Solution: In 2005 astronomers announced the discovery of a

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.

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?

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.

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?

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

Since the moon is constantly attracted toward the earth by the gravitational interaction, why doesnt it crash into the earth?

A spaceship 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) 3T, (b) T13, (c) T, (d) T>13, or (e) T>3?

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 the planet had three times as much mass, what would the new period (in terms of T) be: (a) 3T, (b) T13, (c) T, (d) T>13, or (e) T>3?

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, doesnt the sun take the moon away from the earth?

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

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 stars gravitational force: positive, negative, or zero? What if the planets orbit is an ellipse, so that the speed is not constant? Explain your answers

Does the escape speed for an object at the earths surface 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?

If a projectile is fired straight up from the earths 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.

Discuss whether this statement is correct: In the absence of air resistance, the trajectory of a projectile thrown near the earths surface is an ellipse, not a parabola.

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.

A communications firm wants to place a satellite in orbit so that it is always directly above the earths 45th parallel (latitude 45 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?

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

What would Keplers third law be for circular orbits if an amendment to Newtons law of gravitation made the gravitational force inversely proportional to r3 ? Would this change affect Keplers other two laws? Explain

In the elliptical orbit of Comet Halley shown in Fig. 13.21a, the suns 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?

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

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.

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 m1 = 1.10 kg, m2 = 25.0 kg, and the rod connecting the m1 pairs is 30.0 cm long. If, in each pair, m1 and m2 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 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.

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 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 2m, taking quantities to the right as positive. Include the regions x 6 0, 0 6 x 6 L, and x 7 L. Be especially careful to show the behavior of the graph on either side of x = 0 and x = L.

At what distance above the surface of the earth is the acceleration due to the earth’s gravity \(0.980\mathrm{\ m}/\mathrm{s}^2\) if the acceleration due to gravity at the surface has magnitude \(0.980\mathrm{\ m}/\mathrm{s}^2\)?

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 1 8 the radius of the earth and 1 1700 the mass of the earth. (a) What is the acceleration due to gravity at the surface of Titania? (b) What is the average density of Titania? (This is less than the density of rock, which is one piece of evidence that Titania is made primarily of ice.)

Rhea, one of Saturn’s moons, has a radius of 764 km and an acceleration due to gravity of \(0.265 \mathrm{\ m} / \mathrm{s}^{2}\) at its surface. Calculate its mass and average density.

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

Volcanoes on Io. Jupiters moon Io has active volcanoes (in fact, it is the most volcanically active body in the solar system) that eject material as high as 500 km (or even higher) above the surface. Io has a mass of 8.93 * 1022 kg and a radius of 1821 km. For this calculation, ignore any variation in gravity over the 500-km range of the debris. How high would this material go on earth if it were ejected with the same speed as on Io?

. 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 \(2.87 \times 10^6 \ \mathrm {km}\) from the earth and traveling at \(1.20 \times 10^4 \ \mathrm{km/h}\) relative to the earth. At this time, what were (a) the spacecraft’s kinetic energy relative to the earth and (b) the potential energy of the earth–spacecraft system?

A planet orbiting a distant star has radius 3.24 * 106 m. The escape speed for an object launched from this planets surface is 7.65 * 103 m>s. What is the acceleration due to gravity at the surface of the planet?

An earth satellite moves in a circular orbit with an orbital speed of 6200 m>s. Find (a) the time of one revolution of the satellite; (b) the radial acceleration of the satellite in its orbit.

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

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 km>s) is the Aura spacecraft moving?

Two satellites are in circular orbits around a planet that has radius 9.00 * 106 m. One satellite has mass 68.0 kg, orbital radius 7.00 * 107 m, and orbital speed 4800 m>s. The second satellite has mass 84.0 kg and orbital radius 3.00 * 107 m. What is the orbital speed of this second satellite?

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 \(1.5 \times 10^{15} \mathrm{\ kg}\). 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?

Planet Vulcan. Suppose that a planet were discovered between the sun and Mercury, with a circular orbit of radius equal to 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 Mercurys orbit. It was even given the name Vulcan, although we now have no evidence that it actually exists. Mercurys precession has been explained by general relativity.)

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 earths orbit around the sun. What are (a) the orbital speed and (b) the orbital period of the planet of Rho1 Cancri?

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.

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

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 km/s) 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 r = 0 to r S .

