35. || INT FIGURE P13.35 shows three masses. What are the magnitude and the direction of the net gravitational force on the 20.0 kg mass? Give the direction as an angle cw from the y-axis.

    FIGURE P13.35

    

    FIGURE P13.35 Full Alternative Text

36. || INT What is the net gravitational force on the 20.0 kg mass in FIGURE P13.36? Give your answer using unit vectors.

    FIGURE P13.36

    

    FIGURE P13.36 Full Alternative Text

37. || What is the total gravitational potential energy of the three masses in FIGURE P13.35?

38. || What is the total gravitational potential energy of the three masses in FIGURE P13.36?

39. || Two spherical objects have a combined mass of 150 kg. The gravitational attraction between them is $8.00\times {10}^{-6}\text{N}$ when their centers are 20 cm apart. What is the mass of each?

40. || INT Today, optical techniques can be used to measure displacements of nm. Suppose a 120 kg lead sphere is suspended from a 55.0-cm-long spring, stretching the spring by 15.0 cm. This sphere is then observed to move an additional 1.60 nm when a second 120 kg sphere is slowly raised from below to where the distance between their centers is 28.0 cm. What value of G is determined from this experiment?

41. || An object of mass m is dropped from height h above a planet of mass M and radius R. Find an expression for the object’s speed as it hits the ground.

42. ||| A projectile is shot straight up from the earth’s surface at a speed of $10\text{,}000\text{k}\text{m}\text{/}\text{h}$. How high does it go?

43. || The unexplored planet Alpha Centauri III has a radius of $7.0\times {10}^6\text{m}$. A visiting astronaut drops a rock, from rest, into a 100-m-deep crevasse. She records that it takes 6.0 s for the rock to reach the bottom. What is the mass of Alpha Centauri III?

44. || Two spherical objects have a combined mass of 400 kg. The gravitational attraction between them is $6.00\times {10}^{-7}\text{N}$ and their gravitational potential energy is $-1.20\times {10}^{-6}\text{J}$. What is the mass of each?

45. || INT Suppose that on earth you can jump straight up a distance of 45 cm. Asteroids are made of material with mass density $2800{\text{kg/m}}^3$. What is the maximum diameter of a spherical asteroid from which you could escape by jumping?

46. || CALC A rogue band of colonists on the moon declares war and prepares to use a catapult to launch large boulders at the earth. Assume that the boulders are launched from the point on the moon nearest the earth. For this problem you can ignore the rotation of the two bodies and the orbiting of the moon.

    1. What is the minimum speed with which a boulder must be launched to reach the earth?

       Hint: The minimum speed is not the escape speed. You need to analyze a three-body system.

    2. Ignoring air resistance, what is the impact speed on earth of a boulder launched at this minimum speed?

47. ||| INT Two spherical asteroids have the same radius R. Asteroid 1 has mass M and asteroid 2 has mass 2M. The two asteroids are released from rest with distance 10R between their centers. What is the speed of each asteroid just before they collide?

    Hint: You will need to use two conservation laws.

48. ||| The two stars in a binary star system have masses $2.0\times {10}^{30}\text{kg}$ and $6.0\times {10}^{30}\text{kg}$. They are separated by $2.0\times {10}^{12}\text{m}$. What are

    1. The system’s rotation period, in years?

    2. The speed of each star?

49. || A starship is circling a distant planet of radius R. The astronauts find that the free-fall acceleration at their altitude is half the value at the planet’s surface. How far above the surface are they orbiting? Your answer will be a multiple of R.

50. ||| A 4000 kg lunar lander is in orbit 50 km above the surface of the moon. It needs to move out to a 300-km-high orbit in order to link up with the mother ship that will take the astronauts home. How much work must the thrusters do?

51. ||| The 75,000 kg space shuttle used to fly in a 250-km-high circular orbit. It needed to reach a 610-km-high circular orbit to service the Hubble Space Telescope. How much energy was required to boost it to the new orbit?

52. || In 2000, NASA placed a satellite in orbit around an asteroid. Consider a spherical asteroid with a mass of $1.0\times {10}^{16}\text{kg}$ and a radius of 8.8 km.

