Physics for Scientists and Engineers with Modern Physics

Raymond A. Serway, John W. Jewett, Jr.

Chapter 13

Universal Gravitation - all with Video Answers

Educators

Chapter Questions

Problem 1

Two ocean liners, each with a mass of 40 000 metric tons, are moving on parallel courses 100 m apart. What is the magnitude of the acceleration of one of the liners toward the other due to their mutual gravitational attraction? Model the ships as particles.

Donald Albin

Donald Albin

Numerade Educator

Problem 2

Determine the order of magnitude of the gravitational force that you exert on another person 2 m away. In your solution, state the quantities you measure or estimate and their values.

Donald Albin

Numerade Educator

Problem 3

A 200-kg object and a 500-kg object are separated by 4.00 m. (a) Find the net gravitational force exerted by these objects on a 50.0-kg object placed midway between them. (b) At what position (other than an infinitely remote one) can the 50.0-kg object be placed so as to experience a net force of zero from the other two objects?

Donald Albin

Numerade Educator

Problem 4

During a solar eclipse, the Moon, the Earth, and the Sun all lie on the same line, with the Moon between the Earth and the Sun. (a) What force is exerted by the Sun on the Moon? (b) What force is exerted by the Earth on the Moon? (c) What force is exerted by the Sun on the Earth? (d) Compare the answers to parts (a) and (b). Why doesn’t the Sun capture the Moon away from the Earth?

Donald Albin

Numerade Educator

Problem 5

In introductory physics laboratories, a typical Caven-dish balance for measuring the gravitational constant G uses lead spheres with masses of 1.50 kg and 15.0 g whose centers are separated by about 4.50 cm. Calculate the gravitational force between these spheres, treating each as a particle located at the sphere’s center.

Donald Albin

Numerade Educator

Problem 6

Three uniform spheres of masses $m_{1}=2.00 \mathrm{kg}, m_{2}=$ $4.00 \mathrm{kg},$ and $m_{3}=6.00 \mathrm{kg}$ are placed at the corners of a right triangle as shown in Figure P13.6. Calculate the resultant gravitational force on the object of mass $m_{2},$ assuming the spheres are isolated from the rest of the Universe.

Donald Albin

Numerade Educator

Problem 7

Two objects attract each other with a gravitational force of magnitude $1.00 \times 10^{-8} \mathrm{N}$ when separated by $20.0 \mathrm{cm} .$ If the total mass of the two objects is $5.00 \mathrm{kg},$ what is the mass of each?

Donald Albin

Numerade Educator

Problem 8

Why is the following situation impossible? The centers of two homogeneous spheres are 1.00 $\mathrm{m}$ apart. The spheres are each made of the same element from the periodic table. The gravitational force between the spheres is 1.00 N.

Donald Albin

Numerade Educator

Problem 9

A student proposes to study the gravitational force by suspending two 100.0-kg spherical objects at the lower ends of cables from the ceiling of a tall cathedral and measuring the deflection of the cables from the vertical. The 45.00-m-long cables are attached to the ceiling 1.000 m apart. The first object is suspended, and its position is carefully measured. The second object is suspended, and the two objects attract each other gravitationally. By what distance has the first object moved horizontally from its initial position due to the gravitational attraction to the other object? Suggestion: Keep in mind that this distance will be very small and make appropriate approximations.

Donald Albin

Numerade Educator

Problem 10

When a falling meteoroid is at a distance above the Earth’s surface of 3.00 times the Earth’s radius, what is its acceleration due to the Earth’s gravitation?

Chasen Shaw

Numerade Educator

Problem 11

Miranda, a satellite of Uranus, is shown in Figure P13.11a. It can be modeled as a sphere of radius 242 km and mass $6.68 \times 10^{19} \mathrm{kg} .$ (a) Find the free-fall acceleration on its surface. (b) A cliff on Miranda is 5.00 km high. It appears on the limb at the 11 o’clock position in Figure P13.11a and is magnified in Figure P13.11b. If a devotee of extreme sports runs horizontally off the top of the cliff at 8.50 m/s, for what time interval is he in flight? (c) How far from the base of the vertical cliff does he strike the icy surface of Miranda? (d) What will be his vector impact velocity?

Donald Albin

Numerade Educator

Problem 12

The free-fall acceleration on the surface of the Moon is about one-sixth that on the surface of the Earth. The radius of the Moon is about 0.250$R_{E}\left(R_{E}=\text { Earth's radius }=\right.$ $6.37 \times 10^{6} \mathrm{m}$ ). Find the ratio of their average densities, $\rho_{\text { Moon }} / \rho_{\text { Earth }}$

Donald Albin

Numerade Educator

Problem 13

An artificial satellite circles the Earth in a circular orbit at a location where the acceleration due to gravity is $9.00 \mathrm{m} / \mathrm{s}^{2} .$ Determine the orbital period of the satellite.

