Planets are not uniform inside. Normally, they are densest

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?

Cavendish Experiment. In the Cavendish balance apparatus shown in Fig. 13.4, suppose that and the rod connecting the pairs is 30.0 cm long. If, in each pair, and 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.

Rendezvous in Space! 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?

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

Two uniform spheres, each of mass 0.260 kg, are fixed at points A and B (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

Find the magnitude and direction of the net gravitational force on mass A due to masses B and C in Fig. E13.6. Each mass is 2.00 kg.

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

An 8.00-kg point mass and a 15.0-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.

A particle of mass 3m 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 3m, and (ii) for points along the line passing through M and perpendicular to the line connecting m and 3m?

The point masses m and 2m lie along the x-axis, with m at the origin and 2m at 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 2m, taking quantities to the right as positive. Include the regions and Be especially careful to show the behavior of the graph on either side of and

At what distance above the surface of the earth is the acceleration due to the earths gravity if the acceleration due to gravity at the surface has magnitude

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?

Titania, the largest moon of the planet Uranus, has the radius of the earth and 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.)

Rhea, one of Saturn's moons, has a radius of 765 km and an acceleration due to gravity of \(0.278\mathrm{\ m}/\mathrm{s}^2\) at its surface. Calculate its mass and average density.

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 we say 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?

Volcanoes on Io. 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 \(8.94\times 10^{22}\mathrm{\ kg}\) and a radius of 1815 km. 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?

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 spacecrafts mass?

Ten days after it was launched toward Mars in December 1998, the Mars Climate Orbiter spacecraft (mass 629 kg) was from the earth and traveling at relative to the earth. At this time, what were (a) the spacecrafts kinetic energy relative to the earth and (b) the potential energy of the earthspacecraft system?

For a satellite to be in a circular orbit 780 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)?

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

Two satellites are in circular orbits around a planet that has radius . One satellite has mass 68.0 kg, orbital radius , and orbital speed 4800 m s. The second satellite has mass 84.0 kg and orbital radius . What is the orbital speed of this second satellite?

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

Deimos, a moon of Mars, is about 12 km in diameter with mass 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?

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.) Text Transcription: 2/3

The star \(\mathrm{Rho}^{1}\) 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 \(\mathrm{Rho}^{1}\) 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 \(\mathrm{Rho}^{1}\) Cancri? Text Transcription: Rho^1

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.

(a) Use Fig. 13.18 to show that the sun-planet distance at perihelion is (1 - e)a, the sun-planet distance at aphelion is (1 + e)a, and therefore the sum of these two distances is 2a. (b) When the dwarf planet Pluto was at perihelion in 1989, it was almost 100 million km closer to the sun than Neptune. The semimajor axes of the orbits of Pluto and Neptune are \(5.92\times10^{12}\mathrm{\ m}\) and \(4.50\times10^{12}\mathrm{\ m}\), respectively, and the eccentricities are 0.248 and 0.010. Find Pluto's closest distance and Neptune’s farthest distance from the sun. (c) How many years after being at perihelion in 1989 will Pluto again be at perihelion?

Hot Jupiters. 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 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 suns mass. (b) How fast (in ) is this planet moving?

Planets Beyond the Solar System. 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.)

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 to

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 to

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 (Fig. E13.32). (a) Calculate the gravitational potential energy of the rodsphere system. Take the potential energy to be zero when the rod and sphere are infi- nitely 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 given in Appendix B.) (b) Use 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

Consider the ring-shaped body of Fig. E13.33. 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 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 when Explain why these results make sense.

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 475.0 N in the rope. If Sneezy hangs from a similar rope while delivering presents at the earths 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

The acceleration due to gravity at the north pole of Neptune is approximately Neptune has mass and radius and rotates once around its axis in about 16 h. (a) What is the gravitational force on a 5.0-kg object at the north pole of Neptune? (b) What is the apparent weight of this same object at Neptunes equator? (Note that Neptunes surface is gaseous, not solid, so it is impossible to stand on it.)

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 what would be the mass of a mini black hole?

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?

(a) Show that a black hole attracts an object of mass m with a force of \(m c^{2} R_{\mathrm{S}} /\left(2 r^{2}\right)\), where r is the distance between the object and the center of the black hole. (b) Calculate the magnitude of the gravitational force exerted by a black hole of Schwarzschild radius 14.0 mm on a 5.00-kg mass 3000 km from it. (c) What is the mass of this black hole?

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 (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 suns mass. (c) What is the radius of its event horizon?

Four identical masses of 800 kg each are placed at the corners of a square whose side length is 10.0 cm. What is the net gravitational force (magnitude and direction) on one of the masses, due to the other three?

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?

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 and its diameter is 3138 km

Three uniform spheres are fixed at the positions shown in 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 below the what will the particles speed be when it reaches the origin?

A uniform sphere with mass 60.0 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 (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 (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?

(a) Use data from Appendix F to calculate the acceleration due to gravity on the moon. (b) Calculate the friction force on a walking 65-kg astronaut carrying a 43-kg instrument pack on the moon if the coefficient of kinetic friction at her hip joint is 0.0050. (c) What would be the friction force on earth for this astronaut?

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? Text Transcription: 1.22 x 10^6 km 1.35 x 10^23 kg

The asteroid Toro has a radius of about 5.0 km. Consult Appendix F as necessary. (a) Assuming that the density of Toro is the same as that of the earth \(\left(5.5\mathrm{\ g}/\mathrm{cm}^3\right)\) find its total mass and find the acceleration due to gravity at its surface. (b) Suppose an object is to be placed in a circular orbit around Toro, with a radius just slightly larger than the asteroid’s radius. What is the speed of the object? Could you launch yourself into orbit around Toro by running?

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 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? You can ignore any gravitational effects due to the other planets or the sun.

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?

Some scientists are eager to send a remote-controlled submarine to Jupiters moon Europa to search for life in its oceans below an icy crust. Europas mass has been measured to be its diameter is 3138 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 are 25.0 cm square and can stand a maximum inward force of 9750 N per window, what is the greatest depth to which this submarine can safely dive?

