) reduces to the expected result when $x$ is much larger than $a$. (e) What are the values of $U$ and ${F_x}$ when $x = 0$? Explain why these results make sense.
Figure e13.35 (FIGURE CAN'T COPY)

David Gold

Numerade Educator

Problem 36

A Visit to Santa. You decide to visit Santa Claus at the north pole to put in a good word about your splendid behavior throughout the year. While there, you notice that the elf Sneezy, when hanging from a rope, produces a tension of 395.0 N in the rope. If Sneezy hangs from a similar rope while delivering presents at the earth's equator, what will the tension in it be? (Recall that the earth is rotating about an axis through its north and south poles.) Consult Appendix F and start with a free-body diagram of
Sneezy at the equator.

Jilin Wang

Boston University

Problem 37

The acceleration due to gravity at the north pole of Neptune is approximately 11.2 m/s$^2$. Neptune has mass 1.02 $\times$ 10$^{26}$ kg and radius 2.46 $\times$ 10$^4$ km and rotates once around its
axis in about 16 h. (a) What is the gravitational force on a 3.00-kg object at the north pole of Neptune? (b) What is the apparent weight of this same object at Neptune's equator? (Note that Neptune's "surface" is gaseous, not solid, so it is impossible to stand on it.)

David Gold

Numerade Educator

Problem 38

Cosmologists have speculated that black holes the size of a proton could have formed during the early days of the Big Bang when the universe began. If we take the diameter of a proton to be 1.0 $\times$ 10$^{-15}$ m, what would be the mass of a mini black hole?

Jilin Wang

Boston University

Problem 39

Astronomers have observed a small, massive object at the center of our Milky Way galaxy (see Section 13.8). A ring of material orbits this massive object; the ring has a diameter of about 15 light-years and an orbital speed of about 200 km/s. (a) Determine the mass of the object at the center of the Milky Way galaxy. Give your answer both in kilograms and in solar masses (one solar mass is the mass of the sun). (b) Observations of stars, as well as theories of the structure of stars, suggest that it is impossible for a single star to have a mass of more than about
50 solar masses. Can this massive object be a single, ordinary star? (c) Many astronomers believe that the massive object at the center of the Milky Way galaxy is a black hole. If so, what must the Schwarzschild radius of this black hole be? Would a black hole of this size fit inside the earth's orbit around the sun?

David Gold

Numerade Educator

Problem 40

In 2005 astronomers announced the discovery of a large black hole in the galaxy Markarian 766 having clumps of matter orbiting around once every 27 hours and moving at 30,000 km/s. (a) How far are these clumps from the center of the black hole? (b) What is the mass of this black hole, assuming circular orbits? Express your answer in kilograms and as a multiple of our sun's mass. (c) What is the radius of its event horizon?

Jilin Wang

Boston University

Problem 41

Neutron stars, such as the one at the center of the Crab Nebula, have about the same mass as our sun but have a $much$ smaller diameter. If you weigh 675 N on the earth, what would you weigh at the surface of a neutron star that has the same mass as our sun and a diameter of 20 km?

David Gold

Numerade Educator

Problem 42

Four identical masses of 8.00 kg each are placed at the corners of a square whose side length is 2.00 m. What is the net gravitational force (magnitude and direction) on one of the masses, due to the other three?

Jilin Wang

Boston University

Problem 43

Three uniform spheres are fixed at the positions shown in $\textbf{Fig. P13.43.}$ (a) What are the magnitude and direction of the force on a 0.0150-kg particle placed at $P$? (b) If the spheres are in deep outer space and a 0.0150-kg particle is released from rest 300 m from the origin along a line 45$^\circ$ below the $-x$-axis, what will the particle's speed be when it reaches the origin?
Figure P13.43 (FIGURE CAN'T COPY)

David Gold

Numerade Educator

Problem 44

There is strong evidence that Europa, a satellite of Jupiter, has a liquid ocean beneath its icy surface. Many scientists think we should land a vehicle there to search for life. Before launching it, we would want to test such a lander under the gravity conditions at the surface of Europa. One way to do this is to put the lander at the end of a rotating arm in an orbiting earth satellite. If the arm is 4.25 m long and pivots about one end, at what angular speed (in rpm) should it spin so
that the acceleration of the lander is the same as the acceleration due to gravity at the surface of Europa? The mass of Europa is 4.80 $\times$ 10$^{22}$ kg and its diameter is 3120 km.