Donald Albin

Numerade Educator

Problem 14

Io, a satellite of Jupiter, has an orbital period of 1.77 days and an orbital radius of $4.22 \times 10^{5} \mathrm{km} .$ From these data, determine the mass of Jupiter.

Donald Albin

Numerade Educator

Problem 15

A minimum-energy transfer orbit to an outer planet consists of putting a spacecraft on an elliptical trajectory with the departure planet corresponding to the perihelion of the ellipse, or the closest point to the Sun, and the arrival planet at the aphelion, or the farthest point from the Sun. (a) Use Kepler’s third law to calculate how long it would take to go from Earth to Mars on such an orbit as shown in Figure P13.15. (b) Can such an orbit be undertaken at any time? Explain.

Donald Albin

Numerade Educator

Problem 16

A particle of mass $m$ moves along a straight line with constant velocity $\overrightarrow{\mathbf{v}}_{0}$ in the $x$ direction, a distance $b$ from the $x$ axis (Fig. P13.16). (a) Does the particle possess any angular momentum about the origin? (b) Explain why the amount of its angular momentum should change or should stay constant. (c) Show that Kepler’s second law is satisfied by showing that the two shaded triangles in the figure have the same area when $t_{D}-t_{C}=t_{B}-t_{A}$.

Donald Albin

Numerade Educator

Problem 17

Plaskett’s binary system consists of two stars that revolve in a circular orbit about a center of mass midway between them. This statement implies that the masses of the two stars are equal (Fig. P13.17). Assume the orbital speed of

Donald Albin

Numerade Educator

Problem 18

Comet Halley (Fig. P13.18) approaches the Sun to within 0.570 AU, and its orbital period is 75.6 yr. (AU is the symbol for astronomical unit, where $1 \mathrm{AU}=1.50 \times 10^{11} \mathrm{m}$ is the mean Earth–Sun distance.) How far from the Sun will Halley’s comet travel before it starts its return journey?

Donald Albin

Numerade Educator

Problem 19

Use Kepler’s third law to determine how many days it takes a spacecraft to travel in an elliptical orbit from a point 6 670 km from the Earth’s center to the Moon, 385 000 km from the Earth’s center.

Donald Albin

Numerade Educator

Problem 20

Two planets X and Y travel counterclockwise in circular orbits about a star as shown in Figure P13.20. The radii of their orbits are in the ratio 3:1. At one moment, they are aligned as shown in Figure P13.20a, making a straight line with the star. During the next five years, the angular displacement of planet $\mathrm{X}$ is $90.0^{\circ}$ as shown in Figure $\mathrm{P} 13.20 \mathrm{b}$ . What is the angular displacement of planet $\mathrm{Y}$ at this moment?

Donald Albin

Numerade Educator

Problem 21

A synchronous satellite, which always remains above the same point on a planet’s equator, is put in orbit around Jupiter to study that planet’s famous red spot. Jupiter rotates once every 9.84 h. Use the data of Table 13.2 to find the altitude of the satellite above the surface of the planet.

Donald Albin

Numerade Educator

Problem 22

Neutron stars are extremely dense objects formed from the remnants of supernova explosions. Many rotate very rapidly. Suppose the mass of a certain spherical neutron star is twice the mass of the Sun and its radius is 10.0 km. Determine the greatest possible angular speed it can have so that the matter at the surface of the star on its equator is just held in orbit by the gravitational force.

Donald Albin

Numerade Educator

Problem 23

Suppose the Sun’s gravity were switched off. The planets would leave their orbits and fly away in straight lines as described by Newton’s first law. (a) Would Mercury ever be farther from the Sun than Pluto? (b) If so, find how long it would take Mercury to achieve this passage. If not, give a convincing argument that Pluto is always farther from the Sun than is Mercury.

Donald Albin

Numerade Educator

Problem 24

(a) Given that the period of the Moon’s orbit about the Earth is 27.32 days and the nearly constant distance between the center of the Earth and the center of the Moon is $3.84 \times 10^{8} \mathrm{m},$ use Equation 13.8 to calculate the mass of the Earth. (b) Why is the value you calculate a bit too large?

Donald Albin

Numerade Educator

Problem 25

Three objects of equal mass are located at three corners of a square of edge length $\ell$ as shown in Figure $\mathrm{P} 13.25 .$ Find the magnitude and direction of the gravitational field at the fourth corner due to these objects.

Donald Albin

Numerade Educator

Problem 26

(a) Compute the vector gravitational field at a point P on the perpendicular bisector of the line joining two objects of equal mass separated by a distance 2$a$ as shown in Figure $\mathrm{P} 13.26 .$ (b) Explain physically why the field should approach zero as $r \rightarrow 0 .$ (c) Prove mathematically that the answer to part (a) behaves in this way. (d) Explain physically why the magnitude of the field should approach 2$G M / r^{2}$ as $r \rightarrow \infty$ . (e) Prove mathematically that the answer to part (a) behaves correctly in this limit.

