For each of the examples of two-body gravitational action below, find the missing quantity for the data given. (G = 6.7 x (Nm 2 /kg 2 )

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1 1 For each of the examples of two-body gravitational action below, find the missing quantity for the data given. (G = 6.7 x (Nm 2 /kg 2 ) 7 Using Newton's Universal Law of Gravity (a) What is the gravitational force between two spherical 8.00 kg masses that are 5.0 m apart? (b) What is the gravitational force between them when they are 5.0 x 10 1 m apart? (a) 1.7 x N (b) 1.7 x N (a) 6.7 x 10 6 N (b) 1.94 x kg (c) 1.5 kg (d) 1.06 x m 8 The gravitational force between two electrons 1.00 m apart is 5.42 x l0-71 N. Find the mass of an electron x kg 2 Two satellites are in circular orbits about Earth, one 150 km above the surface, the other 160 km. (a) Which satellite has the larger orbital period? (b) Which one has the larger velocity? 9 Compute the gravitational force the sun exerts on Jupiter. (a) The one at 160 km has the larger period. (b) The one at 150 km has the larger velocity x N Two ships each of 10 8 kg mass are moored near each other. Assume the effective distance between them as far as gravitational attraction is concerned is 100 m. What gravitational force do they exert on one another? 3 The mass of an electron is 9.1 x 10-3l kg. The mass of a proton is 1.7 x kg. They are about 1.0 x m apart in a hydrogen atom. What gravitational force exists between the proton and the electron of a hydrogen atom? 671 N 1.0 x N 11 From the data on the planetary data sheet, calculate the gravitational force on the earth due to the sun. It is this force which holds the earth in its orbit. 4 Two 1.00-kg masses have their centers 1.00 m apart. What is the force of attraction between them? 3.6 x N x N Two large spheres are suspended close to each other. Their centers are 4.0 m apart. One sphere weighs 9.8 x 10 2 N. The other sphere has a weight of 1.96 x 10 2 N. What is the gravitational force between them? 12 A satellite is placed in a circular orbit with a radius of 1.0 x 10 7 m a period of 9.9 x 10 3 s. Calculate the mass of Earth. Hint: Gravity supplies the needed centripetal force for such a satellite. Scientists have actually measured the mass of Earth this way. 8.3 x 10-9 N 6.0 x kg 6 If the centers of Earth and the moon are 3.9 x 10 8 m apart, the gravitational force between them is about 1.9 x N. What is the approximate mass of the moon? (Mass of Earth = 5.9 x ) 13 Find the speed and period of a satellite that would orbit Mars 175 km above its surface. 7.2 x kg V = 3.47 x 10 3 m/s T = 6.45 x 10 3 s or 1.79 h

2 14 The following problem examins some characteristics of the planet Mercury. (a) Find the speed of a satellite in orbit 265 km above Mercury s surface. (b) Find the the period of the satellite orbit 265 km above Mercury s surface. 19 A 10,000 kg spaceship is drifting on a long mission toward the outer edge of the solar system. It has put out a small experimental satellite which revolves around it at a distance of 120 meters under their mutual gravitational attraction. (a) What is the period of revolution of the satellite? (b) What is the speed of the satellite? (a) 2.96 x 10 3 m/s (b) 85.9 min (a) 1.01 x 10 7 s (b) m/s or 7.5 x 10-5 m/s 15 Using Astronomical data. (a) Find the velocity with which Mercury around the sun (b) Also, find the velocity of Saturn. (c) Now, comment on whether or not it mae sense that Mercury is named after a speedy messenger of the gods, while Saturn is named after the father or Jupiter. (a) 4.79 x 10 4 m/s (b) 9.65 x 10 3 m/s (c) about as 1/5 as fast as Mercury 20 Assume the earth is perfectly round and has a radius of 6400 km. (a) What is the weight of a 100 kg man at the North Pole? (b) What is the centripetal force of a 100 kg man at the Equator? (c) How much less does a man with a mass of 100 kg apparently weigh at the equator than at the poles because of the rotation of the earth? (d) How fast would the earth have to spin in order that he would exert no force on a scale at the equator? (e) How many times larger is the speed of rotation in d than the actual speed? 16 Using Astronomical Data (a) Calculate the velocity that a satellite shot from Newton's cannon must have in order to orbit Earth, 150 km above its surface. (b) How long would it take for the satellite to return to the cannon in seconds and minutes? (a) 7.8 x 10 3 m/s (b) 84 min 18 sec (a) 980 N (b) 3.37 N (c) N (d) S or 1.4 hours, or m/s (e) 17 times greater 17 A geosynchronous satellite appears to remain over one spot on Earth. A geosynchronous satellite has an orbital radius of 4.23 x 10 7 m. (a) Calculate its speed in orbit. (b) Calculate its period. 21 If you weigh 637 N on Earth's surface, how much would you weigh on the planet Mars? (Mars has a mass of 6.37 x kg and a radius of 3.43 x 10 5 m.) 235 N 18 (a) 3.07 x 10 3 m/s (b) 24.0 h On July 19, 1969, Apollo li's orbit around the moon was adjusted to an average orbit of 111 km. The radius of the moon is 1785 km and the mass of the moon is 7.3 x kg. (a) How many minutes did it take to orbit once? (b) At what velocity did it orbit the moon? 22 What would be the value of g, acceleration of gravity if: (a) The Earth's mass was double its actual value, but its radius remained the same? (b) The radius was doubled, but the mass remained the same? (c) The both the mass and radius were doubled (a) 19.6 m/s 2 (b) 2.45 m/s 2 (c) 4.9 m/s 2 (a) 1.2 x 10 2 min (b) 1.6 x 10 3 m/s 23 What would be the strength of Earth's gravitational field at a point where an 80.0 kg astronaut would experience a 25% reduction in weight? 7.35 m/s 2

3 24 On the surface of the moon, a 91.0 kg physics teacher weighs only N. What is the value of the moon's gravitational field at its surface? 31 Find the weight of a 100 kg man on Jupiter N 1.60 m/s 2 25 Two satellites of equal mass are put into orbit 30 m apart. The gravitational force between them is 2.0 x 10-7 N. (a) What is the mass of each satellite? (b) What is the initial acceleration given to each satellite by the gravitational force? 32 Calculate the theoretical value for the acceleration due to gravity at a point 1.00 x 10 7 m from the center of the earth m/s 2 (a) 1.6 x 10 3 kg (b) 1.3 x m/s 2 33 At what height above the earth's surface will a rocket experience just a quarter the pull from the earth that it feels at the earth's surface? 26 The asteroid Ceres has a mass 7 x kg and a radius of 500 km. (a) What is g on the surface? (b) How much would a 85-kg astronaut weigh on Ceres? 1.28 x 10 7 m 2 Earth radii from the center of the Earth 1 Earth radius above the surface of the earth 27 (a) 0.2 m/s 2 (b) 2 x 10 1 N The radius of Earth is about 6.40 x 10 3 km. A 7.20 x 10 3 N spacecraft is traveling away from Earth. (a) What is the weight of the spacecraft 6.40 x 10 3 km from the Earth's surface? (b) What is the weight of the spacecraft 1.28 x 10 4 km from the Earth's surface? (a) 1.80 x 10 3 N (b) 800 N 34 The mass and radius of the moon are 7.3 x kg and 1.74 x 10 5 m respectively. (a) Calculate the value of g on the moon (b) Calculate the weight of a 50 kg boy on the earth (c) Calculate the weight of a 50 kg boy on the moon (d) Calculate the time it takes of an object to fall 4.9 m on the moon. (e) Calculate the time it takes of an object to fall 4.9 m both on the earth (f) Many physics students would like to change the magnitude of g from 9.8 m/s 2 to 10 m/s 2. To do this how should the mass and the radius of the earth be changed? (g) How could the same change in g be accomplished, theoretically, without any change in the mass or radius of the earth? 28 How high does a rocket have to go above Earth's surface until its weight is half what it would be on Earth? 9.05 x 10 5 m (a) 1.6 m/s 2 (b) F earth = 490 N (c) F moon = 80 N (d) 2.4 sec Moon (e) 1 second Earth (f) Only if G is not constant g. 29 At what height above the earth's surface will a rocket have half the force of gravitation on it that it would have at sea level? Express your answer in earth radii. 35 Use the Universal Law of Gravity to determine the following: (a) What is the weight of a 1.0 kg mass one earth radius from the surface of the earth (Radius of the earth = 6.4 x 10 5 m or 4000 miles). (b) At what distance from the surface is the weight of any mass reduced one-half? 30 The instrument-carrying payload of a rocket weighs 1058 newtons on the earth. What does it weigh x 10 4 km above the earth? (a) 2.5 N (b) 1.4 Earth radii from the center of the earth.4 radii from the surface of the earth 42 N