    1. What is the speed of a satellite orbiting 5.0 km above the surface?

    2. What is the escape speed from the asteroid?

53. || Two stationary $5.0\times {10}^{18}\text{kg}$ asteroids are 500 km apart, measured between their centers. A Federation spaceship tries to fly between the asteroids when, as luck would have it, the antimatter reactor blows, releasing $1.5\times {10}^{23}\text{J}$ of energy. 2.0% of that energy goes to the asteroids, with each receiving an equal share, driving them apart. How far apart do they get before gravity pulls them back together?

54. || In 2014, the European Space Agency placed a satellite in orbit around comet 67P/Churyumov-Gerasimenko and then landed a probe on the surface. The actual orbit was elliptical, but we’ll approximate it as a 50-km-diameter circular orbit with a period of 11 days.

    1. What was the satellite’s orbital speed around the comet? (Both the comet and the satellite are orbiting the sun at a much higher speed.)

    2. What is the mass of the comet?

    3. The lander was pushed from the satellite, toward the comet, at a speed of 70 cm/s, and it then fell—taking about 7 hours—to the surface. What was its landing speed? The comet’s shape is irregular, but on average it has a diameter of 3.6 km.

55. || Most galaxies, including our own Milky Way, have supermassive black holes at their centers. Recently, astronomers were able to track one star, named S2, as it orbited the black hole at the center of our galaxy. The star’s actual orbit is elliptical, but we’ll model it as a circular orbit. S2 has a period of 15 years and a radius of 1000 au, where 1 au = 1 astronomical unit is the distance of the earth from the sun.

    1. What is the speed of S2? Give your answer as a fraction of c, where c = the speed of light$=3.0\times {10}^8\text{m}\text{/}\text{s}$.

    2. What is the mass of the black hole? Give your answer as a multiple of $M_{\text{s}}$, where $M_{\text{s}}=$1 solar mass$=2.0\times {10}^{30}\text{kg}$.

56. || Figure 13.17 showed a graph of $\text{log }T$ versus $\text{log }r$ for the planetary data given in Table 13.2. Such a graph is called a log-log graph. The scales in Figure 13.17 are logarithmic, not linear, meaning that each division along the axis corresponds to a factor of 10 increase in the value. Strictly speaking, the “correct” labels on the y-axis should be 7, 8, 9, and 10 because these are the logarithms of ${10}^7\text{,}\dots \text{,}{10}^{10}$.

    1. Consider two quantities u and v that are related by the expression $v^p=C{u}^q$, where C is a constant. The exponents p and q are not necessarily integers. Define $x=\text{log }u$ and $y=\text{log }v$. Find an expression for y in terms of x.

    2. What shape will a graph of y versus x have? Explain.

    3. What slope will a graph of y versus x have? Explain.

    4. Use the experimentally determined “best-fit” line in Figure 13.17 to find the mass of the sun.

57. ||| FIGURE P13.57 shows two planets of mass m orbiting a star of mass M. The planets are in the same orbit, with radius r, but are always at opposite ends of a diameter. Find an exact expression for the orbital period T.

    FIGURE P13.57

    

    FIGURE P13.57 Full Alternative Text

    Hint: Each planet feels two forces.

58. || Large stars can explode as they finish burning their nuclear fuel, causing a supernova. The explosion blows away the outer layers of the star. According to Newton’s third law, the forces that push the outer layers away have reaction forces that are inwardly directed on the core of the star. These forces compress the core and can cause the core to undergo a gravitational collapse. The gravitational forces keep pulling all the matter together tighter and tighter, crushing atoms out of existence. Under these extreme conditions, a proton and an electron can be squeezed together to form a neutron. If the collapse is halted when the neutrons all come into contact with each other, the result is an object called a neutron star, an entire star consisting of solid nuclear matter. Many neutron stars rotate about their axis with a period of $\approx 1\text{s}$ and, as they do so, send out a pulse of electromagnetic waves once a second. These stars were discovered in the 1960s and are called pulsars.

    1. Consider a neutron star with a mass equal to the sun, a radius of 10 km, and a rotation period of 1.0 s. What is the speed of a point on the equator of the star?