Many satellites are moving in a circle in the earths equatorial plane. They are at such a height above the earths surface that they always remain above the same point. (a) Find the altitude of these satellites above the earths 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 latitude.

A landing craft with mass 12,500 kg is in a circular orbit 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 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 planets surface?

What is the escape speed from a 300-km-diameter asteroid with a density of

(a) Asteroids have average densities of about and radii from 470 km down to less than a kilometer. Assuming that the asteroid has a spherically symmetric mass distribution, estimate the radius of the largest asteroid from which you could escape simply by jumping off. (Hint: You can estimate your jump speed by relating it to the maximum height that you can jump on earth.) (b) Europa, one of Jupiters four large moons, has a radius of 1570 km. The acceleration due to gravity at its surface is Calculate its average density

(a) Suppose you are at the earths 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 earths surface is the satellites 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 satellites orbit above the surface of the earth?

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?

There are two equations from which a change in the gravitational potential energy U of the system of a mass m and the earth can be calculated. One is (Eq. 7.2). The other is (Eq. 13.9). As shown in Section 13.3, the first equation is correct only if the gravitational force is a constant over the change in height The second is always correct. Actually, the gravitational force is never exactly constant over any change in height, but if the variation is small, we can ignore it. Consider the difference in U between a mass at the earths surface and a distance h above it using both equations, and find the value of h for which Eq. (7.2) is in error by 1%. Express this value of h as a fraction of the earths radius, and also obtain a numerical value for it.

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 returns to the ground in 6.00 s; the circumference of Mongo at the equator is 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?

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 \(8.60\times10^7\mathrm{\ m}\), what is the mass of the planet?

In Example 13.5 (Section 13.3) we ignored the gravitational effects of the moon on a spacecraft en route from the earth to the moon. In fact, we must include the gravitational potential energy due to the moon as well. For this problem, you can ignore the motion of the earth and moon. (a) If the moon has radius and the distance between the centers of the earth and the moon is find the total gravitational potential energy of the particle earth and particlemoon systems when a particle with mass m is between the earth and the moon, and a distance r from the center of the earth. Take the gravitational potential energy to be zero when the objects are far from each other. (b) There is a point along a line between the earth and the moon where the net gravitational force is zero. Use the expression derived in part (a) and numerical values from Appendix F to find the distance of this point from the center of the earth. With what speed must a spacecraft be launched from the surface of the earth just barely to reach this point? (c) If a spacecraft were launched from the earths surface toward the moon with an initial speed of with what speed would it impact the moon?

Calculate the percent difference between your weight in Sacramento, near sea level, and at the top of Mount Everest, which is 8800 m above sea level.

The 0.100-kg sphere in Fig. P13.62 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?

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 If nothing is done to correct its orbit, with what speed (in ) will the spacecraft crash into the lunar surface?

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 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 comets 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?

A hammer with mass m is dropped from rest from a height h above the earths surface. This height is not necessarily small compared with the radius of the earth. If you ignore air resistance, derive an expression for the speed of the hammer when it reaches the surface of the earth. Your expression should involve h, and the mass of the earth.

. (a) Calculate how much work is required to launch a spacecraft of mass m from the surface of the earth (mass radius ) and place it in a circular low earth orbitthat is, an orbit whose altitude above the earths surface is much less than (As an example, the International Space Station is in low earth orbit at an altitude of about 400 km, much less than ) You can ignore the kinetic energy that the spacecraft has on the ground due to the earths 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. You can 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.

A spacecraft is to be launched from the surface of the earth so that it will escape from the solar system altogether. (a) Find the speed relative to the center of the earth with which the spacecraft must be launched. Take into consideration the gravitational effects of both the earth and the sun, and include the effects of the earths orbital speed, but ignore air resistance. (b) The rotation of the earth can help this spacecraft achieve escape speed. Find the speed that the spacecraft must have relative to the earths surface if the spacecraft is launched from Florida at the point shown in Fig. P13.67. The rotation and orbital motions of the earth are in the same direction. The launch facilities in Florida are north of the equator. (c) The European Space Agency (ESA) uses launch facilities in French Guiana (immediately north of Brazil), north of the equator. What speed relative to the earths surface would a spacecraft need to escape the solar system if launched from French Guiana?

Gravity Inside the Earth. Find the gravitational force that the earth exerts on a 10.0-kg mass if it is placed at the following locations. Consult Fig. 13.9, and assume a constant density through each of the interior regions (mantle, outer core, inner core), but not the same density in each of these regions. Use the graph to estimate the average density for each region: (a) at the surface of the earth; (b) at the outer surface of the molten outer core; (c) at the surface of the solid inner core; (d) at the center of the earth.

Hundreds of thousands of asteroids orbit the sun within the asteroid belt, which extends from about to about from the sun. (a) Find the orbital period (in years) of (i) an asteroid at the inside of the belt and (ii) an asteroid at the outside of the belt. Assume circular orbits. (b) In 1867 the American astronomer Daniel Kirkwood pointed out that several gaps exist in the asteroid belt where relatively few asteroids are found. It is now understood that these Kirkwood gaps are caused by the gravitational attraction of Jupiter, the largest planet, which orbits the sun once every 11.86 years. As an example, if an asteroid has an orbital period half that of Jupiter, or 5.93 years, on every other orbit this asteroid would be at its closest to Jupiter and feel a strong attraction toward the planet. This attraction, acting over and over on successive orbits, could sweep asteroids out of the Kirkwood gap. Use this hypothesis to determine the orbital radius for this Kirkwood gap. (c) One of several other Kirkwood gaps appears at a distance from the sun where the orbital period is 0.400 that of Jupiter. Explain why this happens, and find the orbital radius for this Kirkwood gap.