4 36 The figure shows an arrangement of three particles, particle 1 of mass m 1 = 6.0 kg and particles 2 and 3 of mass m 2 = m 3 = 4.0 kg, and distance a = 2.0 cm. What is the net gravitational force F 1 on particle 1 due to the other particles? 39 In the figure here, what is the direction of the net gravitational force on the particle of mass m 1 due to the other particles, each of mass m and arranged symmetrically relative to the y axis? negative y direction 37 The figure shows four arrangements of three particles of equal masses. (a) Rank the arrangements according to the magnitude of the net gravitational force on the particle labeled m, greatest first. (b) In arrangement 2, is the direction of the net force closer to the line of length d or to the line of length D? 40 An astronaut whose height h is 1.70 m floats "feet down" in an orbiting space shuttle at distance r = 6.77 x 10 6 m away from the center of Earth. (a) What is the difference between the gravitational acceleration at her feet and at her head? (b) If the astronaut is now "feet down" at the same orbital radius r = 6.77 x 10 6 m about a black hole of mass M h = 1.99 x kg (10 times our Sun's mass), what is the difference between the gravitational acceleration at her feet and at her head? The black hole has a mathematical surface (event horizon) of radius R h = 2.95 x 10 4 m. Nothing, not even light, can escape from that surface or anywhere inside it. Note that the astronaut is well outside the surface (at r = 229R h ). 38 (a) 1, tie of 2 and 4, then 3; (b) line d The figure shows an arrangement of five particles, with masses m 1 = 8.0 kg, m 2 = m 3 = m 4 = m 5 = 2.0 kg, and with a = 2.0 cm and θ = What is the net gravitational force F l,net on particle 1 due to the other particles? 41 You move a ball of mass m away from a sphere of mass M. (a) Does the gravitational potential energy of the ball-sphere system increase or decrease? (b) Is positive or negative work done by the gravitational force between the ball and the sphere? (a) increase; (b) negative 42 An asteroid, headed directly toward Earth, has a speed of 12 km/s relative to the planet when the asteroid is 10 Earth radii from Earth's center. Neglecting the effects of Earth's atmosphere on the asteroid, find the asteroid's speed v f when it reaches Earth's surface. 43 Satellite 1 is in a certain circular orbit around a planet, while satellite 2 is in a larger circular orbit. (a) Which satellite has the longer period? (b) Which satellite has the greater speed? (a) 2; (b) 1

5 44 Comet Halley orbits the Sun with a period of 76 years and, in 1986, had a distance of closest approach to the Sun, its perihelion distance R P, of 8.9 x m. This is between the orbits of Mercury and Venus. (a) What is the comet's farthest distance from the Sun, which is called its aphelion distance R A? (b) What is the eccentricity e of the orbit of comet Halley? 47 A playful astronaut releases a bowling ball, of mass m = 7.20 kg, into circular orbit about Earth at an altitude h of 350 km. (a) What is the mechanical energy E of the ball in its orbit? (b) What is the mechanical energy E 0 of the ball on the launchpad at Cape Canaveral? From there to the orbit, what is the change E in the ball's mechanical energy? 45 The figure shows the observed orbit of fhe star S2 as the star moves around a mysterious and unobserved object called Sagittarius A* (pronounced "A star"), which is at the center of the Milky Way galaxy. S2 orbits Sagittarius A* with a period of T = 15.2 y and with a semimajor axis of a = 5.50 light-days (= 1.42 x M). What is the mass M of Sagittarius A*? What is Sagittarius A*? 48 In the figure, two particles, of masses m and 2m, are fixed in place on an axis. (a) Where on the axis can a third particle of mass 3m be placed (other than at infinity) so that the net gravitational force on it from the first two particles is zero: to the left of the first two particles, to their right, between them but closer to the more massive particle, or between them but closer to the less massive particle? (b) Does the answer change if the third particle has, instead, a mass of 16m? (c) Is there a point off the axis (other than infinity) at which the net force on the third particle would be zero? (a) between, closer to less massive particle; (b) no; (c) no 46 In the figure here, a space shuttle is initially in a circular orbit of radius r about Earth. At point P, the pilot briefly fires a forward-pointing thruster to decrease the shuttle's kinetic energy K and mechanical energy E. (a) Which of the dashed elliptical orbits shown in the figure will the shuttle then take? (b) Is the orbital period T of the shuttle (the time to return to P) then greater than, less than, or the same as in the circular orbit? 49 The figure shows three situations involving a point particle P with mass m and a spherical shell with a uniformly distributed mass M. The radii of the shells are given. Rank the situations according to the magnitude of the gravitational force on particle P due to the shell, greatest first. b and c tie, then a (zero) (a) path 1 [decreased E (more negative) gives decreased a]; (b) less than (decreased a gives decreased T)

6 50 In the figure, a central particle is surrounded by two circular rings of particles, at radii r and R, with R > r. All the particles have mass m. What are the magnitude and direction of the net gravitational force on the central particle due to the particles in the rings? 53 The figure shows three particles initially fixed in place, with B and C identical and positioned symmetrically about the y axis, at distance d from A. (a) In what direction is the net gravitational force F net on A? (b) If we move C directly away from the origin, does F net change in direction? If so, how and what is the limit of the change? Gm 2 /r 2, upward (a) positive y; (b) yes, rotates counterclock-wise until it points toward particle B 51 In the figure, a central particle of mass M is surrounded by a square array of other particles, separated by either distance d or distance d/2 along the perimeter of the square. What are the magnitude and direction of the net gravitational force on the central pariticle due to the other particles? 54 In the figure, three particles are fixed in place. The mass of B is greater than the mass of C. Can a fourth particle (particle D) be placed somewhere so that the net gravitational force on particle A from particles B, C, and D is zero? If so, in which quadrant should it be placed and which axis should it be near? 3GM 2 /d 2, leftward 52 The figure shows three arrangements of the same identical particles, with three of them placed on a circle of radius 0.20 m and the fourth one placed at the center of the circle. (a) Rank the arrangements according to the magnitude of the net gravitational force on the central particle due to the other three particles, greatest first. (b) Rank them according to the gravitational potential energy of the four-particle-system, least negative first. yes, in the second quadrant, closer to the y axis than to the x axis, at a distance that depends on its mass (a) c, b, a; (b) a, b, c

7 55 The figure gives the gravitational acceleration a g for four planets as a function of the radial distance r from the center of the planet, starting at the surface of the planet (at radius R 1, R 2, R 3, or R 4 ). Plots 1 and 2 coincide for r R 2 ; plots 3 and 4 coincide for r R4. (a) Rank the four planets according to mass greatest first. (b) Rank the four planets according to mass per volume, greatest first. 58 The figure shows six paths by which a rocket orbiting a moon might move from point a to point b. (a) Rank the paths according to the corresponding change in the gravitational potential energy of the rocket-moon system. (b) Rank the paths according to the net work done on the rocket by the gravitational force from the moon, greatest first. (a) all tie; (b) all tie (a) 1 and 2 tie, then 3 and 4 tie; (b) 1, 2, 3, 4 56 The figure shows three uniform spherical planets that are identical in size and mass. The periods of rotation" T for the planets are given, and six lettered points are indicated- three points are on the equators of the planets and three points are on the north poles. Rank the points according to the value of the free-fall acceleration g at them, greatest first. 59 In the figure, a particle of mass m (not shown) is to be moved from an infinite distance to one of the three possible locations a, b, and c. Two other particles, of masses m and 2m, are fixed in place. Rank the three possible locations according to the work, done by the net gravitational force on the moving particle due to the fixed particles, greatest first. b, a, c 60 What must the separation be between a 5.2 kg particle and a 2.4 kg particle for their gravitational attraction to have a magnitude of 2.3 x N? a tie of b, d, and f, then e, c, a 19 m 57 Rank the four systems of equal-mass particles according to the absolute value of the gravitational potential energy of the system, greatest first. 61 The Sun and Earth each exert a gravitational force on the Moon. What is the ratio F sun /F earth of these two forces? (The average Sun-Moon distance is equal to the Sun-Earth distance.) , tie of 2 and 4, then 3

8 62 In the figure, three 5.00 kg spheres are located at distances d 1 = m and d 2 = m. What is the (a) What is the magnitude of the net gravitational force on sphere B due to spheres A and C? (b) What is the direction (relative to the positive direction of the x axis) of the net gravitational force on sphere B due to spheres A and C? 65 In the figure, two point particles are fixed on an x axis separated by distance d. Particle A has mass m A and particle B has mass 3.00 m A. A third particle C, of mass 75.0 m A, is to be placed on the x axis and near particles A and B. In terms of distance d, at what x coordinate should C be placed so that the net gravitational force on particle A from particles B and C is zero? d 63 (a) 2.13 x 10 8 N; (b) How far from Earth must a space probe be along a line toward the Sun so that the Sun's gravitational pull on the probe balances Earth's pull? 2.60 x 10 5 km 66 In figure a, particle A is fixed in place at x = m on the x axis and particle B, with a mass of 1.0 kg, is Fig Problem 8. fixed in place at the origin. Particle C (not shown) can be moved along the x axis, between particle B and x = infinity. Figure b shows the x component F x,net of the net gravitational force on particle B due to particles A and C, as a function of position x of particle C. The plot actually extends to the right, approaching an asymptote of x N as x -- >- infinity. (a) What is the mass of particle A? (b) What is the mass of particle C? 64 In the figure, a square of edge length 20.0 cm is formed by four spheres of masses m 1 = 5.00 g, m 2 = 3.00 g, m 3 = 1.00 g, and m 4 = 5.00 g. In unit- vector notation, what is the net gravitational force from them on a central sphere with mass m 5 = 2.50 g? (a) 0.25 kg; (b) 1.0kg (1.18 x N) i +(1.18 x N) j 67 As seen in the figure, two spheres of mass m and a third sphere of mass M form an equilateral triangle, and a fourth sphere of mass m 4 is at the center of the triangle. The net gravitational force on that central sphere from the three other spheres is zero. (a) What is M in terms of m? (b) If we double the value of m 4, what then is the magnitude of the net gravitational force on the central sphere? (a) M = m; (b) 0

