Chapter 13

Cavendish Experiment. In the Cavendish balance apparatus shown in Fig. \(13.4,\) suppose that \(m_{1}=1.10 \mathrm{kg}, m_{2}=\) \(25.0 \mathrm{kg},\) and the rod connecting the \(m_{1}\) pairs is 30.0 \(\mathrm{cm}\) long. If, in each pair, \(m_{1}\) and \(m_{2}\) are 12.0 \(\mathrm{cm}\) apart center to center, find (a) the net force and (b) the net torque (about the rotation axis) on the rotating part of the apparatus. (c) Does it seem that the torque in part (b) would be enough to easily rotate the rod? Suggest some ways to improve the sensitivity of this experiment.

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

An \(8.00-\mathrm{kg}\) point mass and a 15.0 -kg point mass are held in place 50.0 \(\mathrm{cm}\) apart. A particle of mass \(m\) is released from a point between the two masses 20.0 \(\mathrm{cm}\) from the \(8.00-\mathrm{kg}\) mass along the line connecting the two fixed masses. Find the magnitude and direction of the acceleration of the particle.

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

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

At what distance above the surface of the earth is the acceleration due to the earth's gravity 0.980 \(\mathrm{m} / \mathrm{s}^{2}\) if the acceleration due to gravity at the surface has magnitude 9.80 \(\mathrm{m} / \mathrm{s}^{2}\) ?

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

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

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

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

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

Deimos, a moon of Mars, is about 12 \(\mathrm{km}\) in diameter with mass \(2.0 \times 10^{15} \mathrm{kg} .\) Suppose you are stranded alone on Deimos and want to play a one-person game of baseball. You would be the pitcher, and you would be the batter! (a) With what speed would you have to throw a baseball so that it would go into a circular orbit just above the surface and return to you so you could hit it? Do you think you could actually throw it at this speed? (b) How long (in hours) after throwing the ball should you be ready to hit it? Would this be an action-packed baseball game?

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

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

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

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

A thin, uniform rod has length \(L\) and mass \(M . \mathrm{A}\) small uniform sphere of mass \(m\) is placed a distance \(x\) from one end of the rod, along the axis of the rod (Fig. E13.32). (a) Calculate the gravitational potential energy of the rod-sphere system. Take the potential energy to be zero when the rod and sphere are infinitely far apart. Show that your answer reduces to the expected result when \(x\) is much larger than \(L .\) (Hint: Use the power series expansion for \(\ln (1+x)\) given in Appendix B.) Use \(F_{x}=-d U / d x\) to find the magnitude and direction of the gravitational force exerted on the sphere by the rod (see Section 7.4\()\) . Show that your answer reduces to the expected result when \(x\) is much larger than \(L .\)

The acceleration due to gravity at the north pole of Neptune is approximately 10.7 \(\mathrm{m} / \mathrm{s}^{2} .\) Neptune has mass \(1.0 \times 10^{26} \mathrm{kg}\) and radius \(2.5 \times 10^{4} \mathrm{km}\) and rotates once around its axis in about 16 \(\mathrm{h}\) . (a) What is the gravitational force on a 5.0 -kg object at the north pole of Neptune? (b) What is the apparent weight of this same object at Neptune's equator? (Note that Neptune's "surface" is gaseous, not solid, so it is impossible to stand on it.)

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

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

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

A uniform sphere with mass 60.0 \(\mathrm{kg}\) is held with its center at the origin, and a second uniform sphere with mass 80.0 \(\mathrm{kg}\) is held with its center at the point \(x=0, y=3.00 \mathrm{m} .\) (a) What are the magnitude and direction of the net gravitational force due to these objects on a third uniform sphere with mass 0.500 \(\mathrm{kg}\) placed at the point \(x=4.00 \mathrm{m}, y=0 ?\) (b) Where, other than infinitely far away, could the third sphere be placed such that the net gravitational force acting on it from the other two spheres is equal to zero?

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

Submarines on Europa. Some scientists are eager to send a remote-controlled submarine to Jupiter's moon Europa to search for life in its oceans below an icy crust. Europa's mass has been measured to be \(4.8 \times 10^{22}\) kg, its diameter is \(3138 \mathrm{km},\) and it has no appreciable atmosphere. Assume that the layer of ice at the surface is not thick enough to exert substantial force on the water. If the windows of the submarine you are designing are 25.0 \(\mathrm{cm}\) square and can stand a maximum inward force of 9750 N per window, what is the greatest depth to which this submarine can safely dive?