If a satellite is in a sufficiently low orbit, it will encounter air drag from the earth’s atmosphere. Since air drag does negative work (the force of air drag is directed opposite the motion), the mechanical energy will decrease. According to Eq. (13.13), if E decreases (becomes more negative), the radius r of the orbit will decrease. If air drag is relatively small, the satellite can be considered to be in a circular orbit of continually decreasing radius. (a) According to Eq. (13.10), if the radius of a satellite’s circular orbit decreases, the satellite’s orbital speed v increases. How can you reconcile this with the statement that the mechanical energy decreases? (Hint: Is air drag the only force that does work on the satellite as the orbital radius decreases?) (b) Due to air drag, the radius of a satellite’s circular orbit decreases from r to \(r-\Delta r\), where the positive quantity \(\Delta r\) is much less than r. The mass of the satellite is m. Show that the increase in orbital speed is \(\Delta v=+(\Delta r / 2) \sqrt{G m_{\mathrm{E}} / r^{3}}\); that the change in kinetic energy is \(\Delta K=+\left(G m_{\mathrm{E}} m / 2 r^{2}\right) \Delta r\); that the change in gravitational potential energy is \(\Delta U = -2 \Delta K = -\left(G m_{\mathrm{E}} m / r^{2}\right) \Delta r\); and that the amount of work done by the force of air drag is \(W=-\ \left(Gm_{\mathrm{E}}m/2r^2\right)\ \Delta r\). Interpret these results in light of your comments in part (a). (c) A satellite with mass 3000 kg is initially in a circular orbit 300 km above the earth’s surface. Due to air drag, the satellite’s altitude decreases to 250 km. Calculate the initial orbital speed; the increase in orbital speed; the initial mechanical energy; the change in kinetic energy; the change in gravitational potential energy; the change in mechanical energy; and the work done by the force of air drag. (d) Eventually a satellite will descend to a low enough altitude in the atmosphere that the satellite burns up and the debris falls to the earth. What becomes of the initial mechanical energy? Text Transcription: r - Delta r Delta r Delta v = + (Delta r/2) sqrt Gm_E/r^3 Delta K = + (Gm_Em/2r^2) Delta r Delta U = -2 Delta K = - (Gm_Em/r^2) Delta r W = - (Gm_Em/2r^2) Delta r

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?

Two stars, with masses and are in circular orbits around their center of mass. The star with mass has an orbit of radius the star with mass has an orbit of radius (a) Show that the ratio of the orbital radii of the two stars equals the reciprocal of the ratio of their massesthat is, (b) Explain why the two stars have the same orbital period, and show that the period T is given by (c) The two stars in a certain binary star system move in circular orbits. The first star, Alpha, has an orbital speed of The second star, Beta, has an orbital speed of 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. 13.27). 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 objects 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.

Comets travel around the sun in elliptical orbits with large eccentricities. If a comet has speed when at a distance of from the center of the sun, what is its speed when at a distance of 5.0 x 1010 m?

An astronaut is standing at the north pole of a newly discovered, spherically symmetric planet of radius R. In his hands he holds a container full of a liquid with mass m and volume V. At the surface of the liquid, the pressure is at a depth d below the surface, the pressure has a greater value p. From this information, determine the mass of the planet

The earth does not have a uniform density; it is most dense at its center and least dense at its surface. An approximation of its density is where and Use for the radius of the earth approximated as a sphere. (a) Geological evidence indicates that the densities are and at the earths center and surface, respectively. What values does the linear approximation model give for the densities at these two locations? (b) Imagine dividing the earth into concentric, spherical shells. Each shell has radius r, thickness dr, volume and mass By integrating from to show that the mass of the earth in this model is (c) Show that the given values of A and B give the correct mass of the earth to within 0.4%. (d) We saw in Section 13.6 that a uniform spherical shell gives no contribution to g inside it. Show that inside the earth in this model. (e) Verify that the expression of part (d) gives at the center of the earth and at the surface. (f) Show that in this model g does not decrease uniformly with depth but rather has a maximum

In Example 13.10 (Section 13.6) we saw that inside a planet of uniform density (not a realistic assumption for the earth) the acceleration due to gravity increases uniformly with distance from the center of the planet. That is, where is the acceleration due to gravity at the surface, r is the distance from the center of the planet, and R is the radius of the planet. The interior of the planet can be treated approximately as an incompressible fluid of density (a) Replace the height y in Eq. (12.4) with the radial coordinate r and integrate to find the pressure inside a uniform planet as a function of r. Let the pressure at the surface be zero. (This means ignoring the pressure of the planets atmosphere.) (b) Using this model, calculate the pressure at the center of the earth. (Use a value of equal to the average density of the earth, calculated from the mass and radius given in Appendix F.) (c) Geologists estimate the pressure at the center of the earth to be approximately Does this agree with your calculation for the pressure at What might account for any differences?

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?

The planet Uranus has a radius of 25,560 km and a surface acceleration due to gravity of \(11.1\mathrm{\ m}/\mathrm{s}^2\) 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 \(6.6\times10^{19}\mathrm{\ kg}\) and a radius of 235 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.

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?

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 comets orbit. Compare it to the average sunPluto distance and to the distance to Alpha Centauri, the nearest star to the sun, which is 4.3 light-years distant

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 \(15.0\times10^3\mathrm{\ kg}/\mathrm{m}^3\) at the center and \(2.0\times10^3\mathrm{\ kg}/\mathrm{m}^3\) at the surface. What is the acceleration due to gravity at the surface of this planet?

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.

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. E13.33). 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

A thin, uniform rod has length L and mass M. Calculate the magnitude of the gravitational force the rod exerts on a particle with mass m that is at a point along the axis of the rod a distance x from one end (see Fig. E13.32). Show that your result reduces to the expected result when x is much larger than L.

A shaft is drilled from the surface to the center of the earth (see Fig. 13.24). 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 \(F_{\mathrm{g}}=G m_{\mathrm{E}} m r / R_{\mathrm{E}}^3\) (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?