9 68 Three point particles are fixed in position in an xy plane. Two of them, particle A of mass 6.00 g and particle B of mass 12.0 g, are shown in the figure, with a separation of d AB = m at angle θ = Particle C, with mass 8.00 g, is not shown. The net gravitational force acting on particle A due to particles B and C is 2.77 x N at an angle of from the posilive direction of the x axis. (a) What is the x coordinate of particle C? (b) What is the y coordinate of particle C? 70 In the figure, three point particles are fixed in place in an xy plane. Particle A has mass m A, particle B has mass 2.00m A, and particle C has mass 3.00m A. A fourth particle D, with mass 4.00m A, is to be placed near the other three particles. (a) In terms of distance d, at what x coordinate should particle D be placed so that the net gravitational force on particle A from particle's B, C, and D is zero? (b) In terms of distance d, at what y coordinate should particle D be placed so that the net gravitational force on particle A from particle's B, C, and D is zero? (a) m; (b) m (a) 0.716d; (b) -1.07d 69 The figure shows a spherical hollow inside a lead sphere of radius R = 4.00 cm; the surface of the hollow passes through the center of the sphere and "touches" the right side of the sphere. The mass of the sphere before hollowing was M = 2.95 kg. With what gravitational force does the hollowed-out lead sphere attract a small sphere of mass m = kg that lies at a distance d = 9.00 cm from the center of the lead sphere, on the straight line connecting the centers of the spheres and of the hollow? 71 Three point particles are fixed in place in an xyz coordinate system. Particle A, at the origin, has mass m A. Particle B, at xyz coordinates (2.00d, 1.00d, 2.00d), has mass 2.00m A, and particle C, at coordinates (-1.00d, 2.00d, -3.00d), has mass 3.00m A. A fourth particle D, with mass 4.00m A, is to be placed near the other particles. (a) In terms of distance d, at what x coordinate should D be placed so that the net gravitational force on A from B, C, and D is zero? (b) In terms of distance d, at what y coordinate should D be placed so that the net gravitational force on A from B, C, and D is zero? (c) In terms of distance d, at what z coordinate should D be placed so that the net gravitational force on A from B, C, and D is zero? 8.31 x 10-9 N (a) -1.88d; (b) -3.90d; (c) 0.489d 72 In 1956, Frank Lloyd Wright proposed the construction of a mile-high building in Chicago. Suppose the building had been constructed. Ignoring Earth's rotation, find the change in your weight if you were to ride an elevator from the street level, where you weigh 600 N, to the top of the building N 73 At what altitude above Earth's surface would the gravitational acceleration be 4.9 m/s 2? 2.6 x 10 6 m

10 74 An object weighs 100 N on the Earth s Surface. (a) What will an object weigh on the Moon's surface? (b) How many Earth radii must this same object be from the center of Earth if it is to weigh the same as it does on the Moon? (a) 17 N; (b) Two concentric spherical shells with uniformly distributed masses M 1 and M 2 are situated as shown in the figure. (a) Find the magnitude of the net gravitational force on a particle of mass m, due to the shells, when the particle is located at radial distance a. (b) Find the magnitude of the net gravitational force on a particle of mass m, due to the shells, when the particle is located at radial distance b. (c) Find the magnitude of the net gravitational force on a particle of mass m, due to the shells, when the particle is located at radial distance c. 75 One model for a certain planet has a core of radius R and mass M surrounded by an outer shell of inner radius R, outer radius 2R, and mass 4M. M is 4.1 x kg and R is 6.0 x 10 6 m. (a) What is the gravitational acceleration of a particle at points R from the center of the planet? (b) What is the gravitational acceleration of a particle at points 3R from the center of the planet? (a) 7.6m/s 2 (b) 4.2m/s 2 76 The radius R h and mass M h of a black hole are related by R h = 2GM h /c 2, where c is the speed of light. Assume that the gravitational acceleration a, of an object at a distance r 0 = 1.001R from the center of a black hole is given by a g = GM/r 2 (it is, for large black holes). (a) In terms of M h, find a g at r 0. (b) Does a g at r 0 increase or decrease as M h increases? (c) What is a g at r 0 for a very large black hole whose mass is 1.55 x times the solar mass of 1.99 x kg? (d) An astronaut whose height h is 1.70 m floats "feet down" in an orbiting space shuttle at distance r, with her feet toward this black hole, what is the difference in gravitational acceleration between her head and her feet? (e) Is the tendency to stretch the astronaut severe? 79 (a) G(M 1 +M 2 )m/a 2 ; (b) GM 1 m/b 2 (c) 0 A solid uniform sphere has a mass of 1.0 x 10 4 kg and a radius of 1.0 m. (a) What is the magnitude of the gravitational force due to the sphere on a particle of mass m located at a distance of 1.5 m from the center of the sphere? (b) What is the magnitude of the gravitational force due to the sphere on a particle of mass m located at a distance of 0.50 m from the center of the sphere? (c) Write a general expression for the magnitude of the gravitational force on the particle at a distance r 1.0 m from the center of the sphere. (a) (3.02 x kg.m/s 2 )/Mh; (b) decrease; (c) 9.82 m/s 2 (d) 7.30 x m/s 2 (e) no (a) (3.0 x 10-7 N/kg)m; (b) (3.3 x 10-7 N/kg)m; (c) (6.7 x 10-7 N/kg m)mr 77 Certain neutron stars (extremely dense stars) are believed to be rotating at about 1 rev/s. If such a star has a radius of 20 km, what must be its minimum mass so that material on its surface remains in place during the rapid rotation? 5 x kg

11 80 Assume a planet is a uniform sphere of radius R that (somehow) has a narrow radial tunnel through itl center (see the figure). Also assume we can position an apple anywhere along the tunnel or outside the sphere. Let F R be the magnitude of the gravitational force on the apple when it is located at the planet's surface. (a) How far from the surface is there a point where the magnitude of the gravitational force on the apple is 1/2 F R if we move the apple away from the planet? (b) How far from the surface is there a point where the magnitude of the gravitational force on the apple is 1/2 F R if we move the apple into the tunnel? 82 A 5.2 kg particle and a 2.4 kg particle have a magnitude of 2.3 x N. (a) What is the gravitational potential energy of the two particle system? (b) If you triple the separation between the particles, how much work is done by the gravitational force between the particles and (c) If you triple the separation between the particles, how much work is done by you? (a) -4.4 x J; (b) -2.9 x J; (c) 2.9 x J 83 The mean diameters of Mars and Earth are 6.9 x 10 3 km and 1.3 x 10 4 km, respectively. The mass of Mars is 0.11 times Earth's mass. (a) What is the ratio of the mean density (mass per unit volume) of Mars to that of Earth? (b) What is the value of the gravitational acceleration on Mars? (c) What is the escape speed on Mars? (a) 0.414R; (b) 0.500R (a) 0.74; (b) 3.8 m/s 2 (c) 5.0 km/s 81 The figure shows, not to scale, across section through the interior of Earth. Rather than being uniform throughout, Earth is divided into three zones: an outer crust, a mantle, and an inner core. The dimensions of these zones and the masses contained within them are shown on the figure. Earth has a total mass of 5.98 x kg and a radius of 6370 km. Ignore rotation and assume that Earth is spherical. (a) Calculate a g at the surface. (b) Suppose that a bore hole (the Mohole) is driven to the crust-mantle interface at a depth of 25.0 km; what would be the value of a g at the bottom of the hole? (c) Suppose that Earth were a uniform sphere with the same total mass and size. What would be the value of a g at a depth of 25.0 km? (Precise measurements of a g are sensitive probes of the interior structure of Earth, although results can be clouded by local variations in mass distribution.) A mass M is split into two parts, m and M - m, which are then separated by a certain distance. What ratio m/m gives the least gravitational potential energy for the system? 1/2 the following question. (a) What multiple of the energy needed to escape from Earth gives the energy needed to escape fromthe Moon? (b) What multiple of the energy needed to escape from Earth gives the energy needed to escape from Jupiter? (a) ; (b) 28.5 (a) 9.83 m/s 2 (b) 9.84 m/s 2 (c) 9.79 m/s 2