Jilin Wang

Boston University

Problem 45

A uniform sphere with mass 50.0 kg is held with its center at the origin, and a second uniform sphere with mass 80.0 kg is held with its center at the point $x =$ 0, $y =$ 3.00 m. (a) What are the magnitude and direction of the net gravitational force due to these objects on a third uniform sphere with mass
0.500 kg placed at the point $x =$ 4.00 m, $y =$ 0? (b) Where, other than infinitely far away, could the third sphere be placed such that the net gravitational force acting on it from the other two spheres
is equal to zero?

David Gold

Numerade Educator

Problem 46

On December 25, 2004, the $Huygens$ probe separated from the $Cassini$ spacecraft orbiting Saturn and began a 22-day journey to Saturn's giant moon Titan, on whose surface it landed. Besides the data in Appendix F, it is useful to know that Titan is 1.22 $\times$ 10$^6$ km from the center of Saturn and has a mass of 1.35 $\times$ 10$^{23}$ kg and a diameter of 5150 km. At what distance from Titan should the gravitational pull of Titan just balance the gravitational pull of Saturn?

Jilin Wang

Boston University

Problem 47

An experiment is performed in deep space with two uniform spheres, one with mass 50.0 kg and the other with mass 100.0 kg. They have equal radii, $r =$ 0.20 m. The spheres are released from rest with their centers 40.0 m apart. They accelerate toward each other because of their mutual gravitational attraction. You can ignore all gravitational forces other than that between the two spheres. (a) Explain why linear momentum is conserved. (b) When their centers are 20.0 m apart, find (i) the speed of each
sphere and (ii) the magnitude of the relative velocity with which one sphere is approaching the other. (c) How far from the initial position of the center of the 50.0-kg sphere do the surfaces of the two spheres collide?

Nicholas Mogoi

Numerade Educator

Problem 48

At a certain instant, the earth, the moon, and a stationary 1250-kg spacecraft lie at the vertices of an equilateral triangle whose sides are 3.84 $\times$ 10$^5$ km in length. (a) Find the magnitude
and direction of the net gravitational force exerted on the spacecraft by the earth and moon. State the direction as an angle measured from a line connecting the earth and the spacecraft. In a sketch,
show the earth, the moon, the spacecraft, and the force vector. (b) What is the minimum amount of work that you would have to do to move the spacecraft to a point far from the earth and moon? Ignore any gravitational effects due to the other planets or the sun.

Jilin Wang

Boston University

Problem 49

Many satellites are moving in a circle in the earth's equatorial plane. They are at such a height above the earth's surface that they always remain above the same point. (a) Find the altitude of these satellites above the earth's surface. (Such an orbit is said to be $geosynchronous.$) (b) Explain, with a sketch, why the radio signals from these satellites cannot directly reach receivers on earth that are north of 81.3$^\circ$ N latitude.

David Gold

Numerade Educator

Problem 50

Some scientists are eager to send a remote-controlled submarine to Jupiter's moon Europa to search for life in its oceans below an icy crust. Europa's mass has been measured to be 4.80 $\times$ 10$^{22}$ kg, its diameter is 3120 km, and it has no appreciable atmosphere. Assume that the layer of ice at the surface is not thick enough to exert substantial
force on the water. If the windows of the submarine you are designing each have an area of 625 cm$^2$ and can stand a maximum inward force of 8750 N per window, what is the greatest depth to which this submarine can safely dive?

Jilin Wang

Boston University

Problem 51

What is the escape speed from a 300-km-diameter asteroid with a density of 2500 kg>m$^3$?