(a) When an object is in a circular orbit of radius r around the earth (mass ), the period of the orbit is T, given by Eq. (13.12), and the orbital speed is given by Eq. (13.10). Show that when the object is moved into a circular orbit of slightly larger radius where its new period is and its new orbital speed is where and are all positive quantities and T = 3p r v and v = p r T [Hint: Use the expression valid for ] (b) The International Space Station (ISS) is in a nearly circular orbit at an altitude of 398.00 km above the surface of the earth. A maintenance crew is about to arrive on the space shuttle that is also in a circular orbit in the same orbital plane as the ISS, but with an altitude of 398.10 km. The crew has come to remove a faulty 125-m electrical cable, one end of which is attached to the ISS and the other end of which is floating free in space. The plan is for the shuttle to snag the free end just at the moment that the shuttle, the ISS, and the center of the earth all lie along the same line. The cable will then break free from the ISS when it becomes taut. How long after the free end is caught by the space shuttle will it detach from the ISS? Give your answer in minutes. (c) If the shuttle misses catching the cable, show that the crew must wait a time before they have a second chance. Find the numerical value of t and explain whether it would be worth the wait.

The most efficient way to send a spacecraft from the earth to another planet is by using a Hohmann transfer orbit (Fig. P13.87). 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 oneway trip from the 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 Marss orbit around the sun. At launch, what must the angle between a sunMars line and a sunearth line be? Use data from Appendix F.

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

Mass M is distributed uniformly over a disk of radius a. Find the gravitational force (magnitude and direction) between this diskshaped mass and a particle with mass m located a distance x above the center of the disk (Fig. P13.89). 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 13.33 for the gravitational force due to each ring, and integrate to find the total force.)

Mass M is distributed uniformly along a line of length 2L. A particle with mass m is at a point that is a distance a above the center of the line on its perpendicular bisector (point P in Fig. P13.90). For the gravitational force that the line exerts on the particle, calculate the components perpendicular and parallel to the line. Does your result reduce to the correct expression as a becomes very large?

Problem 1DQ A student wrote: “The only reason an apple falls downward to meet the earth instead of the earth rising upward to meet the apple is that the earth is much more massive and so exerts a much greater pull.” Please comment .

Problem 1E 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?

A planet makes a circular orbit with period \(T\) around a star. If it were to orbit, at the same distance, a star with three times the mass of the original star, would the new period (in terms of \(T\)) be (a) \(3 T\), (b) \(T \sqrt{3}\) (c) \(T\) (d) \(T / \sqrt{3}\)or (e) \(T / 3\) ? Equation Transcription: Text Transcription: T T 3T T \sqrt3 T \T / \sqrt3 T/3

Cavendish Experiment. 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, \(m_{1} \text { and } m_{2}\) 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.

Problem 3DQ If all planets had the same average density, how would the acceleration due to gravity at the surface of a planet depend on its radius?

Problem 3E Rendezvous in Space! 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?

Two uniform spheres, each of mass 0.260 kg, are fixed at points and (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 and acted on only by forces of gravitational attraction of the spheres at and . Equation Transcription: Text Transcription: 0.260 kg A and B 0.010 kg P A and B 0.010 kg 10.0 cm 0.260 kg

Problem 6DQ When will you attract the sun more: today at noon, or tonight at midnight? Explain.

Find the magnitude and direction of the net gravitational force on mass due to masses B and C in Fig. E13.6. Each mass is 2.00 kg

Since the moon is constantly attracted toward the earth by the gravitational interaction, why doesn’t it crash into the earth?

Problem 7E 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.

A planet makes a circular orbit with period \(T\) around a star. If the planet were to orbit at the same distance around this star, but had three times as much mass, what would the new period (in terms of \(T\)) be: (a) \(3 T\), (b) \(T \sqrt{3}\) (c) \(\text { T/ } \sqrt{3}\)or (e) \(\text { T/3 }\)? Equation Transcription: Text Transcription: T T 3T T squart 3 T/ squart 3 T/3

Problem 8E 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.

The sun pulls on the moon with a force that is more than twice the magnitude of the force with which the earth attracts the moon. Why, then, doesn’t the sun take the moon away from the earth?

Problem 9E A particle of mass 3m 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?

Problem 10DQ As defined in Chapter 7, gravitational potential energy is U = mgy and is positive for a body of mass m above the earth’s surface (which is at y = 0). But in this chapter, gravitational potential energy is U = ?GmEm/r. which is negative for a body of mass m above the earth’s surface (which is at r = RE). How can you reconcile these seemingly incompatible descriptions of gravitational potential energy?

Problem 10E 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.

Problem 11DQ A planet is moving at constant speed in a circular orbit around a star. In one complete orbit, what is the net amount of work done on the planet by the star’s gravitational force: positive, negative, or zero? What if the planet’s orbit is an ellipse, so that the speed is not constant? Explain your answers.

Problem 11E At what distance above the surface of the earth is the acceleration due to the earth’s gravity 0.980 m/s2 if the acceleration due to gravity at the surface has magnitude 9.80 m/s2?

Problem 12DQ Does the escape speed for an object at the earth’s sur-face depend on the direction in which it is launched? Explain. Does your answer depend on whether or not you include the effects of air resistance?

Problem 12E 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?

If a projectile is fired straight up from the earth’s surface, what would happen if the total mechanical energy (kinetic plus potential) is (a) less than zero, and (b) greater than zero? In each case, ignore air resistance and the gravitational effects of the sun, the moon, and the other planets.

Titania, the largest moon of the planet Uranus, has \(\frac{1}{8}\) the radius of the earth \(\frac{1}{1700}\) and 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.)

Problem 14DQ Discuss whether this statement is correct: “In the absence of air resistance, the trajectory of a projectile thrown near the earth’s surface is an ellipse, not a parabola.”

Problem 14E Rhea, one of Saturn’s m ns, has a radius of 765 km and an acceleration due to gravity of 0.278 m/s2 at its surface. Calculate its mass and average density.

The earth is closer to the sun in November than in May. In which of these months does it move faster in its orbit? Explain why.

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 we say 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?