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

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. E13.34). (a) Calculate the gravitational potential energy of the rodsphere 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 ln11 + x2 given in Appendix B.) (b) Use Fx = -dU>dx to find the magnitude and direction of the gravitational force exerted on the sphere by the rod (see Section 7.4). Show that your answer reduces to the expected result when x is much larger than L.

Consider the ringshaped body of Fig. E13.35. A particle with mass m is placed a distance x from the center of the ring, along the line through the center of the ring and perpendicular to its plane. (a) Calculate the gravitational potential energy U of this system. Take the potential energy to be zero when the two objects are far apart. (b) Show that your answer to part (a) reduces to the expected result when x is much larger than the radius a of the ring. (c) Use Fx = -dU>dx to find the magnitude and direction of the force on the particle (see Section 7.4). (d) Show that your answer to part (c) reduces to the expected result when x is much larger than a. (e) What are the values of U and Fx when x = 0? Explain why these results make sense.

A Visit to Santa. You decide to visit Santa Claus at the north pole to put in a good word about your splendid behavior throughout the year. While there, you notice that the elf Sneezy, when hanging from a rope, produces a tension of 395.0 N in the rope. If Sneezy hangs from a similar rope while delivering presents at the 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 \(11.2 \mathrm{~m} / \mathrm{s}^{2}\). Neptune has mass \(1.02 \times 10^{26}\mathrm{\ kg}\) and radius \(2.46 \times 10^4\mathrm{\ km}\) and rotates once around its axis in about 16 h. (a) What is the gravitational force on a 3.00-kg object at the north pole of Neptune? (b) What is the apparent weight of this same object at Neptune’s equator? (Note that Neptune’s “surface” is gaseous, not solid, so it is impossible to stand on it.)

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 * 10-15 m, what would be the mass of a mini black hole?

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?

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

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?

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

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 45 below the -x@axis, what will the particles speed be when it reaches the origin?

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 4.80 * 1022 kg and its diameter is 3120 km.

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

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 \(1.22 \times 10^{6} \mathrm{\ km}\) from the center of Saturn and has a mass of \(1.35 \times 10^{23} \mathrm{\ kg}\) and a diameter of 5150 km. At what distance from Titan should the gravitational pull of Titan just balance the gravitational pull of Saturn?

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?

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 * 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

Geosynchronous Satellites. 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 81.3 N latitude.

Submarines on Europa. 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 4.80 * 1022 kg, its diameter is 3120 km, and it has no appreciable atmosphere. Assume that the layer of ice at the surface is not thick enough to exert substantial force on the water. If the windows of the submarine you are designing each have an area of 625 cm2 and can stand a maximum inward force of 8750 N per window, what is the greatest depth to which this submarine can safely dive?

What is the escape speed from a 300-km-diameter asteroid with a density of 2500 kg>m3 ?

A landing craft with mass 12,500 kg is in a circular orbit 5.75 * 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 * 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 planets surface?

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?

(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?

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 * 107 m, what is the mass of the planet?

Your starship, the Aimless Wanderer, lands on the mysterious planet Mongo. As chief scientist-engineer, you make the following measurements: A 2.50-kg stone thrown upward from the ground at 12.0 m>s returns to the ground in 4.80 s; the circumference of Mongo at the equator is 2.00 * 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?

You are exploring a distant planet. When your spaceship is in a circular orbit at a distance of 630 km above the planets surface, the ships orbital speed is 4900 m>s. By observing the planet, you determine its radius to be 4.48 * 106 m. You then land on the surface and, at a place where the ground is level, launch a small projectile with initial speed 12.6 m>s at an angle of 30.8 above the horizontal. If resistance due to the planets atmosphere is negligible, what is the horizontal range of the projectile?

The 0.100-kg sphere in Fig. P13.58 is released from rest at the position shown in the sketch, with its center 0.400 m from the center of the 5.00-kg mass. Assume that the only forces on the 0.100-kg sphere are the gravitational forces exerted by the other two spheres and that the 5.00-kg and 10.0-kg spheres are held in place at their initial positions. What is the speed of the 0.100-kg sphere when it has moved 0.400 m to the right from its initial position?

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 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?

Falling Hammer. 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 RE of the earth. Ignoring air resistance, derive an expression for the speed y of the hammer when it reaches the earths surface. Your expression should involve h, RE, and mE (the earths mass).