David Gold

Numerade Educator

Problem 52

A landing craft with mass 12,500 kg is in a circular
orbit 5.75 $\times$ 10$^5$ m above the surface of a planet. The period of
the orbit is 5800 s. The astronauts in the lander measure the
diameter of the planet to be 9.60 $\times$ 10$^6$ m. The lander sets down
at the north pole of the planet. What is the weight of an 85.6-kg
astronaut as he steps out onto the planet's surface?

Jilin Wang

Boston University

Problem 53

Planet X rotates in the same manner as the earth, around an axis through its north and south poles, and is perfectly spherical. An astronaut who weighs 943.0 N on the earth weighs 915.0 N at the north pole of Planet X and only 850.0 N at its equator. The distance from the north pole to the equator is 18,850 km, measured along the surface of Planet X. (a) How long is the day on Planet X? (b) If a 45,000-kg satellite is placed in a circular orbit 2000 km above the surface of Planet X, what will be its orbital period?

David Gold

Numerade Educator

Problem 54

(a) Suppose you are at the earth's equator and observe a satellite passing directly overhead and moving from west to east in the sky. Exactly 12.0 hours later, you again observe this satellite to be directly overhead. How far above the earth's surface is the satellite's orbit? (b) You observe another satellite directly overhead and traveling east to west. This satellite is again overhead in 12.0 hours. How far is this satellite's orbit above the surface of the earth?

Jilin Wang

Boston University

Problem 55

An astronaut, whose mission is to go where no one has gone before, lands on a spherical planet in a distant galaxy. As she stands on the surface of the planet, she releases a small rock from rest and finds that it takes the rock 0.480 s to fall 1.90 m. If the radius of the planet is 8.60 $\times$ 10$^7$ m, what is the mass of the planet?

David Gold

Numerade Educator

Problem 56

Your starship, the $Aimless$ $Wanderer$, lands on the mysterious planet Mongo. As chief scientist-engineer, you make the following measurements: A 2.50-kg stone thrown upward from the ground at 12.0 m/s returns to the ground in 4.80 s; the circumference of Mongo at the equator is 2.00 $\times$ 10$^5$ km; and there is no appreciable atmosphere on Mongo. The starship commander, Captain Confusion, asks for the following information: (a) What is the mass of Mongo? (b) If the $Aimless$ $Wanderer$ goes into a circular orbit 30,000 km above the surface of Mongo, how many hours will it take the ship to complete one orbit?

Jilin Wang

Boston University

Problem 57

You are exploring a distant planet. When your spaceship is in a circular orbit at a distance of 630 km above the planet's surface, the ship's orbital speed is 4900 m/s. By observing the planet, you determine its radius to be 4.48 $\times$ 10$^6$ m. You then land on the surface and, at a place where the ground is level, launch a small projectile with initial speed 12.6 m/s at an angle of 30.8$^\circ$ above the horizontal. If resistance due to the planet's atmosphere is negligible, what is the horizontal range of the projectile?

David Gold

Numerade Educator

Problem 58

The 0.100-kg sphere in $\textbf{Fig. P13.58}$ is released from rest at the position shown in the sketch, with its center 0.400 m from the center of the 5.00-kg mass. Assume that the only forces on the 0.100-kg sphere are the gravitational forces exerted by the other two spheres and that the 5.00-kg and 10.0-kg spheres are held in place at their initial positions. What is the speed of the 0.100-kg sphere when it has moved 0.400 m to the right from its initial position?
Figure P13.58 (FIGURE CAN'T COPY)

Sheh Lit Chang

University of Washington

Problem 59

An unmanned spacecraft is in a circular orbit around the moon, observing the lunar surface from an altitude of 50.0 km (see Appendix F). To the dismay of scientists on earth, an electrical fault causes an on-board thruster to fire, decreasing the speed of the spacecraft by 20.0 m/s. If nothing is done to correct its orbit, with what speed (in km/h) will the spacecraft crash into the lunar surface?