A communications firm wants to place a satellite in orbit so that it is always directly above the earth’s 45th parallel (latitude \(45^{\circ}\) north). This means that the plane of the orbit will not pass through the center of the earth. Is such an orbit possible? Why or why not?

Problem 16E Volcanoes on Io. Jupiter’s moon to 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, to has a mass of 8.94 × 1022 kg and a radius of 1815 km. 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?

Problem 17DQ At what point in an elliptical orbit is the acceleration maximum? At what point is it minimum? Justify your answers.

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?

Which takes more fuel: a voyage from the earth to the moon or from the moon to the earth? Explain.

Ten days after it was launched toward Mars in December 1998, the Mars Climate Orbiter spacecraft (mass 629 kg) was \(2.87 \times 10^6 \mathrm{\ km}\)from the earth and traveling at \(1.20 \times 10^4 \mathrm{\ km}/ \mathrm{h}\) 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?

Aura Mission. 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?

Problem 21DQ Many people believe that orbiting astronauts feel weightless because they are “beyond the pull of the earth’s gravity.” How far from the earth would a spacecraft have to travel to be truly beyond the earth’s gravitational influence? If a spacecraft were really unaffected by the earth’s gravity, would it remain in orbit? Explain. What is the real reason astronauts in orbit feel weightless?

Two satellites are in circular orbits around a planet that has radius \(9.00 \times 10^{6} \mathrm{\ m}\). One satellite has mass 68.0 kg, orbital radius \(5.00 \times 10^{7} \mathrm{\ m}\), and orbital speed 4800 m/s. The second satellite has mass 84.0 kg and orbital radius \(3.00 \times 10^{7} \mathrm{\ m}\). What is the orbital speed of this second satellite?

Problem 22DQ As part of their training before going into orbit, astronauts ride in an airliner that is flown along the same parabolic trajectory as a freely falling projectile. Explain why this gives the same experience of apparent weightlessness as being in orbit.

Problem 22E International Space Station. 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?

Deimos, a moon of Mars, is about 12 km in diameter with mass \(2.0 \times 10^{15}\mathrm{\ kg}\). 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? Text Transcription: 2.0 times 10^15 kg

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.) Equation Transcription: Text Transcription: 2 over 3

Problem 25E The star Rho1 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 Rho1 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 Rho1 Cancri?

Problem 26E 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.

(a) Use Fig. 13.18 to show that the sun–planet distance at perihelion is (1 - e)a, the sun–planet distance at aphelion is (1 + e)a and therefore the sum of these two distances is 2a. (b) When the dwarf planet Pluto was at perihelion in 1989, it was almost 100 million km closer to the sun than Neptune. The semi-major axes of the orbits of Pluto and Neptune are \(5.92 \times 10^{12} \mathrm{\ m}\) and \(4.50 \times 10^{12} \mathrm{\ m}\), respectively, and the eccentricities are 0.248 and 0.010. Find Pluto’s closest distance and Neptune’s farthest distance from the sun. (c) How many years after being at perihelion in 1989 will Pluto again be at perihelion?

Hot Jupiters. 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 \(\mathrm{km} / \mathrm{s}\)) is this planet moving? Equation Transcription: Text Transcription: 1 over 9 km/s

Problem 29E Planets Beyond the Solar System. 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.)

A uniform, spherical, \(1000.0-\mathrm{kg}\) sphere has a radius of \(5.00 \mathrm{~m}\). (a) Find the gravitational force this sphere exerts on a \(2.00-\mathrm{kg}\) point mass placed at the following distances from the center of the sphere: (i) \(5.01 \mathrm{~m}\), (ii) \(4.99 \mathrm{~m}\), (iii) \(2.72 \mathrm{~m}\). (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 \(r=0 \text { to } r \rightarrow \infty\) Equation Transcription: Text Transcription: 1000.0-kg 5.00 m 2.00-kg 5.01 m 4.99 m 2.72 m m r of m r=0 \text to r \rightarrow \infty

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 \(r=0 \text { to } r \rightarrow \infty\).

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 (Fig. E13.32). (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 ln \((1+x)\) given in Appendix B.) (b) Use \(F_{x}=-d U / d x\) 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 is much larger than Equation Transcription: Text Transcription: L M m x x (1+x) F_x = -dU/dx

Problem 34E 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 475.0 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.

Problem 35E The acceleration due to gravity at the north pole of Neptune is approximately 10.7 m/s2. Neptune has mass 1.0 × 1026 kg and radius 2.5 × 104 km and rotates once around its axis in about 16 h. (a) What is the gravitational force on a 5.0-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.)

Problem 36E Mini Black Holes. 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 X 10-15 m, what would be the mass of a mini black hole?

Problem 37E At the Galaxy’s Core. 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?

(a) Show that a black hole attracts an object of mass m with a force of \(m c^{2} R_{\mathrm{S}} /\left(2 r^{2}\right)\), where r is the distance between the object and the center of the black hole. (b) Calculate the magnitude of the gravitational force exerted by a black hole of Schwarzschild radius 14.0 mm on a 5.00-kg mass 3000 km from it. (c) What is the mass of this black hole?

Problem 39E 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?

Problem 40P Four identical masses of 800 kg each are placed at the corners of a square whose side length is 10.0 cm. What is the net gravitational force (magnitude and direction) on one of the masses, due to the other three?

Problem 41P 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?

Problem 42P Exploring Europa. 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 4.8 × 1022 kg and its diameter is 3138 km.

Problem 43P Three uniform spheres are fixed at the positions shown in 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 45o below the – x-axis, what will the particle’s speed be when it reaches the origin?

Problem 44P A uniform sphere with mass 60.0 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?

Problem 48P 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 X 105 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.

Submarines on Europa. 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 \(4.8 \times 10^{22} \mathrm{\ kg}\), its diameter is 3138 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 are 25.0 cm square and can stand a maximum inward force of 9750 N per window, what is the greatest depth to which this submarine can safely dive?

Problem 51P Geosynchronous Satellites. 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° N latitude.

Problem 52P A landing craft with mass 12,500 kg is in a circular orbit 5.75 X 105 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 X 106 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?