12 86 The figure gives the potential energy function U(r) of a projectile, plotted outward from the surface of a planet of radius R s. (a) If the projectile is launched radially outward from the surface with a mechanical energy of -2.0 x 10 9 J, what is its kinetic energy at radius r = 1.25R? (b) If the projectile is launched radially outward from the surface with a mechanical energy of -2.0 x 10 9 J, what is its turning point in terms of R s? 89 The three spheres in the figure, with masses m A = 80 g, m B = 10 g, and m C = 20 g, have their centers on a common line, with L = 12 cm and d = 4.0 cm. You move sphere B along the line until its center-to-center separation from C is d = 4.0 cm. (a) How much work is done on sphere B by you? (b) How much work is done on sphere B by the net gravitational force on B due to spheres A and C? (a) 0.50 pj; (b) -0.5pJ (a) 2.0 x 10 9 J; (b) 2.5 Rs 87 The figure gives the potential energy function U(r) of a projectile, plotted outward from the surface of a planet of radius R,. What least kinetic energy is required of a projectile launched at the surface if the projectile is to "escape" the planet? 90 Zero, a hypothetical planet, has a mass of 5.0 x kg, a radius of 3.0 x 10 6 in, and no atmosphere. A 10 kg space probe is to be launched vertically from its surface. (a) If the probe is launched with an initial energy of 5.0 x 10 7 J, what will be its kinetic energy when it is 4.0 x 10 6 m from the center of Zero? (b) If the probe is to achieve a maximum distance of 8.0 x 10 6 m from the center of Zero, with what initial kinetic energy must it be launched from the surface of Zero? (a) 2.2 x 10 7 J; (b) 6.9 x 10 7 J x 10 9 J A projectile is shot directly away from Earth's surface. Neglect the rotation of Earth. (a) What multiple of Earth's radius R E gives the radial distance a projectile reaches if its initial speed is of the escape speed from Earth? (b) What multiple of Earth's radius R E gives the radial distance a projectile reaches if its initial kinetic energy is of the kinetic energy required to escape Earth? (c) What is the least initial mechanical energy required at launch if the projectile is to escape Earth? (a) 1.33; (b) 2.00; (c) the following questions. (a) What is the escape speed on a spherical asteroid whose radius is 500 km and whose gravitational acceleration at the surface is 3.0 m/s 2? (b) How far from the surface will a particle go if it leaves the asteroid's surface with a radial speed of 1000 m/s? (c) With what speed will an object hit the asteroid if it is dropped from 1000 km above the surface? (a) 1.7 km/s (b) 2.5 x 10 5 m; (c) 1.4 km/s In deep space, sphere A of mass 20 kg is located at the origin of an x axis and sphere B of mass 10 kg is located on the axis at x = 0.80 m. Sphere B is released from rest while sphere A is held at the origin. (a) What is the gravitational potential energy of the two-sphere system just as B is released? (b) What is the kinetic energy of B when it has moved 0.20 m toward A? (a) -1.7 x 10-8 J (b) 0.56 x 10-8 J

13 93 Two neutron stars are separated by a distance of 1.0 x m. They each have a mass of 1.0 x kg and a radius of 1.0 x 10 5 m. They are initially at rest with respect to each other. (a) As measured from that rest frame, how fast are they moving when their separation has decreased to one-half its initial value? (b) As measured from that rest frame, how fast are they moving when they are about to collide? (a) 82 km/s; (b) 1.8 x 10 4 km/s 96 The first known collision between space debris and a functioning satellite occurred in 1996: At an altitude of 700 km, a year-old French spy satellite was hit by a piece of an Ariane rocket that had been in orbit for 10 years. A stabilizing boom on the satellite was demolished, and the satellite was sent spinning out of control. (a) Just before the collision and in kilometers per hour, what was the speed of the rocket piece relative to the satellite if both were in circular orbits and the collision was head-on? (b) Just before the collision and in kilometers per hour, what was the speed of the rocket piece relative to the satellite if both were in circular orbits and the collision was along perpendicular paths? (a) 5.4 x 10 4 km/h; (b) 3.8 x 10 4 km/h 94 Figure a shows a particle A that can be moved along a y axis from an infinite distance to the origin. That origin lies at the midpoint between particles B and C, which have identical masses, and the y axis is a perpendicular bisector between them. Distance D is m. Figure b shows the potential energy U of the three-particle system as a function of the position of particle A along the y axis. The curve actually extends rightward and approaches an asymptote of -2.7 x J as y --> infinity. (a) What is the mass of particles B and C? (b) What is the mass of particle A? 97 The Martian satellite Phobos travels in an approximately circular orbit of radius 9.4 x 10 6 in with a period of 7 h 39 min. Calculate the mass of Mars from this information. 6.5 x kg 98 The mean distance of Mars from the Sun is 1.52 times that of Earth from the Sun. From Kepler's law of periods, calculate the number of years required for Mars to make one revolution around the Sun; compare your answer with the value given in Appendix C of your text book y (a) 0.50 kg; (b) 1.5kg 95 The figure shows four particles, each of mass 20.0 g, that form a square with an edge length of d = m. If d is reduced to m, what is the change in the gravitational potential energy of the four-particle system? 99 The Sun, which is 2.2 x M from the center of the Milky Way galaxy, revolves around that center once every 2.5 x 10 8 years. Assuming each star in the Galaxy has a mass equal to the Sun's mass of 2.0 x kg, the stars are distributed uniformly in a sphere about the galactic center, and the Sun is at the edge of that sphere, estimate the number of stars in the Galaxy. 5 x stars x J 100 A satellite is put in a circular orbit about Earth with a radius equal to one-half the radius of the Moon's orbit. What is its period of revolution in lunar months? (A lunar month is the period of revolution of the Moon.) 0.35 lunar months

14 101 An Earth sattelite is 160 km above the Earth s surface. (a) What linear speed must the Earth satellite have to be in a circular orbit? (b) What is the period of revolution? (a) 7.82km/s (b) 87.5 min 106 In 1993 the spacecraft Galileo sent home an image (see the figure) of asteroid 243 Ida and a tiny orbiting moon (now known as Dactyl), the first confirmed example of an asteroid moon system. In the image, the moon, which is 1.5 km wide, is 100 km from the center of the asteroid, which is 55 km long. The shape of the moon's orbit is not well known; assume it is circular with a period of 27 h. (a) What is the mass of the asteroid? (b) The volume of the asteroid, measured from the Galileo images, is km 3. What is the density (mass per unit volume) of the asteroid? 102 The Sun's center is at one focus of Earth's orbit. The eccentricity of Earth s orbit is , and the semimajor axis is 1.50 x m. (a) How far from this focus is the other focus, in meters and (b) How far from this focus is the other focus, in terms of the solar radius, 6.96 x 10 8 m? (a) 5.01 x 10 9 m; (b) 7.20 solar radii (a) 6 x kg (b) 4 x 10 3 kg/m A satellite, moving in an elliptical orbit, is 360 km above Earth's surface at its farthest point and 180 km above at its closest point. (a) Calculate the semi-major axis of the orbit. (b) Calculate the eccentricity of the orbit. (a) 6.64 x 10 3 km; (b) An orbiting satellite stays over a certain spot on the equator of (rotating) Earth. What is the altitude of the orbit (called a geosynchronous orbit)? 107 In 1610, Galileo used his telescope to discover four prominent moons around Jupiter. Their mean orbital radii a and periods T are given in the table. (a) Plot log a (y axis) against log T (x axis) and show that you get a straight line. (b) Measure the slope of the line and compare it with the value that you expect from Kepler's third law. (c) Find the mass of Jupiter from the intercept of this line with the y axis x 10 4 km 105 A comet that was seen in April 574 by Chinese astronomers on a day known by them as the Woo Woo day was spotted again in May Assume the time between observations is the period of the Woo Woo day comet and take its eccentricity as (a) What is the semimajor axis of the comet's orbit? (b) What is its greatest distance from the Sun in terms of the mean orbital radius R p of Pluto? (a) 1.9 x m; (b) 3.5RP 108 A 20 kg satellite has a circular orbit with a period of 2.4 h and a radius of 8.0 x 10 6 m around a planet of unknown mass. If the magnitude of the gravitational acceleration on the surface of the planet is 8.0 m/s 2, what is the radius of the planet? 5.8 x 10 6 m

15 109 In a certain binary-star system, each star has the same mass as our Sun, and they revolve about their center of mass. The distance between them is the same as the distance between Earth and the Sun. What is their period of revolution in years? 0.71 y 113 An asteroid, whose mass is 2.0 x 10-4 times the mass of Earth, revolves in a circular orbit around the Sun at a distance that is twice Earth's distance from the Sun. (a) Calculate the period of revolution of the asteroid in years. (b) What is the ratio of the kinetic energy of the asteroid to the kinetic energy of Earth? (a) 2.8 y; (b) 1.0 x The presence of an unseen planet orbiting a distant star can sometimes be inferred from the motion of the star as we see it. As the star and planet orbit the center of mass of the star-planet system, the star moves toward and away from us with what is called the line of sight velocity, a motion that can be detected. The figure shows a graph of the line of sight velocity versus time for the star 14 Herculis. The star's mass is believed to be 0.90 of the mass of our Sun. Assume that only one planel orbits the star and that our view is along the plane of the orbit. (a) Approximate the planet's mass in terms of Jupiter's mass m j. (b) Approximatethe planet's orbital radius in terms of Earth's orbital radius r E. 114 Two Earth satellites, A and B, each of mass m, are to be launched into circular orbits about Earth's center. Satellite A is to orbit at.an altitude of 6370 km. Satellite B is to orbit at an altitude of km. The radius of Earth R E is 6370 km. (a) What is the ratio of the potential energy of satellite B to that of satellite A, in orbit? (b) What is the ratio of the kinetic energy of satellite B to that of satellite A, in orbit? (c) Which satellite has the greater total energy if each has a mass of 14.6 kg? (d) By how much? (a) 1/2 (b) 1/2 (c) B; (d) 1.1 x 10 8 (a) 3.7mJ (b) 2.5rg 115 the following questions. (a) At what height above Earth's surface is the energy required to lift a satellite to that height equal to the kinetic energy required for the satellite to be in orbit at that height? (b) For greater heights, which is greater, the energy for lifting or the kinetic energy for orbiting? 111 Three identical stars of mass M form an equilateral triangle that rotates around the triangle's center as the stars move in a common circle about that center. The triangle has edge length L. What is the speed of the stars? (a) 3.19 x 10 3 km; (b) lifting 112 A satellite orbits a planet of unknown mass in a circle of radius 2.0 x 10 7 m. The magnitude of the gravitational force on the satellite from the planet is F = 80 N. (a) What is the kinetic energy of the satellite in this orbit? (b) What would F be if the orbit radius were increased to 3.0 x 10 7 M? (a) 8.0 x 10 8 J (b) 36N