David Gold

Numerade Educator

Problem 60

On July 4, 2005, the NASA spacecraft $Deep$ $Impact$ fired a projectile onto the surface of Comet Tempel 1. This comet is about 9.0 km across. Observations of surface debris released by the impact showed that dust with a speed as low as 1.0 m/s was able to escape the comet. (a) Assuming a spherical shape, what is the mass of this comet? ($Hint:$ See Example 13.5 in Section 13.3.) (b) How far from the comet's center will this debris be when it has lost (i) 90.0% of its initial kinetic
energy at the surface and (ii) all of its kinetic energy at the surface?

Jilin Wang

Boston University

Problem 61

A hammer with mass $m$ is dropped from rest from a height $h$ above the earth's surface. This height is not necessarily small compared with the radius $R_E$ of the earth.
Ignoring air resistance, derive an expression for the speed y of the hammer when it reaches the earth's surface. Your expression should involve $h$, $R_E$, and $m_E$ (the earth's mass).

David Gold

Numerade Educator

Problem 62

(a) Calculate how much work is required to launch a spacecraft of mass $m$ from the surface of the earth (mass $m_E$, radius $R_E$) and place it in a circular $low$ $earth$ $orbit-$that is, an orbit whose altitude above the earth's surface is much less than $R_E$. (As an example, the International Space Station is in low earth orbit at an altitude of about 400 km, much less than $R_E =$ 6370 km.) Ignore the kinetic energy that the spacecraft has on the ground due to the earth's rotation. (b) Calculate the minimum amount of additional work required to move the spacecraft from low earth orbit to a very great distance from the earth. Ignore the gravitational effects of the sun, the moon, and the other planets. (c) Justify the statement "In terms of energy, low earth orbit is halfway to the edge of the universe."

Jilin Wang

Boston University

Problem 63

Two identical stars with mass $M$ orbit around their center of mass. Each orbit is circular and has radius $R$, so that the two stars are always on opposite sides
of the circle. (a) Find the gravitational force of one star on the other. (b) Find the orbital speed of each star and the period of the orbit. (c) How much energy would be required to separate the two stars to infinity?

David Gold

Numerade Educator

Problem 64

Two stars, with masses ${M_1}$ and ${M_2}$, are in circular orbits around their center of mass. The star with mass ${M_1}$ has an orbit of radius ${R_1}$; the star with mass ${M_2}$ has an orbit of radius ${R_2}$. (a) Show that the ratio of the orbital radii of the two stars equals the reciprocal of the ratio of their masses$-$that is, ${R_1}$/${R_2}$ $=$ ${M_2}$/${M_1}$. (b) Explain why the two stars have the same orbital period, and show that the period $T$ is given by $T = 2\pi$(R1 + R2)$^{3/2}$/$\sqrt{G(M1 + M2)}$. (c) The two stars in a certain binary star system move in circular orbits. The first star, Alpha, has an orbital speed of 36.0 km/s. The second star, Beta, has an orbital speed of 12.0 km/s. The orbital period
is 137 d. What are the masses of each of the two stars? (d) One of the best candidates for a black hole is found in the binary system called A0620-0090. The two objects in the binary system are an
orange star, V616 Monocerotis, and a compact object believed to be a black hole (see Fig. 13.28). The orbital period of A0620-0090 is 7.75 hours, the mass of V616 Monocerotis is estimated to be
0.67 times the mass of the sun, and the mass of the black hole is estimated to be 3.8 times the mass of the sun. Assuming that the orbits are circular, find the radius of each object's orbit and the
orbital speed of each object. Compare these answers to the orbital radius and orbital speed of the earth in its orbit around the sun.

Jilin Wang

Boston University

Problem 65

Comets travel around the sun in elliptical orbits with large eccentricities. If a comet has speed 2.0 $\times$ 10$^4$ m/s when at a distance of 2.5 $\times$ 10$^{11}$ m from the center of the sun, what is its speed when at a distance of 5.0 $\times$ 10$^{10}$ m?