(a) Calculate how much work is required to launch a spacecraft of mass m from the surface of the earth (mass mE, radius RE) and place it in a circular low earth orbitthat is, an orbit whose altitude above the earths surface is much less than RE. (As an example, the International Space Station is in low earth orbit at an altitude of about 400 km, much less than RE = 6370 km.) 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. 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.

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 StarDifferent Masses. Two stars, with masses M1 and M2, are in circular orbits around their center of mass. The star with mass M1 has an orbit of radius R1; the star with mass M2 has an orbit of radius R2. (a) Show that the ratio of the orbital radii of the two stars equals the reciprocal of the ratio of their massesthat is, R1>R2 = M2>M1. (b) Explain why the two stars have the same orbital period, and show that the period T is given by T = 2p1R1 + R223>2>1G1M1 + M22. (c) The two stars in a certain binary star system move in circular orbits. The first star, Alpha, has an orbital speed of 36.0 km>s. The second star, Beta, has an orbital speed of 12.0 km>s. The orbital period is 137 d. What are the masses of each of the two stars? (d) One of the best candidates for a black hole is found in the binary system called A0620-0090. The two objects in the binary system are an orange star, V616 Monocerotis, and a compact object believed to be a black hole (see Fig. 13.28). The orbital period of A0620-0090 is 7.75 hours, the mass of V616 Monocerotis is estimated to be 0.67 times the mass of the sun, and the mass of the black hole is estimated to be 3.8 times the mass of the sun. Assuming that the orbits are circular, find the radius of each 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 2.0 * 104 m>s when at a distance of 2.5 * 1011 m from the center of the sun, what is its speed when at a distance of 5.0 * 1010 m?

The planet Uranus has a radius of 25,360 km and a surface acceleration due to gravity of 9.0 m>s 2 at its poles. Its moon Miranda (discovered by Kuiper in 1948) is in a circular orbit about Uranus at an altitude of 104,000 km above the planets surface. Miranda has a mass of 6.6 * 1019 kg and a radius of 236 km. (a) Calculate the mass of Uranus from the given data. (b) Calculate the magnitude of Mirandas acceleration due to its orbital motion about Uranus. (c) Calculate the acceleration due to Mirandas gravity at the surface of Miranda. (d) Do the answers to parts (b) and (c) mean that an object released 1 m above Mirandas surface on the side toward Uranus will fall up relative to Miranda? Explain

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 earths surface; at the high point, or apogee, it is 4000 km above the earths surface. (a) What is the period of the spacecrafts orbit? (b) Using conservation of angular momentum, find the ratio of the spacecrafts 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 spacecrafts 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?

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

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?

A satellite with mass 848 kg is in a circular orbit with an orbital speed of 9640 m>s around the earth. What is the new orbital speed after friction from the earths upper atmosphere has done -7.50 * 109 J of work on the satellite? Does the speed increase or decrease?

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 * 103 kg>m3 at the center and 2.0 * 103 kg>m3 at the surface. What is the acceleration due to gravity at the surface of this planet?

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.

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

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

A shaft is drilled from the surface to the center of the earth (see Fig. 13.25). As in Example 13.10 (Section 13.6), make the unrealistic assumption that the density of the earth is uniform. With this approximation, the gravitational force on an object with mass m, that is inside the earth at a distance r from the center, has magnitude \(F_{\mathrm{g}}=G m_{\mathrm{E}} m r / R_{\mathrm{E}}^3\) (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?

For each of the eight planets Mercury to Neptune, the semi-major axis a of their orbit and their orbital period T are as follows: Planet Semi-major Axis 1106 km2 Orbital Period (days) Mercury 57.9 88.0 Venus 108.2 224.7 Earth 149.6 365.2 Mars 227.9 687.0 Jupiter 778.3 4331 Saturn 1426.7 10,747 Uranus 2870.7 30,589 Neptune 4498.4 59,800 (a) Explain why these values, when plotted as T2 versus a3 , fall close to a straight line. Which of Keplers laws is being tested? However, the values of T2 and a3 cover such a wide range that this plot is not a very practical way to graph the data. (Try it.) Instead, plot log1T2 (with T in seconds) versus log1a2 (with a in meters). Explain why the data should also fall close to a straight line in such a plot. (b) According to Keplers laws, what should be the slope of your log1T2 versus log1a2 graph in part (a)? Does your graph have this slope? (c) Using G = 6.674 * 10-11 N # m2>kg2 , calculate the mass of the sun from the y-intercept of your graph. How does your calculated value compare with the value given in Appendix F? (d) The only asteroid visible to the naked eye (and then only under ideal viewing conditions) is Vesta, which has an orbital period of 1325.4 days. What is the length of the semi-major axis of Vestas orbit? Where does this place Vestas orbit relative to the orbits of the eight major planets? Some scientists argue that Vesta should be called a minor planet rather than an asteroid.