16 116 In the figure, two satellites, A and B, both of mass m = 125 kg, move in the same circular orbit of radius r = 7.87 x 10 6 m around Earth but in opposite senses of rotation and therefore on a collision course. (a) Find the total mechanical energy E A + E B of the two satellites + Earth system before the collision. (b) If the collision is completely inelastic so that the wreckage remains as one piece of tangled material (mass = 2m), find the total mechanical energy immediately after the collision. (c) Just after the collision, is the wreckage falling directly toward Earth's center or orbiting around Earth? (a) x 10 9 J; (b) x 10 9 (c) falling 119 A 220 kg satellite in an approximately circular orbit 640 km above the surface of Earth. (a) What is the speed of a 220 kg satellite in an approximately circular orbit 640 km above the surface of Earth? (b) What is the period of a 220 kg satellite in an approximately circular orbit 640 km above the surface of Earth? Suppose the satellite loses mechanical energy at the average rate of 1.4 x 10 5 J per orbital revolution. (c) Adopting the reasonable approximation that the satellite's orbit becomes a "circle of slowly diminishing radius," determine the satellite's altitude at the end of its 1500th revolution. (d) Adopting the reasonable approximation that the satellite's orbit becomes a "circle of slowly diminishing radius," determine the satellite's speed at the end of its 1500th revolution. (e) Adopting the reasonable approximation that the satellite's orbit becomes a "circle of slowly diminishing radius," determine the satellite's period at the end of its 1500th revolution. (f) What is the magnitude of the average retarding force on the satellite? (g) Is angular momentum around Earth's center conserved for the satellite? (h) Is angular momentum around Earth's center conserved for the satellite-earth system (assuming that system is isolated)? (a) 7.5 km/s; (b) 97 min; (c) (d) 7.7 km/s; (e) 93 min; (f) 3.2 x 10- (g) no; (h) yes 117 A satellite is in a circular Earth orbit of radius r. The area A enclosed by the orbit depends on r 2 because A = πr 2. Determine how the following properties of the satellite depends on r. (a) Determine how the period of the satellite depends on r. (b) Determine how the kinetic energy of the satellite depends on r. (c) Determine how the angular momentum of the satellite depends on r. (d) Determine how the speed of the satellite depends on r. (a) r 3 /2; (b) 1/r; (c) r; (d) 1/ r 120 A physicist in a box resting on Earth sees a cantaloupe falling with acceleration a = 9.8 m/s 2. If he and the box accelerate in deep space at 9.8 m/s 2 the cantaloupe has the same acceleration relative to him. In the figure, the scale on which the 60 kg physicist stands reads 220 N. How long will the cantaloupe take to reach the floor if the physicist drops it (from rest relative to himself) at a height of 2.1 m above the floor? 118 One way to attack a satellite in Earth orbit is to launch a swarm of pellets in the same orbit as the satellite but in the opposite direction. Suppose a satellite in a circular orbit 500 km above Earth's surface collides with a pellet having mass 4.0 g. (a) What is the kinetic energy of the pellet in the reference frame of the satellite just before the collision? (b) What is the ratio of this kinetic energy to the kinetic energy of a 4.0 g bullet from a modern army rifle with a muzzle speed of 950 m/s? (a) 4.6 x 10 5 J; (b) 2.6 x s

17 121 The figure is a graph of the kinetic energy K of an asteroid versus its distance r from Earth's center, as the asteroid falls directly in toward that center. (a) What is the (approximate) mass of the asteroid? (b) What is its speed at r = x 10 7 m? 124 the following questions. (a) If the legendary apple of Newton could be released from rest at a height of 2 m from the surface of a neutron star with a mass 1.5 times that of our Sun and a radius of 20 km, what would be the apple's speed when it reached the surface of the star? (b) If the apple could rest on the surface of the star, what would be the approximate difference between the gravitational acceleration at the top and at the bottom of the apple? (Choose a reasonable size for an apple; the answer indicates that an apple would never survive near a neutron star.) (a) 1.4 x 10 6 m/s; (b) 3 x 10 6 m/s 2 (a) 1.0 x 10 3 kg; (b) 1.5 km/s 125 Sphere A with mass 80 kg is located at the origin of an xy coordinate system; sphere B with mass 60 kg is located at coordinates (0.25 m, 0); sphere C with mass 0.20 kg is located in the first quadrant 0.20 m from A and 0.15 m from B. In unitvector notation, what is the gravitational force on C due to A and B? 122 The radius R h of a black hole is the radius of a mathematical sphere, called the event horizon, that is centered on the black hole. Information from events inside the event horizon cannot reach the outside world. According to Einstein's general theory of relativity, R h = 2GM/c 2, where M is the mass of the black hole and c is the speed of light. Suppose that you wish to study a black hole near it, at a radial distance of 50R h. However, you do not want the difference in gravitational acceleration between your feet and your head to exceed 10 m/s 2 when you are feet down (or head down) toward the black hole. (a) As a multiple of our Sun's mass M S, approximately what is the limit to the mass of the black hole you can tolerate at the given radial distance? (You need to estimate your height.) (b) Is the limit an upper limit (you can tolerate smaller masses) or a lower limit (you can tolerate larger masses)? 126 -(0.044 µnˆj A satellite is in elliptical orbit with a period of 8.00 x 10 4 s about a planet of mass 7.00 x kg. At aphelion, at radius 4.5 x 10 7 m, the satellite's angular speed is x 10-5 rad/s. What is its angular speed at perihelion? 9.2 x 10-5 rad/s 123 (a) (1 x 10 2 ) Ms; (b) lower Four identical 1.5 kg particles are placed at the corners of a square with sides equal to 20 cm. What is the magnitude of the net gravitational force on any one of the particles due to the others? 7.2 x 10-9 N 127 In a shuttle draft of mass m = 3000 kg, Captain Janeway orbits a planet of mass M = 9.50 x kg, in a circular orbit of radius r = 4.20 x 10 7 M. (a) What is the period of the orbit of the shuttle craft? (b) What is the speed of the shuttle craft? Janeway briefly fires a forward-pointing thruster, reducing her speed by 2.00%. (c) Just then, what is the speed of the shuttle craft? (d) Just then, what is the kinetic energy of the shuttle craft? (e) Just then, what is the gravitational potential energy of the shuttle craft? (f) Just then, what is the mechanical energy of the shuttle craft? (g) What is the sernimajor axis of the elliptical orbit now taken by the craft? (h) What is the difference between the period of the original circular orbit and that of the new elliptical orbit? (i) Which orbit has the smaller period? (a) 2.15 x 10 4 s; (b) 12.3 km/s; (c) 12.0 km/s; (d) 2.17 x J; (e) x J; (f) x J; (g) 4.04 x 10 7 m; (h) 1.22 x 10 3 s; (i) elliptical

18 The mysterious visitor that appears in the enchanting story The Little Prince was said to come from a planet that "was scarcely any larger than a house!" Assume that the mass per unit volume of the planet is about that of Earth and that the planet does not appreciably spin. (a) Approximate the free-fall acceleration on the planet's surface. (b) Approximatethe escape speed from the planet. (a) 2 x 10-5 m/s 2 ; (b) 2 cm/s A typical neutron star may have a mass equal to that of the Sun but a radius of only 10 km. (a) What is the gravitational acceleration at the surface of such a star? (b) How fast would an object be moving if it fell from rest through a distance of 1.0 m on such a star? (Assume the star does not rotate.) (a) 1.3 x m/s 2 ; (b) 1.6 x 10 6 m/s 130 We watch two identical astronomical bodies A and B, each of mass m, fall toward each other from rest because of the gravitational force on each from the other. Their initial centerto-center separation is R i. Assume that we are in an inertial reference frame that is stationary with respect to the center of mass of this two-body system. (a) Use the principle of conservation of mechanical energy (K f + U f = K i + U i ) to find the total kinetic energy of the system when the center-to-center separation is 0.5R i. (b) Use the principle of conservation of mechanical energy (K f + U f = K i + U i ) to find the kinetic energy of each body when the center-to-center separation is 0.5R i. (c) Use the principle of conservation of mechanical energy (K f + U f = K i + U i ) to find the speed of each body relative to us when the center-to-center separation is 0.5R i.d (d) Use the principle of conservation of mechanical energy (K f + U f = K i + U i ) to find the speed of body B relative to body A when the center-to-center separation is 0.5R i. Next assume that we are in a reference frame attached to body A (we ride on the body). Now we see body B fall from rest toward us. (e) From this reference frame, again use K f + U f = K i + U i to find the kinetic energy of body B when the center-to-center separation is 0.5R i. (f) From this reference frame, again use K f + U f = K i + U i to find the speed of body B relative to body A when the centerto-center separation is 0.5R i. (g) Why are the answers to (d) and (f) different? Which answer is correct? (a) Gm 2 /R i ; (b) Gm 2 /2R i ; (c) Gm/R i ; (d) 22Gm/R i (e) Gm 2 /R i ; (f) 2Gm/R i ; (g) the center-of-mass frame is an inertial frame and in it the conservation of energy principle may be written as in Chapter 8; the reference frame attached to body A is noninertial and the principle cannot be written as in Chapter 8; answer (d) is correct 131 Four uniform spheres, with masses m A = 40 kg, m B = 35 kg, m C = 200 kg, and m D = 50 kg, have (x, y) coordinates of (0, 50 cm), (0, 0), (-80 cm, 0), and (40 cm, 0), respectively. In unitvector notation, what is the net gravitational force on sphere B due to the other spheres? (0.37 µn)ˆj