David Gold

Numerade Educator

Problem 66

The planet Uranus has a radius of 25,360 km and a surface acceleration due to gravity of 9.0 m/s$^2$ at its poles. Its moon Miranda (discovered by Kuiper in 1948) is in a circular orbit about Uranus at an altitude of 104,000 km above the planet's surface. Miranda has a mass of 6.6 $\times$ 10$^{19}$ kg and a radius of 236 km. (a) Calculate the mass of Uranus from the given data. (b) Calculate the magnitude of Miranda's acceleration due to its orbital motion about Uranus. (c) Calculate the acceleration due to Miranda's gravity at the surface of Miranda. (d) Do the answers to parts (b) and (c) mean that an object released 1 m above Miranda's surface on the side toward Uranus will fall $up$ relative to Miranda? Explain.

Jilin Wang

Boston University

Problem 67

Consider a spacecraft in an elliptical orbit around the earth. At the low point, or perigee, of its orbit, it is 400 km above the earth's surface; at the high point, or apogee, it is 4000 km
above the earth's surface. (a) What is the period of the spacecraft's orbit? (b) Using conservation of angular momentum, find the ratio of the spacecraft's speed at perigee to its speed at apogee. (c) Using conservation of energy, find the speed at perigee and the speed at apogee. (d) It is necessary to have the spacecraft escape from the earth completely. If the spacecraft's rockets are fired at perigee, by how much would the speed have to be increased to achieve this? What if the rockets were fired at apogee? Which point in the orbit is more efficient to use?

David Gold

Numerade Educator

Problem 68

A rocket with mass 5.00 $\times$ 10$^3$ kg is in a circular orbit of radius 7.20 $\times$ 10$^6$ m around the earth. The rocket's engines fire for a period of time to increase that radius to 8.80 $\times$ 10$^6$ m, with the orbit again circular. (a) What is the change in the rocket's kinetic energy? Does the kinetic energy increase or decrease? (b) What is the change in the rocket's gravitational potential energy? Does the potential energy increase or decrease? (c) How much work is done by the rocket engines in changing the orbital radius?

Jilin Wang

Boston University

Problem 69

A 5000-kg spacecraft is in a circular orbit 2000 km above the surface of Mars. How much work must the spacecraft engines perform to move the spacecraft to a circular orbit that is 4000 km above the surface?

David Gold

Numerade Educator

Problem 70

A satellite with mass 848 kg is in a circular orbit with an orbital speed of 9640 m/s around the earth. What is the new orbital speed after friction from the earth's upper atmosphere has done $-$7.50 $\times$ 10$^9$ J of work on the satellite? Does the speed increase or decrease?

Jilin Wang

Boston University

Problem 71

Planets are not uniform inside. Normally, they are densest at the center and have decreasing density outward toward the surface. Model a spherically symmetric planet, with the same radius as the earth, as having a density that decreases linearly with distance from the center. Let the density be 15.0 $\times$ 10$^3$ kg/m$^3$ at the center and 2.0 $\times$ 10$^3$ kg/m$^3$ at the surface. What is the acceleration due to gravity at the surface of this planet?

David Gold

Numerade Educator

Problem 72

One of the brightest comets of the 20th century was Comet Hyakutake, which passed close to the sun in early 1996. The orbital period of this comet is estimated to be about 30,000 years. Find the semi-major axis of this comet's orbit. Compare it to the average sun$-$Pluto distance and to the distance to Alpha Centauri, the nearest star to the sun, which is 4.3 light-years distant.

Brandy Heflin

Numerade Educator

Problem 73

An object in the shape of a thin ring has radius $a$ and mass $M$. A uniform sphere with mass m and radius $R$ is placed with its center at a distance $x$ to the right of the center of the ring, along a line through the center of the ring, and perpendicular to its plane (see Fig. E13.35). What is the gravitational
force that the sphere exerts on the ring-shaped object? Show that your result reduces to the expected result when $x$ is much larger than $a$.

Penny Riley

Numerade Educator

Problem 74

A uniform wire with mass $M$ and length $L$ is bent into a semicircle. Find the magnitude and direction of the gravitational force this wire exerts on a point with mass $m$ placed at the center of curvature of the semicircle.