Problem 53P What is the escape speed from a 300-km-diameter asteroid with a density of 2500 kg/m3?

Problem 54P (a) Asteroids have average densities of about 2500 kg/m3 and radii from 470 km down to less than a kilometer. Assuming that the asteroid has a spherically symmetric mass distribution estimate the radius of the largest asteroid from which you could escape simply by jumping off. (Hint: You can estimate your jump speed by relating it to the maximum height that you can jump on earth.) (b) Europa, one of Jupiter’s four large moons, has a radius of 1570 km. The acceleration due to gravity at its surface is 1.33 m/s2. Calculate its average density.

Problem 55P (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?

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?

Problem 57P There are two equations from which a change in the gravitational potential energy U of the system of a mass m and the earth can be calculated. One is U = mgy (Eq. 7.2). The other is U = ?GmEm/r (Eq. 13.9). As shown in Section 13.3, the first equation is correct only if the gravitational force is a constant over the change in height ?y. The second is always correct. Actually, the gravitational force is never exactly constant over any change in height, but if the variation is small, we can ignore it. Consider the difference in U between a mass at the earth’s surface and a distance h above it using both equations, and find the value of h for which Eq (7.2) is in error by 1%. Express this value of h as a fraction of the earth’s radius, and also obtain a numerical value for it.

Problem 58P 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 6.00 s; the circumference of Mongo at the equator is 2.00 × 105 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?

Problem 59P CP 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 X 107 m, what is the mass of the planet?

In Example 13.5 (Section 13.3) we ignored the gravitational effects of the moon on a spacecraft en route from the earth to the moon. In fact, we must include the gravitational potential energy due to the moon as well. For this problem, you can ignore the motion of the earth and moon. (a) If the moon has radius \(R_{\mathrm{M}}\) and the distance between the centers of the earth and the moon is \(R_{\mathrm{EM}}\), find the total gravitational potential energy of the particle– earth and particle–moon systems when a particle with mass m is between the earth and the moon, and a distance r from the center of the earth. Take the gravitational potential energy to be zero when the objects are far from each other. (b) There is a point along a line between the earth and the moon where the net gravitational force is zero. Use the expression derived in part (a) and numerical values from Appendix F to find the distance of this point from the center of the earth. With what speed must a spacecraft be launched from the surface of the earth just barely to reach this point? (c) If a spacecraft were launched from the earth’s surface toward the moon with an initial speed of 11.2 km/swith what speed would it impact the moon?

Problem 61P Calculate the percent difference between your weight in Sacramento, near sea level, and at the top of Mount Everest, which is 8800 m above sea level.

The \(0.100-\mathrm{kg}\) sphere in Fig. P13.62 is released from rest at the position shown in the sketch, with its center \(0.400 \mathrm{~m}\) from the center of the \(\text { 5.00-kg }\) mass. Assume that the only forces on the \(\text { 0.100-kg }\) sphere are the gravitational forces exerted by the other two spheres and that the \(5.00-\mathrm{kg}\) and \(\text { 10.0-kg }\) spheres are held in place at their initial positions. What is the speed of the \(0.100-\mathrm{kg}\) sphere when it has moved \(0.400 \mathrm{~m}\) to the right from its initial position? Equation Transcription: Text Transcription: 0.100-kg 0.400 m 5.00-kg 0.100-kg 5.00-kg 10.0-kg 0.100-kg

Problem 63P 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?

Mass of a Comet. 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?

Problem 65P Falling Hammer. 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 RE of the earth. Ignoring air resistance, derive an expression for the speed v of the hammer when it reaches the earth’s surface. Your expression should involve h, RE, and mE (the earth’s mass).

Gravity Inside the Earth. Find the gravitational force that the earth exerts on a \(\text { 10.0-kg }\) mass if it is placed at the following locations. Consult Fig. 13.9, and assume a constant density through each of the interior regions (mantle, outer core, inner core), but the same density in each of these regions. Use the graph to estimate the average density for each region: (a) at the surface of the earth; (b) at the outer surface of the molten outer core; (c) at the surface of the solid inner core; (d) at the center of the earth. Equation Transcription: Text Transcription: 10.0-kg

Problem 69P Kirkwood Gaps. Hundreds of thousands of asteroids orbit the sun within the asteroid belt, which extends from about 3 × 108 km to about 5 × 108 km from the sun. (a) Find the orbital period (in years) of (i) an asteroid at the inside of the belt and (ii) an asteroid at the outside of the belt. Assume circular orbits. (b) In 1867 the American astronomer Daniel Kirkwood pointed out that several gaps exist in the asteroid belt where relatively few asteroids are found It is now understand that these Kirkwood gaps are caused by the gravitational attraction of Jupiter, the largest planet, which orbits the sun once every 11.86 years. As an example, if an asteroid has an orbital period half that of Jupiter, or 5.93 years, on every other orbit this asteroid would be at its closest to Jupiter and feel a strong attraction toward the planet. This attraction, acting over and over on successive orbits, could sweep asteroids out of the Kirkwood gap. Use this hypothesis to determine the orbital radius for this Kirkwood gap. (c) One of several other Kirkwood gaps appears at a distance from the sun where the orbital period is 0.400 that of Jupiter. Explain why this happens, and find the orbital radius for this Kirkwood gap.