19 132 Four uniform spheres, with masses m A = 40 kg, m B = 35 kg, m C = 200 kg, and m D = 50 kg, have (x, y) coordinates of (0, 50 cm), (0, 0), (-80 cm, 0), and (40 cm, 0), respectively. (a) If you remove sphere A and calculate the gravitational potential energy of the remaining three-particle system. (b) If A is then put back in place, is the potential energy of the four-particle system more or less than that of the system in (a)? (c) In (a), is the work done by you to remove A positive or negative? (d) In (b), is the work done by you to replace A positive or negative? (a) -1.3 x 10-4 J; (b) less; (c) positive; (d) negative 137 In a double-star system, two stars of mass 3.0 x kg each rotate about the system's center of mass at radius 1.0 x m. (a) What is their common angular speed? (b) If a meteoroid passes through the system's center of mass perpendicular to their orbital plane, what minimum speed must it have at the center of mass if it is to escape to "infinity" from the two-star system? (a) 2.2 x 10-7 rad/s; (b) 89 km/s 133 A very early, simple satellite consisted of an inflated spherical aluminum balloon 30 m in diameter and of mass 20 kg. Suppose a meteor having a mass of 7.0 kg passes within 3.0 m of the surface of the satellite. What is the magnitude of the gravitational force on the meteor from the satellite at the closest approach? 29 pn 138 An object lying on Earth's equator is accelerated (a) toward the centert of Earth because Earth rotates, (b) toward the Sun because Earth revolves around the Sun in an almost circular orbit, and (c) toward the center of our galaxy because the Sun moves around the galactic center. For the latter, the period is 2.5 x 10 8 y and the radius is 2.2 x m. Calculate these three accelerations as multiples of g = 9.8 m/s 2. (a) (3.4 x 10-3 )g; (b) (6.1 x 10-4 )g; (c) (1.4 x )g 134 A uniform solid sphere of radius R produces a gravitational acceleration of a g on its surface. (a) At what distance from the sphere's center are there points inside the sphere where the gravitational acceleration is a g /3? (b) At what distance from the sphere's center are there points outside the sphere where the gravitational acceleration is a g /3? 139 The masses and coordinates of three spheres are as follows: 20 kg, x = 0.50 m, y = 1.0 m; 40 kg, x = -1.0 m, y = -1.0 m; 60 kg, x = 0 m, y = m. What is the magnitude of the gravitational force on a 20 kg sphere located at the origin due to these three spheres? 3.2 x 10-7 N (a) R/3; (b) 3R A projectile is fired vertically from Earth's surface with an initial speed of 10 km/s. Neglecting air drag, how far above the surface of Earth will it go? 2.5 x 10 4 km A 50 kg satellite circles planet Cruton every 6.0 h. The magnitude of the gravitational force exerted on the satellite by Cruton is 80 N. (a) What is the radius of the orbit? (b) What is the kinetic energy of the satellite? (c) What is the mass of planet Cruton? (a) 1.9 x 10 7 m; (b) 7.6 x 10 8 J; (c) 8.6 x kg 140 In his 1865 science fiction novel From the Earth to the Moon, Jules Verne described how three astronauts are shot to the Moon by means of a huge gun. According to Verne, the aluminum capsule containing the astronauts is accelerated by ignition of nitrocellulose to a speed of 11 km/s along the gun barrel's length of 220 m. (a) In g units, what is the average acceleration of the capsule and astronauts in the gun barrel? (b) Is that acceleration tolerable or deadly to the astronauts? A modern version of such gun-launched spacecraft (although without passengers) has been proposed. In this modern version, called the SHARP (Super High Altitude Research Project) gun, ignition of methane and air shoves a piston down the gun's tube, compressing hydrogen gas that then launches a rocket. During this launch, the rocket moves 3.5 km and reaches a speed of 7.0 km/s. Once launched, the rocket can be fired to gain additional speed. (c) In g units, what would be the average acceleration of the rocket within the launcher? (d) How much additional speed is needed (via the rocket engine) if the rocket is to orbit Earth at an altitude of 700 km? (a) (2.8 x 10 4 )g; (b) deadly; (c) 714 g; (d) 1.5 km/s

20 141 A kg rocket moving radially outward from Earth has a speed of 3.70 km/s when its engine shuts off 200 km above Earth's surface. (a) Assuming negligible air drag, find the rocket's kinetic energy when the rocket is 1000 km above Earth's surface. (b) What maximum height above the surface is reached by the rocket? (a) 38.3 MJ; (b) 1.03 x 10 3 km 144 In your calculator, make a list of the periods T for the planets in the table and a separate list of the semi major axes a. Multiply the list of T values by an appropriate factor so that the unit of T is seconds. (a) Store values of T 2 and a 3 in two new lists. Have the calculator do a linear regression fit of T 2 versus a 3. From the parameters of the fit and using the known value of G, determine the mass of the Sun. (b) Store values of log T in one list and log a in another list. Have the calculator plot log T versus log a and then do a linear regression fit of the plot. From the parameters of this fit and using the known value of G, again determine the mass of the Sun. 142 Planet Roton, with a mass of 7.0 x kg and a radius of 1600 km, gravitationally attracts a meteorite that is initially at rest relative to the planet, at a distance great enough to take as infinite. The meteorite falls toward the planet. Assuming the planet is airless, find the speed of the meteorite when it reaches the planet's surface. 2.4 x 10 4 m/s 143 The orbit of Earth around the Sun is almost circular: The closest and farthest distances are 1.47 x 10 8 km and 1.52 x 10 8 km respectively. (a) Determine the corresponding variations in total energy. (b) Determine the corresponding variations in gravitational potential energy. (c) Determine the corresponding variations in kinetic energy. (d) Determine the corresponding variations in orbital speed. (Hint: Use conservation of energy and conservation of angular momentum.) 145 (a) 1.98 x kg; (b) 2.00 x kg A planet requires 300 (Earth) days to complete its circular orbit around its sun, which has a mass of 6.0 x kg. (a) What is the planet's orbital radius? (b) What is the planet's orbital speed? (a) 0; (b) 1.8 x J; (c) 1.8 x J; (d) 0.99 km/s (a) 1.9 x m; (b) 4.6 x 10 4 m/s 146 Two 20 kg spheres are fixed in place on a y axis, one at y = 0.40 m and the other at y = m. A 10 kg ball is then released from rest at a point on the x axis that is at a great distance (effectively infinite) from the spheres. Assume the only forces acting on the ball are the gravitational forces from the spheres. (a) When the ball reaches the (x, y) point (0.30 m, 0), what is its kinetic energy? (b) When the ball reaches the (x, y) point (0.30 m, 0), what is the net force on it from the spheres, in unit-vector notation? (a) 5.3 x 10-8 J; (b) (-6.4 x 10-8 N)ˆi

21 147 In Pole to Pole, an early science fiction story by George Griffith, three explorers attempt to travel by capsule through a naturally formed (and, of course, fictional) tunnel directly from the south pole to the north pole (see the figure). According to the story, as the capsule approaches Earth's center, the gravitational force on the explorers becomes alarmingly large and then, exactly at the center, it suddenly but only momentarily disappears. Then the capsule travels through the second half of the tunnel, to the north pole. Assume that Earth is a sphere of uniform density p (mass per unit volume). With what speed would mail pass through the center of Earth if falling in the tunnel? 151 What is the percentage change in the acceleration of Earth toward the Sun when the alignment of Earth, Sun, and Moon changes from an eclipse of the Sun (with the Moon between Earth and Sun) to an eclipse of the Moon (Earth between Moon and Sun)? 1.1% km/s A satellite of mass 125 kg is in a circular orbit of radius 7.00x 10 6 m around a planet. If the period is 8050 s, what is the mechanical energy of the satellite? 152 The most valued orbit for a communications satellite is a geosynchronous orbit, in which the satellite remains above a certain point on Earth's equator as it orbits. Such a "geo satellite" would always be in the same position in the sky as seen from the transmitting or receiving equipment of, say, the BBC in London. Unfortunately, room along a geosynchronous orbit is limited because the satellites in such an orbit cannot be too close to one another or the signal quality is degraded. The least separation along the of the circular orbit is (a) What is the maximum number of geo satellites the orbit should contain? (b) What is their orbital radius? (c) What then is the minimum separation d between any two geo satellites along the orbital path? A magnetic storm can disrupt a satellite as well as move it, either toward or away from Earth. Suppose a magnetic storm turns off geo satellite Hot Bird 2, and when groundbased engineers turn it back on, it is east of its assigned orbital spot. (d) Is the period of Hot Bird 2 now greater than or smaller than that of the other geo satellites? (e) Did the storm move the satellite toward or away from Earth? (a) 120; (b) 4.23 x 10 4 km; (c) 2.2 x 10 6 m; (d) smaller; (e) toward GJ 149 A particle of comet dust with mass m is a distance R from Earth's center and a distance r from the Moon's center. If Earth's mass is M E and the Moon's mass is M m, what is the sum of the gravitational potential energy of the particle-earth system and the gravitational potential energy of the particlemoon system? -Gm(ME/R +Mm/r) 153 An object of mass m is initially held in place at radial distance r = 3R E from the center of Earth, where R E is the radius of Earth. Let M E be the mass of Earth. A force is applied to the object to move it to a radial distance r = 4R E, where it again is held in place. Calculate the work done by the applied force during the move by integrating the force magnitude. GM E m/12r E 150 A spaceship is on a straight-line path between Earth and the Moon. At what distance from Earth is the net gravitational force on the spaceship zero? 3.4 x 10 5 km