A landing craft with mass \(12,500 \mathrm{kg}\) is in a circular orbit \(5.75 \times 10^{3} \mathrm{m}\) above the surface of a planet. The period of the orbit is 5800 s. The astronauts in the lander measure the diameter of the planet to be \(9.60 \times 10^{6} \mathrm{m} .\) The lander sets down at the north pole of the planet. What is the weight of an 85.6 -kg astronaut as he steps out onto the planet's surface?

What is the escape speed from a 300-km-diameter asteroid with a density of 2500 \(\mathrm{kg} / \mathrm{m}^{3} ?\)

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

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

There are two equations from which a change in the gravitational potential energy \(U\) of the system of a mass \(m\) and the earth can be calculated. One is \(U=m g y(\) Eq. 7.2\() .\) The other is \(U=-G m_{\mathrm{E}} m / r(\mathrm{Eq} .13 .9) .\) As shown in Section \(13.3,\) the first equation is correct only if the gravitational force is a constant over the change in height \(\Delta y .\) The second is always correct. Actually, the gravitational force is never exactly constant over any change in height, but if the variation is small, we can ignore it. Consider the difference in \(U\) between a mass at the earth's surface and a distance \(h\) above it using both equations, and find the value of \(h\) for which Eq. \((7.2)\) is in error by 1\(\% .\) Express this value of \(h\) as a fraction of the earth's radius, and also obtain a numerical value for it.

Your starship, the Aimless Wanderer, lands on the mysterious planet Mongo. As chief scientist-engineer, you make the following measurements: \(A 2.50-\) -kg stone thrown upward from the ground at 12.0 \(\mathrm{m} / \mathrm{s}\) returns to the ground in 6.00 s; the circumference of Mongo at the equator is \(2.00 \times 10^{5} \mathrm{km} ;\) and there is no appreciable atmosphere on Mongo. The starship commander, Captain Confusion, asks for the following information: (a) What is the mass of Mongo? (b) If the Aimless Wanderer goes into a circular orbit \(30,000 \mathrm{km}\) above the surface of Mongo, how many hours will it take the ship to complete one orbit?

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

Falling Hammer. A hammer with mass \(m\) is dropped from rest from a height \(h\) above the earth's surface. This height is not necessarily small compared with the radius \(R_{\mathrm{E}}\) of the earth. If you ignore air resistance, derive an expression for the speed \(v\) of the hammer when it reaches the surface of the earth. Your expression should involve \(h, R_{\mathrm{E}},\) and \(m_{\mathrm{E}},\) the mass of the earth.

Kirkwood Gaps. Hundreds of thousands of asteroids orbit the sun within the asteroid belt, which extends from about \(3 \times 10^{8} \mathrm{km}\) to about \(5 \times 10^{8} \mathrm{km}\) from the sun. (a) Find the orbital period (in years) of (i) an asteroid at the inside of the belt and (ii) an asteroid at the outside of the belt. Assume circular orbits. (b) In 1867 the American astronomer Daniel Kirkwood pointed out that several gaps exist in the asteroid belt where relatively few asteroids are found. It is now understood that these Kirkwood gaps are caused by the gravitational attraction of Jupiter, the largest planet, which orbits the sun once every 11.86 years. As an example, if an asteroid has an orbital period half that of Jupiter, or 5.93 years, on every other orbit this asteroid would be at its closest to Jupiter and feel a strong attraction toward the planet. This attraction, acting over and over on successive orbits, could sweep asteroids out of the Kirkwood gap. Use this hypothesis to determine the orbital radius for this Kirkwood gap. (c) One of several other Kirkwood gaps appears at a distance from the sun where the orbital period is 0.400 that of Jupiter. Explain why this happens, and find the orbital radius for this Kirkwood gap.