    2. What is g at the surface of this neutron star?

    3. A stationary 1.0 kg mass has a weight on earth of 9.8 N. What would be its weight on the star?

    4. How many revolutions per minute are made by a satellite orbiting 1.0 km above the surface?

    5. What is the radius of a geosynchronous orbit?

59. || The solar system is 25,000 light years from the center of our Milky Way galaxy. One light year is the distance light travels in one year at a speed of $3.0\times {10}^8\text{m}\text{/}\text{s}$. Astronomers have determined that the solar system is orbiting the center of the galaxy at a speed of $230\text{km}\text{/}\text{s}$.

    1. Assuming the orbit is circular, what is the period of the solar system’s orbit? Give your answer in years.

    2. Our solar system was formed roughly 5 billion years ago. How many orbits has it completed?

    3. The gravitational force on the solar system is the net force due to all the matter inside our orbit. Most of that matter is concentrated near the center of the galaxy. Assume that the matter has a spherical distribution, like a giant star. What is the approximate mass of the galactic center?

    4. Assume that the sun is a typical star with a typical mass. If galactic matter is made up of stars, approximately how many stars are in the center of the galaxy?

    Astronomers have spent many years trying to determine how many stars there are in the Milky Way. The number of stars seems to be only about 10% of what you found in part d. In other words, about 90% of the mass of the galaxy appears to be in some form other than stars. This is called the dark matter of the universe. No one knows what the dark matter is. This is one of the outstanding scientific questions of our day.

60. || Three stars, each with the mass of our sun, form an equilateral triangle with sides $1.0\times {10}^{12}\text{m}$ long. (This triangle would just about fit within the orbit of Jupiter.) The triangle has to rotate, because otherwise the stars would crash together in the center. What is the period of rotation?

61. || Comets move around the sun in very elliptical orbits. At its closet approach, in 1986, Comet Halley was $8.79\times {10}^7\text{km}$ from the sun and moving with a speed of $54.6\text{k}\text{m}\text{/}\text{s}$. What was the comet’s speed when it crossed Neptune’s orbit in 2006?

62. || A 55,000 kg space capsule is in a 28,000-km-diameter circular orbit around the moon. A brief but intense firing of its engine in the forward direction suddenly decreases its speed by 50%. This causes the space capsule to go into an elliptical orbit. What are the space capsule’s (a) maximum and (b) minimum distances from the center of the moon in its new orbit?

    Hint: You will need to use two conservation laws.

63. $\begin{array}{l}{\displaystyle \frac{(6.67\times {10}^{-11}{\text{ N m}}^{\text{2}}{\text{/kg}}^2)(5.68\times {10}^{26}\text{ kg})}{r^2}}\hfill \\ ={\displaystyle \frac{(6.67\times {10}^{-11}{\text{ N m}}^{\text{2}}{\text{/kg}}^2)(5.97\times {10}^{24}\text{ kg})}{{(6.37\times {10}^6\text{ m)}}^2}}\hfill \end{array}$

64. $\begin{array}{l}{\displaystyle \frac{(6.67\times {10}^{-11}{\text{ N m}}^{\text{2}}{\text{/kg}}^2)(5.97\times {10}^{24}\text{ kg)(1000 kg})}{r^2}}\hfill \\ ={\displaystyle \frac{(1000{\text{ kg)(1997m/s)}}^2}{r}}\hfill \end{array}$

65. $\begin{array}{l}\frac{1}{2}(100\text{ kg})v_2{}^2\hfill \\ -{\displaystyle \frac{(6.67\times {10}^{-11}{\text{ N m}}^{\text{2}}{\text{/kg}}^2)(7.34\times {10}^{22}\text{ kg)(100 kg})}{1.74\times {10}^6\text{ m}}}\hfill \\ =0-{\displaystyle \frac{(6.67\times {10}^{-11}{\text{ N m}}^{\text{2}}{\text{/kg}}^2)(7.34\times {10}^{22}\text{ kg)(100 kg})}{3.48\times {10}^6\text{ m}}}\hfill \end{array}$

66. ||| A rogue black hole with a mass 12 times that of the sun is 250 earth radii from the center of the earth. What is g_eff, the effective free-fall acceleration, at a point on the earth's surface nearest the black hole? The earth itself accelerates, so g_eff is the acceleration of a mass at the surface relative to the earth's acceleration.