Jilin Wang

Boston University

Problem 75

A shaft is drilled from the surface to the center of the earth (see Fig. 13.25). As in Example 13.10 (Section 13.6), make the unrealistic assumption that the density of the earth is uniform. With this approximation, the gravitational force on an object with mass $m$, that is inside the earth at a distance $r$ from the center, has magnitude $F_g = GmE mr/R_E{^3}$ (as shown in Example 13.10) and points toward the center of the earth. (a) Derive an expression for the gravitational potential energy $U(r)$ of the object$-$earth system as a function of the object's distance from the center of the earth. Take the potential energy to be zero when the object is at the center of the earth. (b) If an object is released in the shaft at the earth's surface, what speed will it have when it reaches the center of the earth?

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 76

For each of the eight planets Mercury to Neptune, the semi-major axis $a$ of their orbit and their orbital period $T$ are as follows: (a) Explain why these values, when plotted as ${T^2}$ versus ${a^3}$, fall close to a straight line. Which of Kepler's laws is being tested? However, the values of ${T^2}$ and ${a^3}$ cover such a wide range that this plot is not a very practical way to graph the data. (Try it.) Instead, plot log$(T)$ (with $T$ in seconds) versus log($a$) (with $a$ in meters). Explain why the data should also fall close to a straight line in such a plot. (b) According to Kepler's laws, what should be the slope of your log$(T)$ versus log$(a)$ graph in part (a)? Does your graph have this slope? (c) Using $G =$ 6.674 $\times$ 10$^{-11}$ N $\cdot$ m$^2$/kg$^2$, calculate the mass of the sun from the $y$-intercept of your graph. How does your calculated value compare with the value given in Appendix F? (d) The only asteroid visible to the naked eye (and then only under ideal viewing conditions) is Vesta, which has an orbital period of 1325.4 days. What is the length of the semi-major axis of Vesta's orbit? Where does this place Vesta's orbit relative to the orbits of the eight major planets? Some scientists argue that Vesta should be called a minor planet rather than an asteroid.

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 77

For a spherical planet with mass $M$, volume $V$, and radius $R$, derive an expression for the acceleration due to gravity at the planet's surface, $g$, in terms of the average density of the planet, $\rho =$ $M/V$, and the planet's diameter, $D = 2R$. The table gives the values of $D$ and $g$ for the eight major planets: (a) Treat the planets as spheres. Your equation for $g$ as a function of $\rho$ and $D$ shows that if the average density of the planets is constant, a graph of $g$ versus $D$ will be well represented by a straight line. Graph g as a function of $D$ for the eight major planets. What does the graph tell you about the variation in average density? (b) Calculate the average density for each major planet. List the planets in order of decreasing density, and give the calculated
average density of each. (c) The earth is not a uniform sphere and has greater density near its center. It is reasonable to assume this might be true for the other planets. Discuss the effect this
nonuniformity has on your analysis. (d) If Saturn had the same average density as the earth, what would be the value of $g$ at Saturn's surface?

David Gold

Numerade Educator

Problem 78

For a planet in our solar system, assume that the axis of orbit is at the sun and is circular. Then the angular momentum about that axis due to the planet's orbital motion is $L = M$$\upsilon$$R$.
(a) Derive an expression for $L$ in terms of the planet's mass $M$, orbital radius $R$, and period $T$ of the orbit. (b) Using Appendix F, calculate the magnitude of the orbital angular momentum for each
of the eight major planets. (Assume a circular orbit.) Add these values to obtain the total angular momentum of the major planets due to their orbital motion. (All the major planets orbit in the same
direction in close to the same plane, so adding the magnitudes to get the total is a reasonable approximation.) (c) The rotational period of the sun is 24.6 days. Using Appendix F, calculate the
angular momentum the sun has due to the rotation about its axis. (Assume that the sun is a uniform sphere.) (d) How does the rotational angular momentum of the sun compare with the total orbital
angular momentum of the planets? How does the mass of the sun compare with the total mass of the planets? The fact that the sun has most of the mass of the solar system but only a small fraction of its total angular momentum must be accounted for in models of how the solar system formed. (e) The sun has a density that decreases with distance from its center. Does this mean that your calculation
in part (c) overestimates or underestimates the rotational angular momentum of the sun? Or doesn't the nonuniform density have any effect?