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

Binary Star-Different Masses. Two stars, with masses \(M_{1}\) and \(M_{2},\) are in circular orbits around their center of mass. The star with mass \(M_{1}\) has an orbit of radius \(R_{1} ;\) the star with mass \(M_{2}\) has an orbit of radius \(R_{2} .\) (a) Show that the ratio of the orbital radii of the two stars equals the reciprocal of the ratio of their masses - that is, \(R_{1} / R_{2}=M_{2} / M_{1}\) . (b) Explain why the two stars have the same orbital period, and show that the period \(T\) is given by \(T=2 \pi\left(R_{1}+R_{2}\right)^{3 / 2} / \sqrt{G\left(M_{1}+M_{2}\right)}\) . (c) The two stars in a certain binary star system move in circular orbits. The first star, Alpha, has an orbital speed of 36.0 \(\mathrm{km} / \mathrm{s}\) . The second star, Beta, has an orbital speed of 12.0 \(\mathrm{km} / \mathrm{s}\) . The orbital period is 137 \(\mathrm{d}\) . What are the masses of each of the two stars? (d) One of the best candidates for a black hole is found in the binary system called A \(0620-0090 .\) The two objects in the binary system are an orange star, V616 Monocerotis, and a compact object believed to be a black hole (see Fig. 13.27\() .\) The orbital period of \(\mathrm{A} 0620-0090\) is 7.75 hours, the mass of V616 Monocerotis is estimated to be 0.67 times the mass of the sun, and the mass of the black hole is estimated to be 3.8 times the mass of the sun. Assuming that the orbits are circular, find the radius of each object's orbit and the orbital speed of each object. Compare these answers to the orbital radius and orbital speed of the earth in its orbit around the sun.

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

An astronaut is standing at the north pole of a newly discovered, spherically symmetric planet of radius \(R .\) In his hands he holds a container full of a liquid with mass \(m\) and volume \(V .\) At the surface of the liquid, the pressure is \(p_{0} ;\) at a depth \(d\) below the surface, the pressure has a greater value \(p .\) From this information, determine the mass of the planet.

The earth does not have a uniform density; it is most dense at its center and least dense at its surface. An approximation of its density is \(\rho(r)=A-B r,\) where \(A=12,700 \mathrm{kg} / \mathrm{m}^{3}\) and \(B=\) \(1.50 \times 10^{-3} \mathrm{kg} / \mathrm{m}^{4}\) . Use \(R=6.37 \times 10^{6} \mathrm{m}\) for the radius of the earth approximated as a sphere. (a) Geological evidence indicates that the densities are \(13,100 \mathrm{kg} / \mathrm{m}^{3}\) and \(2,400 \mathrm{kg} / \mathrm{m}^{3}\) at the earth's center and surface, respectively. What values does the linear approximation model give for the densities at these two locations? (b) Imagine dividing the earth into concentric, spherical shells. Each shell has radius \(r\) , thickness \(d r\) , volume \(d V=4 \pi r^{2} d r,\) and mass \(d m=\rho(r) d V .\) By integrating from \(r=0\) to \(r=R,\) show that the mass of the earth in this model is \(M=\frac{4}{3} \pi R^{3}\left(A-\frac{3}{4} B R\right)\) (c) Show that the given values of \(A\) and \(B\) give the correct mass of the earth to within 0.4\(\%\) (d) We saw in Section 13.6 that a uniform spherical shell gives no contribution to \(g\) inside it. Show that \(g(r)=\frac{4}{3} \pi G r\left(A-\frac{3}{4} B r\right)\) inside the earth in this model. (e) Verify that the expression of part (d) gives \(g=0\) at the center of the earth and \(g=9.85 \mathrm{m} / \mathrm{s}^{2}\) at the surface. (f) Show that in this model \(g\) does not decrease uniformly with depth but rather has a maximum of \(4 \pi G A^{2} / 9 B=10.01 \mathrm{m} / \mathrm{s}^{2}\) at \(r=2 A / 3 B=5640 \mathrm{km} .\)

The planet Uranus has a radius of \(25,560 \mathrm{km}\) and a surface acceleration due to gravity of 11.1 \(\mathrm{m} / \mathrm{s}^{2}\) at its poles. Its moon Miranda (discovered by Kuiper in 1948 is in a circular orbit about Uranus at an altitude of \(104,000 \mathrm{km}\) above the planet's surface. Miranda has a mass of \(6.6 \times 10^{19} \mathrm{kg}\) and a radius of 235 \(\mathrm{km}\). (a) Calculate the mass of Uranus from the given data. (b) Calculate the magnitude of Miranda's acceleration due to its orbital motion about Uranus. (c) Calculate the acceleration due to Miranda's gravity at the surface of Miranda. (d) Do the answers to parts (b) and (c) mean that an object released 1 \(\mathrm{m}\) above Miranda's surface on the side toward Uranus will fall \(u p\) relative to Miranda? Explain.

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

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

An object in the shape of a thin ring has radius \(a\) and mass \(M .\) A uniform sphere with mass \(m\) and radius \(R\) is placed with its center at a distance \(x\) to the right of the center of the ring, along a line through the center of the ring, and perpendicular to its plane (see Fig. E13.33). What is the gravitational force that the sphere exerts on the ring-shaped object? Show that your result reduces to the expected result when \(x\) is much larger than \(a\).

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