22 154 Several planets Jupiter, Saturn, Uranus) are encircled by rings, perhaps composed of material that failed to form a satellite. In addition, many galaxies contain ring-like structures. Consider a homogeneous thin ring of mass M and outer radius R (see figure). (a) What gravitational attraction does it exert on a particle of mass m located on the ring's central axis a distance x from the ring center? (b) Suppose the particle falls from rest as a result of the attraction of the ring of matter. What is the speed with which it passes through the center of the ring? 158 A projectile is launched directly away from the surface of a planet of mass M and radius R; the launch speed is (GM/R) 1/2. Use the principle of conservation of energy to determine the maximum distance from the center of the planet achieved by the projectile. Express your result in terms of R. 2R 159 Show how, guided by Kepler's law of periods, Newton could deduce that the force holding the Moon in its orbit, assumed circular, depends on the inverse square of the Moon's distance from the center of Earth. 155 (a) GMmx(x 2 + R 2 )-3/2; Show that, at the bottom of a vertical mine shaft dug to depth D, the gravitational acceleration a g is given in the equation where a, is the surface value. Assume that Earth is a uniform sphere of radius R. 160 A sphere of matter, of mass M and radius a, has a concentric cavity of radius b, as shown in cross section in the figure. (a) Sketch a curve of the magnitude of the gravitational force F from the sphere on a particle of mass m located a distance r from the center of the sphere, as a function of r in the range 0 r inifinity. Consider r = 0, b, a, and ininingy in particular. (b) Sketch the corresponding curve for the potential energy U (r) of the system. 156 The fastest possible rate of rotation of a planet is that for which the gravitational force on material at the equator just barely provides the centripetal force needed for the rotation. (Why?) (a) Show that the corresponding shortest period of rotation is given in the equation where ρ is the uniform density (mass per unit volume) of the spherical planet. (b) Calculate the rotation period assuming a density of 3.0 g/cm 3, typical of many planets, satellites, and asteroids. No astronomical object has ever been found to be spinning with a period shorter than that determined by this analysis. 161 A certain triple-star system consists of two stars, each of mass m, revolving in the same circular orbit of radius r around a central star of mass M (see figure). The two orbiting stars are always at opposite ends of a diameter of the orbit, Derive an expression for the period of revolution of the stars. (a) (b) 1.9h 157 The gravitational force between two particles with masses m and M, initially at rest at great separation, pulls them together. Show that at any instant the speed of either particle relative to the other is {2G(M + m)/d} where d is their separation at that instant. (Hint: Use the laws of conservation of energy and conservation of linear momentum.) 2πr3/2/ G(M + m/4)

23 162 Some people believe that the positions of the planets at the time of birth influence the newborn. Others deride this belief and claim that the gravitational force exerted on a baby by the obstetrician is greater than the force exerted by the planets. (a) To check this claim, calculate the magnitude of the gravitational force exerted on a 3 kg baby by a 70 kg obstetrician who is 1 m away and roughly approximated as a point mass. (b) To check this claim, calculate the magnitude of the gravitational force exerted on a 3 kg baby by the massive planet Jupiter (m = 2 x kg) at its closest approach to Earth (= 6 x m), and (c) To check this claim, calculate the magnitude of the gravitational force exerted on a 3 kg baby by Jupiter at its greatest distance from Earth (= 9 x m). (d) Is the claim correct? If the Earth s mass were double what it is, in what ways would the Moon s orbit be different? the following questions. (a) Which pulls harder gravitationally, the Earth on the Moon, or the Moon on the Earth? (b) Which accelerates the Earth on the Moon, or the Moon on the Earth? (a) 1 x 10-8 N; (b) 1 x 10-6 N; (c) 5 x 10-7 N; (d) no (a)the gravitational pull is the same in each case, by Newton s 3rd law (b)since the Moon has the smaller mass, it will have the larger acceleration. 163 Some people believe that the positions of the planets at the time of birth influence the newborn. Others deride this belief and claim that the gravitational force exerted on a baby by the obstetrician is greater than the force exerted by the planets. (a) To check this claim, calculate the magnitude of the gravitational force exerted on a 3 kg baby by a 70 kg obstetrician who is 1 m away and roughly approximated as a point mass. (b) To check this claim, calculate the magnitude of the gravitational force exerted on a 3 kg baby by the massive planet Jupiter (m = 2 x kg) at its closest approach to Earth (= 6 x m), and (c) To check this claim, calculate the magnitude of the gravitational force exerted on a 3 kg baby by Jupiter at its greatest distance from Earth (= 9 x m). (d) Is the claim correct? The Sun s gravitational pull on the Earth is much larger than the Moon s. Yet the Moon s is mainly responsible for the tides. Explain. [Hint: Consider the difference in gravitational pull from one side of the Earth to the other.] the following questions: (a) Will an object weigh more at the equator or at the poles? (b) What two effects are at work? (c) Do they oppose each other? 164 (a) 1 x 10-8 N; (b) 1 x 10-6 N; (c) 5 x 10-7 N; (d) no Does an apple exert a gravitational force on the Earth? If so, how large a force? (a) Consider an apple attached to a tree (b) Consider an apple falling. 169 The gravitational force on the Moon due to the Earth is only about half the force on the Moon due to the Sun. Why isn t the Moon pulled away from the Earth? (a) The apple does exert a gravitational force on the Earth. By Newton s 3rd law, the force on the Earth due to the apple is the same magnitude as the force on the apple due to the Earth the weight of the apple. 170 Is the centripetal acceleration of Mars in its orbit around the Sun larger or smaller than the centripetal acceleration of the Earth? (b) The force is also independent of the state of motion of the apple. So for both a hanging apple and a falling apple, the force on the Earth due to the apple is equal to the weight of the apple. 171 Would it require less speed to launch a satellite (a) toward the east or (b) toward the west? Consider the Earth s rotation direction.

24 172 What keeps a satellite up in its orbit around the Earth? 177 Suppose the space shuttle is in orbit 400 km from the Earth s surface, and circles the Earth about once every 90 minutes. Find the centripetal acceleration of the space shuttle in its orbit. Express your answer in terms of g, the gravitational acceleration at the Earth s surface. 173 Astronauts who spend long periods in outer space could be adversely affected by weightlessness. One way to simulate gravity is to shape the spaceship like a cylindrical shell that rotates, with the astronauts walking on the inside surface. (a) Explain how this simulates gravity when you consider how objects fall, (b) Explain how this simulates gravity when you consider the force we feel on our feet (c) Explain how this simulates gravity when you consider any other aspects of gravity you can think of. 178 Calculate the force of Earth s gravity on a spacecraft 12,800 km (2 Earth radii) above the Earth s surface if its mass is 1350 kg. 179 At the surface of a certain planet, the gravitational acceleration g has a magnitude of A 21.0-kg brass ball is transported to this planet. (a) What is the mass of the brass ball on the Earth and on the planet (b) What is the weight of the brass ball on the Earth and on the planet? (a) (b) (c) 180 Calculate the acceleration due to gravity on the Moon. The Moon s radius is 1.74 x 10 6 m and its mass is 7.35 x kg. 174 The Earth moves faster in its orbit around the Sun in January than in July. Is the Earth closer to the Sun in January, or in July? Explain. [Note: This is not much of a factor in producing the seasons the main factor is the tilt of the Earth s axis relative to the plane of its orbit.] 181 A hypothetical planet has a radius 1.5 times that of Earth, but has the same mass. What is the acceleration due to gravity near its surface? 175 The mass of Pluto was not known until it was discovered to have a moon. Explain how this discovery enabled an estimate of Pluto s mass. 182 A hypothetical planet has a mass 1.66 times that of Earth, but the same radius. What is g near its surface? 176 Calculate the centripetal acceleration of the Earth in its orbit around the Sun, and the net force exerted on the Earth. What exerts this force on the Earth? Assume that the Earth s orbit is a circle of radius 183 Two objects attract each other gravitationally with a force of 2.5 x N when they are 0.25 m apart. Their total mass is 4.0 kg. Find their individual masses.