If a satellite is in a sufficiently low orbit, it will encounter air drag from the earth’s atmosphere. Since air drag does negative work (the force of air drag is directed opposite the motion), the mechanical energy will decrease. According to Eq. (13.13), if E decreases (becomes more negative), the radius r of the orbit will decrease. If air drag is relatively small, the satellite can be considered to be in a circular orbit of continually decreasing radius. (a) According to Eq. (13.10), if the radius of a satellite’s circular orbit decreases, the satellite’s orbital speed v increases. How can you reconcile this with the statement that the mechanical energy decreases? (Hint: Is air drag the only force that does work on the satellite as the orbital radius decreases?) (b) Due to air drag, the radius of a satellite’s circular orbit decreases from r to \(r-\Delta r\), where the positive quantity \(\Delta r\) is much less than r. The mass of the satellite is m. Show that the increase in orbital speed is \(\Delta v=+(\Delta r / 2) \sqrt{G m_{\mathrm{E}} / r^{3}}\); that the change in kinetic energy is \(\Delta K=+\left(G m_{\mathrm{E}} m / 2 r^{2}\right) \ \Delta r\); that the change in gravitational potential energy is \(\Delta U=-2 \Delta K=-\left(G m_{\mathrm{E}} m / r^{2}\right) \ \Delta r\); and that the amount of work done by the force of air drag is \(W=-\left(Gm_{\mathrm{E}}m/2r^2\right)\ \Delta r\). Interpret these results in light of your comments in part (a). (c) A satellite with mass 3000 kg is initially in a circular orbit 300 km above the earth’s surface. Due to air drag, the satellite’s altitude decreases to 250 km. Calculate the initial orbital speed; the increase in orbital speed; the initial mechanical energy; the change in kinetic energy; the change in gravitational potential energy; the change in mechanical energy; and the work done by the force of air drag. (d) Eventually a satellite will descend to a low enough alti-tude in the atmosphere that the satellite burns up and the debris falls to the earth. What becomes of the initial mechanical energy?

Binary Star—Equal Masses. 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?

Binary Star—Different Masses. 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 \(T=2 \pi\left(R_{1}+R_{2}\right)^{3 / 2} / \sqrt{G\left(M_{1}+M_{2}\right)}\). (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. 13.27). 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 cir-cular, 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.

Problem 73P Comets travel around the sun in elliptical orbits with large eccentricities. If a comet has speed 2.0 X 104 m/s when at a distance of 2.5 X 1011 m from the center of the sun, what is its speed when at a distance of 5.0 X 1010 m?

Problem 74P An astronaut is standing at the north pole of a newly discovered, spherically symmetric planet of radius R. In his hands he holds a container full of a liquid with mass m and volume V. At the surface of the liquid, the pressure is P0; at a depth d below surface, the pressure has a greater value p. From this information, determine the mass of the planet.

Problem 78P The planet Uranus has a radius of 25,560 km and a surface acceleration due to gravity of 11.1 m/s2 at its poles. Its m n Miranda (discovered by Kuiper in 1948) is in a circular orbit about Uranus at an altitude of 104,000 km above the planet’s surfaceMiranda has a mass of 6.6 × 1019 kg and a radius of 235 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 tall up relative to Miranda? Explain.

Problem 79P 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?

Problem 80P 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.

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 \(15.0 \times 10^3 \mathrm{\ kg}/ \mathrm{m}^3\) at the center and \(2.0 \times 10^3 \mathrm{\ kg}/\mathrm{m}^3\) at the surface. What is the acceleration due to gravity at the surface of this planet?

Problem 82P CALC 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.

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. E13.33). What is the gravitational force that the sphere exerts on the ring-shaped object? Show that your result reduces to the expected result when is much larger than \(a\). Equation Transcription: Text Transcription: a M m R x x a

A thin, uniform rod has length \(L\) and mass \(M\). Calculate the magnitude of the gravitational force the rod exerts on a particle with mass \(m\) that is at a point along the axis of the rod a distance \(x\) from one end (see Fig. E13.32). Show that your result reduces to the expected result when \(x\) is much larger than \(L\). Equation Transcription: Text Transcription: L M m x x L

A shaft is drilled from the surface to the center of the earth (see Fig. 13.24). 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 \(F_{g}=G m_{E} m r / R_{E}^{3}\) (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? Equation Transcription: Text Transcription: m r F_g=G m_E m r / R_E^3 U(r)

(a) When an object is in a circular orbit of radius \(r\) around the earth (mass \(m_{E}\)), the period of the orbit is \(T\) given by Eq. (13.12), and the orbital speed is \(v) given by Eq. (13.10). Show that when the object is moved into a circular orbit of slightly larger radius \(r+\Delta r\) where \(\Delta r<<r\) its new period is \(T+\Delta T\) and its new orbital speed is \(v-\Delta v\) where \(\Delta r, \Delta T\) and \(\Delta v\) are all positive quantities and \(\Delta T=\frac{3 \pi \Delta r}{v} \text { and } \Delta v=\frac{\pi \Delta r}{T}\) [Hint: Use the expression \((1+x)^{n} \approx 1 n x\), valid for \(|x| \ll 1\). ] (b) The International Space Station (ISS) is in a nearly circular orbit at an altitude of 398.00 km above the surface of the earth. A maintenance crew is about to arrive on the space shuttle that is also in a circular orbit in the same orbital plane as the ISS, but with an altitude of 398.10 km. The crew has come to remove a faulty 125-m electrical cable, one end of which is attached to the ISS and the other end of which is floating free in space. The plan is for the shuttle to snag the free end just at the moment that the shuttle, the ISS, and the center of the earth all lie along the same line. The cable will then break free from the ISS when it becomes taut. How long after the free end is caught by the space shuttle will it detach from the ISS? Give your answer in minutes. (c) If the shuttle misses catching the cable, show that the crew must wait a time \(t \approx T^{2} / \Delta T\) before they have a second chance. Find the numerical value of t and explain whether it would be worth the wait. Equation Transcription: and Text Transcription: r m_E T v r+\Delta r \Delta r<<r T+\Delta T v-\Delta v \Delta r, \Delta T \Delta v \Delta T=\frac 3 \pi \Delta r v \text and \Delta v=\frac \pi \Delta r T (1+x)^n \approx 1 n x |x| \ll 1 t \approx T^{2} / \Delta T

Interplanetary Navigation. The most efficient way to send a spacecraft from the earth to another planet is by using a (Fig. P13.87). 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 oneway trip from the 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 data from Appendix F.

Problem 88CP CP Tidal Forces near a Black Hole. 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.