67. ||| Two Jupiter-size planets are released from rest $1.0\times {10}^{11}\text{m}$ apart. What are their speeds as they crash together?

68. ||| A satellite in a circular orbit of radius r has period T. A satellite in a nearby orbit with radius $r+\text{Δ}r$, where $\text{Δ}r\ll r$, has the very slightly different period $T+\text{Δ}T$.

    1. Show that

       $${\displaystyle \frac{\text{Δ}T}{T}}={\displaystyle \frac{3}{2}}{\displaystyle \frac{\text{Δ}r}{r}}$$

    2. Two earth satellites are in parallel orbits with radii 6700 km and 6701 km. One day they pass each other, 1 km apart, along a line radially outward from the earth. How long will it be until they are again 1 km apart?

69. ||| While visiting Planet Physics, you toss a rock straight up at $11\text{m}\text{/}\text{s}$ and catch it 2.5 s later. While you visit the surface, your cruise ship orbits at an altitude equal to the planet’s radius every 230 min. What are the (a) mass and (b) radius of Planet Physics?

70. ||| Let’s look in more detail at how a satellite is moved from one circular orbit to another. FIGURE CP13.70 shows two circular orbits, of radii $r_1$ and $r_2$, and an elliptical orbit that connects them. Points 1 and 2 are at the ends of the semimajor axis of the ellipse.

    1. A satellite moving along the elliptical orbit has to satisfy two conservation laws. Use these two laws to prove that the velocities at points 1 and 2 are

       $$v^{\prime }{}_1=\sqrt{{\displaystyle \frac{2GM\left(r_2\text{/}{r}_1\right)}{r_1+r_2}}}\text{and}{v}^{\prime }{}_2=\sqrt{{\displaystyle \frac{2GM\left(r_1\text{/}{r}_2\right)}{r_1+r_2}}}$$

       The prime indicates that these are the velocities on the elliptical orbit. Both reduce to Equation 13.22 if $r_1=r_2=r$.

    2. Consider a 1000 kg communications satellite that needs to be boosted from an orbit 300 km above the earth to a geosynchronous orbit 35,900 km above the earth. Find the velocity $v_1$ on the inner circular orbit and the velocity $v^{\prime }{}_1$ at the low point on the elliptical orbit that spans the two circular orbits.

    3. How much work must the rocket motor do to transfer the satellite from the circular orbit to the elliptical orbit?

    4. Now find the velocity $v^{\prime }{}_2$ at the high point of the elliptical orbit and the velocity $v_2$ of the outer circular orbit.

    5. How much work must the rocket motor do to transfer the satellite from the elliptical orbit to the outer circular orbit?

    6. Compute the total work done and compare your answer to the result of Example 13.6.

    FIGURE CP13.70

    

    FIGURE CP13.70 Full Alternative Text

71. ||| CALC FIGURE CP13.71 shows a particle of mass m at distance x from the center of a very thin cylinder of mass M and length L. The particle is outside the cylinder, so $x>L\text{/}2$.

    1. Calculate the gravitational potential energy of these two masses.

    2. Use what you know about the relationship between force and potential energy to find the magnitude of the gravitational force on m when it is at position x.

    FIGURE CP13.71

    

    FIGURE CP13.71 Full Alternative Text

72. ||| September 2015 saw the historic discovery of gravitational waves, almost exactly 100 years after Einstein predicted their existence as a consequence of his theory of general relativity. Gravitational waves are a literal stretching and compressing of the fabric of space. Even the most sensitive instruments—capable of sensing that the path of a 4-km-long laser beam has lengthened by one-thousandth the diameter of a proton—can detect waves created by only the most extreme cosmic events. The first detection was due to the collision of two black holes more than 750 million light years from earth. Although a full description of gravitational waves requires knowledge of Einstein’s general relativity, a surprising amount can be understood with the physics you’ve already learned.

    1. Consider two equal masses M that interact gravitationally and revolve at angular velocity $\omega$ about their center of mass. Let the distance between them be 2r, so that each is distance r from the center of mass. Apply Newton’s second law for circular motion to one of the masses to find an expression for ${\omega }^2$ (not $\omega )$ in terms of G, M, and r.