25 184 the following questions: (a) Calculate the effective value of g, the acceleration of gravity, at 3200 m, above the Earth s surface. (b) Calculate the effective value of g, the acceleration of gravity, at3200 km, above the Earth s surface. 190 Every few hundred years most of the planets line up on the same side of the Sun. Calculate the total force on the Earth due to Venus, Jupiter, and Saturn, assuming all four planets are in a line. The masses are M V = 0.815M E, M J = 318M E, M S -= 95.1M E and their mean distances from the Sun are 108, 150, 778, and 1430 million km, respectively. What fraction of the Sun s force on the Earth is this? 185 What is the distance from the Earth s center to a point outside the Earth where the gravitational acceleration due to the Earth is 1/10 of its value at the Earth s surface? 186 A certain neutron star has five times the mass of our Sun packed into a sphere about 10 km in radius. Estimate the surface gravity on this monster. 191 Given that the acceleration of gravity at the surface of Mars is 0.38 of what it is on Earth, and that Mars radius is 3400 km, determine the mass of Mars. 187 A typical white-dwarf star, which once was an average star like our Sun but is now in the last stage of its evolution, is the size of our Moon but has the mass of our Sun. What is the surface gravity on this star? 192 Determine the mass of the Sun using the known value for the period of the Earth and its distance from the Sun. Hint: Use Kepler s Law. 188 You are explaining why astronauts feel weightless while orbiting in the space shuttle. Your friends respond that they thought gravity was just a lot weaker up there. Convince them and yourself that it isn t so by calculating the acceleration of gravity 250 km above the Earth s surface in terms of g. 193 Calculate the speed of a satellite moving in a stable circular orbit about the Earth at a height of 3600 km. 189 Four 9.5-kg spheres are located at the corners of a square of side 0.60 m. Calculate the magnitude and direction of the total gravitational force exerted on one sphere by the other three. 194 The space shuttle releases a satellite into a circular orbit 650 km above the Earth. How fast must the shuttle be moving (relative to Earth) when the release occurs?

26 195 At what rate must a cylindrical spaceship rotate if occupants are to experience simulated gravity of 0.60 g? Assume the spaceship s diameter is 32 m, and give your answer as the time needed for one revolution. 200 the following questions (a) What is the apparent weight of a 75-kg astronaut 4200 km from the center of the Earth s Moon in a space vehicle moving at constant velocity, (b) What is the apparent weight of a 75-kg astronaut 4200 km from the center of the Earth s Moon in a space vehicle accelerating toward the Moon at State the direction in each case. 201 Suppose that a binary-star system consists of two stars of equal mass. They are observed to be separated by 360 million km and take 5.7 Earth years to orbit about a point midway between them. What is the mass of each? 196 the following questions: (a) Determine the time it takes for a satellite to orbit the Earth in a circular near-earth orbit. A near-earth orbit is one at a height above the surface of the Earth which is very small compared to the radius of the Earth. (b) Does your result depend on the mass of the satellite? Explain 202 the following questions (a) Show that if a satellite orbits very near the surface of a planet with period T, the density of the planet is ρ = m/v = 3π/GT 2 (b) Estimate the density of the Earth, given that a satellite near the surface orbits with a period of about 85 min. 197 At what horizontal velocity would a satellite have to be launched from the top of Mt. Everest to be placed in a circular orbit around the Earth? 203 Use Kepler s laws and the period of the Moon (27.4 days) to determine the period of an artificial satellite orbiting very near the Earth s surface. 198 During an Apollo lunar landing mission, the command module continued to orbit the Moon at an altitude of about 100 km. How long did it take to go around the Moon once? 204 The asteroid Icarus, though only a few hundred meters across, orbits the Sun like the planets. Its period is 410 d. What is its mean distance from the Sun? 199 The rings of Saturn are composed of chunks of ice that orbit the planet. The inner radius of the rings is 73,000 km, while the outer radius is 170,000 km. Find the period of an orbiting chunk of ice at the inner radius and the period of a chunk at the outer radius. Compare your numbers with Saturn s mean rotation period of 10 hours and 39 minutes. The mass of Saturn is 5.7 x kg. 205 Neptune is an average distance of from the Sun. Estimate the length of the Neptunian year given that the Earth is from the Sun on the average.

27 206 Halley s comet orbits the Sun roughly once every 76 years. It comes very close to the surface of the Sun on its closest approach (a) Estimate the greatest distance of the comet from the Sun. Is it still in the Solar System? (b) What planet s orbit is nearest when it is out there? [Hint: The mean distance s in Kepler s third law is half the sum of the nearest and farthest distance from the Sun.] 209 Determine the mass of the Earth from the known period and distance of the Moon. 210 Determine the mean distance from Jupiter for each of Jupiter s moons, using Kepler s third law. Use the distance of Io and the periods given in Table 5 3. Compare to the values in the Table. 207 Our Sun rotates about the center of the Galaxy at a distance of about light-years What is the period of our orbital motion about the center of the Galaxy? 208 The table gives the mass, period, and mean distance for the four largest moons of Jupiter (those discovered by Galileo in 1609). (a) Determine the mass of Jupiter using the data for Io. (b) Determine the mass of Jupiter using data for each of the other three moons. Are the results consistent? 211 The asteroid belt between Mars and Jupiter consists of many fragments (which some space scientists think came from a planet that once orbited the Sun but was destroyed). (a) If the center of mass of the asteroid belt (where the planet would have been) is about three times farther from the Sun than the Earth is, how long would it have taken this hypothetical planet to orbit the Sun? (b) Can we use these data to deduce the mass of this planet?

28 212 A science-fiction tale describes an artificial planet in the form of a band completely encircling a sun. The inhabitants live on the inside surface (where it is always noon). Imagine that this sun is exactly like our own, that the distance to the band is the same as the Earth Sun distance (to make the climate temperate), and that the ring rotates quickly enough to produce an apparent gravity of g as on Earth. What will be the period of revolution, this planet s year, in Earth days? 216 A projected space station consists of a circular tube that will rotate about its center (like a tubular bicycle tire). The circle formed by the tube has a diameter of about 1.1 km. What must be the rotation speed (revolutions per day) if an effect equal to gravity at the surface of the Earth (1.0 g) is to be felt? 213 How far above the Earth s surface will the acceleration of gravity be half what it is on the surface? 217 Derive a formula for the mass of a planet in terms of its radius r, the acceleration due to gravity at its surface and the gravitational constant G. 214 Because the Earth rotates once per day, the apparent acceleration of gravity at the equator is slightly less than it would be if the Earth didn t rotate. (a) Estimate the magnitude of this effect. (b) What fraction of g is this? 218 How long would a day be if the Earth were rotating so fast that objects at the equator were apparently weightless? 215 At what distance from the Earth will a spacecraft traveling directly from the Earth to the Moon experience zero net force because the Earth and Moon pull with equal and opposite forces? 219 Two equal-mass stars maintain a constant distance apart of 8.0 x m and rotate about a point midway between them at a rate of one revolution every 12.6 yr. (a) Why don t the two stars crash into one another due to the gravitational force between them? (b) What must be the mass of each star? 220 Jupiter is about 320 times as massive as the Earth. Thus, it has been claimed that a person would be crushed by the force of gravity on a planet the size of Jupiter since people can t survive more than a few g s. Calculate the number of g s a person would experience at the equator of such a planet. Use the following data for Jupiter: mass = 1.9 x kg, equitorial radius = 7.1 x 10 4 km, equatorial rotation period = 9 hr 55 min. Take the centripetal acceleration into account.

29 221 Astronomers using the Hubble Space Telescope deduced the presence of an extremely massive core in the distant galaxy M87, so dense that it could be a black hole (from which no light escapes). They did this by measuring the speed of gas clouds orbiting the core to be at a distance of 60 light-years (5. x ) from the core. Deduce the mass of the core, and compare it to the mass of our Sun. 225 The comet Hale-Bopp has a period of 3000 years. (a) What is its mean distance from the Sun? (b) At its closest approach, the comet is about 1 A.U. from the Sun (1 A.U = distance from earth to the sun) from Earth to the Sun). What is the farthest distance? (c) What is the ratio of the speed at the closest point to the speed at the farthest point? [Hint: Use Kepler s second law and estimate areas by a triangle (as in the figure, but smaller distance travelled.] 222 The Navstar Global Positioning System (GPS) utilizes a group of 24 satellites orbiting the Earth. Using triangulation and signals transmitted by these satellites, the position of a receiver on the Earth can be determined to within an accuracy of a few centimeters. The satellite orbits are distributed evenly around the Earth, with four satellites in each of six orbits, allowing continuous navigational fixes. The satellites orbit at an altitude of approximately 11,000 nautical miles [1 nautical mile = km = 6076 ft ]. (a) Determine the speed of each satellite. (b) Determine the period of each satellite The Near Earth Asteroid Rendezvous (NEAR), after traveling 2.1 billion km, is meant to orbit the asteroid Eros at a height of about 15 km. Eros is roughly 40 km x 6 km x 6 km. Assume Eros has a density (mass/volume) of about 2.3 x 10 3 kg/m 3. (a) What will be the period of NEAR as it orbits Eros? (b) If Eros were a sphere with the same mass and density, what would its radius be? (c) What would g be at the surface of this spherical Eros? You are an astronaut in the space shuttle pursuing a satellite in need of repair. You are in a circular orbit of the same radius as the satellite (400 km above the Earth), but 25 km behind it. (a) How long will it take to overtake the satellite if you reduce your orbital radius by 1.0 km? (b) By how much must you reduce your orbital radius to catch up in 7.0 hours? Estimate what the value of G would need to be if you could actually feel yourself gravitationally attracted to someone near you. Make reasonable assumptions, like The Sun rotates around the center of the Milky Way Galaxy at a distance of about 30,000 light-years from the center (1 ly = 9.5 x m) If it takes about 200 million years to make one rotation, estimate the mass of our Galaxy. Assume that the mass distribution of our Galaxy is concentrated mostly in a central uniform sphere. If all the stars had about the mass of our Sun (2 x kg ) how many stars would there be in our Galaxy? 228 Four 1.0-kg masses are located at the corners of a square 0.50 m on each side. Find the magnitude and direction of the gravitational force on a fifth 1.0-kg mass placed at the midpoint of the bottom side of the square.