João Gabriel Alencar Caribé

João Gabriel Alencar Caribé

Numerade Educator

Problem 79

The most efficient way to send a spacecraft from the earth to another planet is to use a Hohmann transfer orbit ($\textbf{Fig. P13.79}$). If the orbits of the departure and destination planets are circular, the Hohmann transfer orbit is an elliptical orbit whose perihelion and aphelion are tangent to the orbits of the two planets. The rockets are fired briefly at the departure planet to put the spacecraft into the transfer orbit; the spacecraft then coasts until it reaches the destination planet. The rockets are then fired again to put the spacecraft into the same orbit about the sun as the destination planet. (a) For a flight from earth to Mars, in what direction must the rockets be fired at the earth and at Mars: in the direction of motion or opposite the direction of motion? What about for a flight from Mars to the earth? (b) How long does a one-way trip from the earth to Mars take, between the firings of the rockets? (c) To reach Mars from the earth, the launch must be timed so that Mars will be at the right spot when the spacecraft reaches Mars's orbit around the sun. At launch, what must the angle between a sun$-$Mars line and a sun$-$earth line be? Use Appendix F.
Figure P13.79 (FIGURE CAN'T COPY)

David Gold

Numerade Educator

Problem 80

An astronaut inside a spacecraft, which protects her from harmful radiation, is orbiting a black hole at a distance of 120 km from its center. The black hole is 5.00 times the mass of the sun and has a Schwarzschild radius of 15.0 km. The astronaut is positioned inside the spaceship such that one of her 0.030-kg ears is 6.0 cm farther from the black hole than the center of mass of the spacecraft and the other ear is 6.0 cm closer. (a) What is the tension between her ears? Would the astronaut find it difficult to keep from being torn apart by the gravitational forces? (Since her whole body orbits with the same angular velocity, one ear is moving too slowly for the radius of its orbit and the other is moving too fast. Hence her head must exert forces on her ears to keep them in their orbits.) (b) Is the center of gravity of her head at the same point as the center of mass? Explain.

Salamat Ali

Numerade Educator

Problem 81

Mass $M$ is distributed uniformly over a disk of radius $a$. Find the gravitational force (magnitude and direction) between this disk-shaped mass and a particle with mass $m$ located a distance $x$ above the center of the disk ($\textbf{Fig. P13.81}$). Does your result reduce to the correct expression as $x$ becomes very large? ($Hint:$ Divide the disk into infinitesimally thin concentric rings, use the expression derived in Exercise 13.35 for the gravitational force due to each ring, and integrate to find the total force.) $\textbf{EXOPLANETS.}$ As planets with a wide variety of properties are being discovered outside our solar system, astrobiologists are considering whether and how life could evolve on planets that might be very different from earth. One recently discovered extrasolar planet, or exoplanet, orbits a star whose mass is 0.70 times the mass of our sun. This planet has been found to have 2.3 times the earth's diameter and 7.9 times the earth's mass. For planets in this size range, computer models indicate a relationship between the planet's density and composition:
Figure P13.81 (FIGURE CAN'T COPY)

David Gold

Numerade Educator

Problem 82

Based on these data, what is the most likely composition of this planet? (a) Mostly iron; (b) iron and rock; (c) iron and rock with some lighter elements; (d) hydrogen and helium gases.

Jilin Wang

Boston University

Problem 83

How many times the acceleration due to gravity $g$ near the earth's surface is the acceleration due to gravity near the surface of this exoplanet? (a) About 0.29$g$; (b) about 0.65$g$; (c) about 1.5$g$; (d) about 7.9$g$.

David Gold

Numerade Educator

Problem 84

Observations of this planet over time show that it is in a nearly circular orbit around its star and completes one orbit in only 9.5 days. How many times the orbital radius $r$ of the earth around our sun is this exoplanet's orbital radius around its sun? Assume that the earth is also in a nearly circular orbit. (a) 0.026$r$; (b) 0.078$r$; (c) 0.70$r$; (d) 2.3$r$.

Salamat Ali

Numerade Educator