    2. The total energy E of the system is the kinetic energy of the two masses, which can be written in terms of ${\omega }^2$, plus their gravitational potential energy. Find an expression for E in terms of G, M, and r. This is a bound system, so the total energy should be negative.

    We’re going to consider two black holes that spiral in until they collide and merge. Because your expression for E is negative, it will become even more negative as $r\to 0$. That is, the system loses energy, and that energy is radiated away as gravitational waves. The details are complex, but we’re interested in only $E_{\text{i}}$, which we can take to be ≈ 0 because initially the black holes are far apart, and $E_{\text{f}}$ at the instant they merge.

    3. A black hole is a mass so dense that nothing, not even light, can escape from it. In other words, the escape speed exceeds the speed of light, designated c. We define the Schwarzchild radius $R_{\text{Sch}}$ to be the distance from the center of a black hole to a point where $v_{\text{escape}}=c$. Find an expression for the Schwarzchild radius of a black hole of mass M. To do this, consider a particle of mass m that is distance R from a much larger mass M. What is the minimum speed the particle needs to escape to infinity? Equate this escape speed to c and solve for $R_{\text{Sch}}$, the distance from the center of the mass at which the escape speed is c.

    4. Two black holes collide and merge when their Schwarzchild radii overlap; that is, they merge when their separation, which we’ve defined as 2r, equals $2R_{\text{Sch}}$. Find an expression for $\text{Δ}E=E_{\text{f}}-E_{\text{i}}$, where $E_{\text{i}}\approx 0$ because initially the black holes are far apart and $E_{\text{f}}$ is their total energy at the instant they merge. This is the energy radiated away as gravitational waves. Your answer will be a fraction of $M{c}^2$, and you probably recognize that this is related to Einstein’s famous $E=m{c}^2$. The quantity $M{c}^2$ is the amount of energy that would be released if an entire star of mass M were suddenly converted entirely to energy.

    5. We still need to determine M, and that’s where the experimental data enter. First, combine your expression for ${\omega }^2$ with your expression for $R_{\text{Sch}}$ to find an equation for the angular velocity ${\omega }_{\text{merge}}$ of two revolving black holes at the instant of merger. Then solve this to get an expression for the mass M in terms of G, c, and ${\omega }_{\text{merge}}$.

    6. FIGURE CP13.72 shows the experimental data from LIGO, the Laser Interferometer Gravitational-Wave Observatory in Hanford, Washington. The graph shows the strain, which is the fractional change in length of a 4-km-long laser beam. Even at the peak, the strain is an astoundingly small 1 part in ${10}^{21}$. The blue curve is the measured strain, while the red curve is the theoretical prediction from general relativity—an almost perfect match. The oscillations correspond to the last few orbits of the two black holes that revolve around their center of mass at angular velocity $\omega$, which is speeding up as they spiral in. Careful measurements of the final revolution before they merge—the peak of the graph—give ${\omega }_{\text{merge}}\approx 9000\text{rpm}$. Using that value, calculate the black hole mass M, giving your answer as a multiple of $M_{\text{s}}$, where $M_{\text{s}}=1\text{ solar mass}=2.0\times {10}^{30}\text{ kg}$.

       FIGURE CP13.72

       

       FIGURE CP13.72 Full Alternative Text

    7. With M known, calculate $R_{\text{Sch}}$ (in km) and the speed v (as a fraction of c) of the black holes as they merged. You should find that there are many solar masses moving at a significant fraction of the speed of light within an area about the size of a major U.S. city. That is certainly an epic cosmic event.

    8. How much energy $E_{\text{grav wave}}$ was radiated away as gravitational waves? Give your answer as a multiple of $M_{\text{s}}{c}^2$. You should find that several solar masses were converted to energy and radiated away as gravitational waves during the $\approx 0.2\text{s}$ duration of this merger.

A full analysis of the situation, using general relativity, found that the gravitational waves were created by the merger of two somewhat unequal black holes, one $29M_{\text{s}}$ and one $36M_{\text{s}}$, and that the radiated gravitational-wave energy was $3M_{\text{s}}{c}^2$. It’s gratifying to see how much of this we can understand from a careful application of introductory physics.