Mass is distributed uniformly over a disk of radius . Find the gravitational force (magnitude and direction) between this diskshaped mass and a particle with mass located a distance above the center of the disk (Fig. P13.89). Does your result reduce to the correct expression as becomes very large? (Hint: Divide the disk into infinitesimally thin concentric rings, use the expression derived in Exercise 13.33 for the gravitational force due to each ring, and integrate to find the total force.)

Mass \(M\) is distributed uniformly along a line of length 2. A particle with mass \(m\) is at a point that is a distance \(a\) above the center of the line on its perpendicular bisector (point P in Fig. P13.90). For the gravitational force that the line exerts on the particle, calculate the components perpendicular and parallel to the line. Does your result reduce to the correct expression as \(a\) becomes very large? Equation Transcription: Text Transcription: M 2L m a a

Problem 4DQ Is a pound of butter on the earth the same amount as a pound of butter on Mars? What about a kilogram of butter? Explain.

Problem 4E Two uniform spheres, each with mass M and radius R, touch each other. What is the magnitude of their gravitational force of attraction?

Example 13.2 (Section 13.1) shows that the acceleration of each sphere caused by the gravitational force is inversely proportional to the mass of that sphere. So why does the force of gravity give all masses the same acceleration when they are dropped near the surface of the earth?

What would Kepler’s third law be for circular orbits if an amendment to Newton’s law of gravitation made the gravitational force inversely proportional to \(r^{3} ?\) Would this change affect Kepler’s other two laws? Explain.

For a satellite to be in a circular orbit 780 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)?

In the elliptical orbit of Comet Halley shown in Fig. 13.20a, the sun’s gravity is responsible for making the comet fall inward from aphelion to perihelion. But what is responsible for making the comet move from perihelion back outward to aphelion?

Problem 45P Hip Wear on the Moon. (a) Use data from Appendix F to calculate the acceleration due to gravity on the moon. (b) Calculate the friction force oil a walking 65-kg astronaut carrying a 43-kg instrument pack on the moon if the coefficient of kinetic friction at her hip joint is 0.0050. (c) What would be the friction force oil earth for this astronaut?

Problem 46P Mission to Titan. 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 X 106 km from the center of Saturn and has a mass of 1.35 X 1023 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?

The asteroid Toro has a radius of about 5.0 km. Consult Appendix F as necessary (a) Assuming that the density of Toro is the same as that of the earth (5.5 g/cm3), find its total mass and find the acceleration due to gravity at its surface. (b) Suppose an object is to be placed in a circular orbit around Toro, with a radius just slightly larger than the asteroid’s radius. What is the speed of the object? Could you launch yourself into orbit around Toro by running?

CP 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?

The earth does not have a uniform density; it is most dense at its center and least dense at its surface. An approximation of its density is \(\rho(r)=A-B r\), where \(A=12,700 \mathrm{~kg} / \mathrm{m}^{3}\) and \(B=1.50 \times 10^{-3} \mathrm{~kg} / \mathrm{m}^{4}\). Use \(R=6.37 \times 10^{6}\) m for the radius of the earth approximated as a sphere. (a) Geological evidence indicates that the densities are 13,100 \(\mathrm{kg} / \mathrm{m}^{3}\) and 2400 \(\mathrm{kg} / \mathrm{m}^{3}\) at the earth’s center and surface, respectively. What values does the linear approximation model give for the densities at these two locations? (b) Imagine dividing the earth into concentric, spherical shells. Each shell has radius , thickness \(d r\), volume \(d V=4 \pi r^{2} d r\), and mass \(d m=\rho(r) d V\). By integrating from \(r=0\) to \(r=R\), show that the mass of the earth in this model is \(M=\frac{4}{3} \pi R^{3}\left(A-\frac{3}{4} B R\right)\). (c) Show that the given values of A and B give the correct mass of the earth to within 0.4%. (d) We saw in Section 13.6 that a uniform spherical shell gives no contribution to inside it. Show that \(g(r)=\frac{4}{3} \pi G r\left(A-\frac{3}{4} B R\right)\) inside the earth in this model. (e) Verify that the expression of part (d) gives \(g=0\) at the center of the earth and \(g=9.85 \mathrm{~m} / \mathrm{s}^{2}\) at the surface. (f) Show that in this model does not decrease uniformly with depth but rather has a maximum of \(4 \pi G A^{2} / 9 B=10.01 \mathrm{~m} / \mathrm{s}^{2}\) at \(r=2 A / 3 B=5640\) km. Equation Transcription: Text Transcription: rho(r)=A-Br A=12700 kg/m^3 B=1.50x10^-3 kg/m^4 R=6.37x10^6 kg/m^3 kg/m^3 dr dV=4pi r^2 dr dm=rho(r)dV r=0 r=R M=4 over 3 piR^3(A-3over4 Br) g(r)=4over3 piGr(A-3over4 Br) g=0 g=9.85 m/s^2 4piGA^2/9B=10.01 m/s^2 r=2A/3B=5640

In Example 13.10 (Section 13.6) we saw that inside a planet of uniform density (not a realistic assumption for the earth) the acceleration due to gravity increases uniformly with distance from the center of the planet. That is, \(g(r)=g_{s} r / R\), where \(g_{s}\) is the acceleration due to gravity at the surface, r is the distance from the center of the planet, and R is the radius of the planet. The interior of the planet can be treated approximately as an incompressible fluid of density \(\rho\).(a) Replace the height y in Eq. (12.4) with the radial coordinate r and integrate to find the pressure inside a uniform planet as a function of r. Let the pressure at the surface be zero. (This means ignoring the pressure of the planet’s atmosphere.) (b) Using this model, calculate the pressure at the center of the earth. (Use a value of \(\rho\) equal to the average density of the earth, calculated from the mass and radius given in Appendix F.) (c) Geologists estimate the pressure at the center of the earth to be approximately \(4 \times 10^{11}\) Pa. Does this agree with your calculation for the pressure at \(r=0\)? What might account for any differences? Equation Transcription: Text Transcription: g(r)=g_sr/R G_s Rho rho 4x10^11 r=0

Problem 77P CP 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?