For a spherical planet with mass M, volume V, and radius R, derive an expression for the acceleration due to gravity at the planets surface, g, in terms of the average density of the planet, r = M>V, and the planets diameter, D = 2R. The table gives the values of D and g for the eight major planets: Planet D 1km2 g 1m,s 2 2 Mercury 4879 3.7 Venus 12,104 8.9 Earth 12,756 9.8 Mars 6792 3.7 Jupiter 142,984 23.1 Saturn 120,536 9.0 Uranus 51,118 8.7 Neptune 49,528 11.0 (a) Treat the planets as spheres. Your equation for g as a function of r and D shows that if the average density of the planets is constant, a graph of g versus D will be well represented by a straight line. Graph g as a function of D for the eight major planets. What does the graph tell you about the variation in average density? (b) Calculate the average density for each major planet. List the planets in order of decreasing density, and give the calculated average density of each. (c) The earth is not a uniform sphere and has greater density near its center. It is reasonable to assume this might be true for the other planets. Discuss the effect this nonuniformity has on your analysis. (d) If Saturn had the same average density as the earth, what would be the value of g at Saturns surface?

For a planet in our solar system, assume that the axis of orbit is at the sun and is circular. Then the angular momentum about that axis due to the planets orbital motion is L = MvR. (a) Derive an expression for L in terms of the planets mass M, orbital radius R, and period T of the orbit. (b) Using Appendix F, calculate the magnitude of the orbital angular momentum for each of the eight major planets. (Assume a circular orbit.) Add these values to obtain the total angular momentum of the major planets due to their orbital motion. (All the major planets orbit in the same direction in close to the same plane, so adding the magnitudes to get the total is a reasonable approximation.) (c) The rotational period of the sun is 24.6 days. Using Appendix F, calculate the angular momentum the sun has due to the rotation about its axis. (Assume that the sun is a uniform sphere.) (d) How does the rotational angular momentum of the sun compare with the total orbital angular momentum of the planets? How does the mass of the sun compare with the total mass of the planets? The fact that the sun has most of the mass of the solar system but only a small fraction of its total angular momentum must be accounted for in models of how the solar system formed. (e) The sun has a density that decreases with distance from its center. Does this mean that your calculation in part (c) overestimates or underestimates the rotational angular momentum of the sun? Or doesnt the nonuniform density have any effect?

Interplanetary Navigation. The most efficient way to send a spacecraft from the earth to another planet is to use a Hohmann transfer orbit (Fig. P13.79). If the orbits of the departure and destination planets are circular, the Hohmann transfer orbit is an elliptical orbit whose perihelion and aphelion are tangent to the orbits of the two planets. The rockets are fired briefly at the departure planet to put the spacecraft into the transfer orbit; the spacecraft then coasts until it reaches the destination planet. The rockets are then fired again to put the spacecraft into the same orbit about the sun as the destination planet. (a) For a flight from earth to Mars, in what direction must the rockets be fired at the earth and at Mars: in the direction of motion or opposite the direction of motion? What about for a flight from Mars to the earth? (b) How long does a one-way trip from the earth to Mars take, between the firings of the rockets? (c) To reach Mars from the earth, the launch must be timed so that Mars will be at the right spot when the spacecraft reaches Marss orbit around the sun. At launch, what must the angle between a sunMars line and a sunearth line be? Use Appendix F.

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 M is distributed uniformly over a disk of radius a. Find the gravitational force (magnitude and direction) between this disk-shaped mass and a particle with mass m located a distance x above the center of the disk (Fig. P13.81). Does your result reduce to the correct expression as x becomes very large? (Hint: Divide the disk into infinitesimally thin concentric rings, use the expression derived in Exercise 13.35 for the gravitational force due to each ring, and integrate to find the total force.)

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

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

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