Donald Albin

Numerade Educator

Problem 27

A spacecraft in the shape of a long cylinder has a length of 100 m, and its mass with occupants is 1 000 kg. It has strayed too close to a black hole having a mass 100 times that of the Sun (Fig. P13.27). The nose of the spacecraft points toward the black hole, and the distance between the nose and the center of the black hole is 10.0 km. (a) Determine the total force on the spacecraft. (b) What is the difference in the gravitational fields acting on the occupants in the nose of the ship and on those in the rear of the ship, farthest from the black hole? (This difference in accelerations grows rapidly as the ship approaches the black hole. It puts the body of the ship under extreme tension and eventually tears it apart.)

Donald Albin

Numerade Educator

Problem 28

A satellite in Earth orbit has a mass of 100 $\mathrm{kg}$ and is at an altitude of $2.00 \times 10^{6} \mathrm{m} .$ (a) What is the potential energy of the satellite-Earth system? (b) What is the magnitude of the gravitational force exerted by the Earth on the satellite? (c) What If? What force, if any, does the satellite exert on the Earth?

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 29

How much work is done by the Moon’s gravitational field on a 1 000-kg meteor as it comes in from outer space and impacts on the Moon’s surface?

Donald Albin

Numerade Educator

Problem 30

A system consists of three particles, each of mass 5.00 g, located at the corners of an equilateral triangle with sides of 30.0 cm. (a) Calculate the potential energy of the system. (b) Assume the particles are released simultaneously. Describe the subsequent motion of each. Will any collisions take place? Explain.

Donald Albin

Numerade Educator

Problem 31

After the Sun exhausts its nuclear fuel, its ultimate fate will be to collapse to a white dwarf state. In this state, it would have approximately the same mass as it has now, but its radius would be equal to the radius of the Earth. Calculate (a) the average density of the white dwarf, (b) the surface free-fall acceleration, and (c) the gravitational potential energy associated with a 1.00-kg object at the surface of the white dwarf.

Donald Albin

Numerade Educator

Problem 32

An object is released from rest at an altitude $h$ above the surface of the Earth. (a) Show that its speed at a distance $r$ from the Earth's center, where $R_{\mathrm{E}} \leq r \leq R_{E}+h,$ is
$$v=\sqrt{2 G M_{E}\left(\frac{1}{r}-\frac{1}{R_{E}+h}\right)}$$
(b) Assume the release altitude is 500 km. Perform the integral
$$\Delta t=\int_{i}^{f} d t=-\int_{i}^{f} \frac{d r}{v}$$
to find the time of fall as the object moves from the release point to the Earth’s surface. The negative sign appears because the object is moving opposite to the radial direction, so its speed is $v=-d r / d t .$ Perform the integral numerically.

Donald Albin

Numerade Educator

Problem 33

At the Earth’s surface, a projectile is launched straight up at a speed of 10.0 km/s. To what height will it rise? Ignore air resistance and the rotation of the Earth.

Donald Albin

Numerade Educator

Problem 34

A space probe is fired as a projectile from the Earth’s surface with an initial speed of $2.00 \times 10^{4} \mathrm{m} / \mathrm{s}$ . What will its speed be when it is very far from the Earth? Ignore atmospheric friction and the rotation of the Earth.

Donald Albin

Numerade Educator

Problem 35

A 1 000-kg satellite orbits the Earth at a constant altitude of 100 km. (a) How much energy must be added to the system to move the satellite into a circular orbit with altitude 200 km? What are the changes in the system’s (b) kinetic energy and (c) potential energy?

Donald Albin

Numerade Educator

Problem 36

A. “treetop satellite” moves in a circular orbit just above the surface of a planet, assumed to offer no air resistance. Show that its orbital speed $v$ and the escape speed from the planet are related by the expression $v_{\mathrm{esc}}=\sqrt{2} v.$

Donald Albin

Numerade Educator

Problem 37

A comet of mass $1.20 \times 10^{10} \mathrm{kg}$ moves in an elliptical orbit around the Sun. Its distance from the Sun ranges between 0.500 $\mathrm{AU}$ and 50.0 $\mathrm{AU}$ . ( a) What is the eccentricity of its orbit? (b) What is its period? (c) At aphelion, what is the potential energy of the comet-Sun system? Note: $1 \mathrm{AU}=$ one astronomical unit $=$ the average distance from the Sun to the Earth $=1.496 \times 10^{11} \mathrm{m} .$

Donald Albin

Numerade Educator

Problem 38

(a) What is the minimum speed, relative to the Sun, necessary for a spacecraft to escape the solar system if it starts at the Earth's orbit? (b) Voyager 1 achieved a maximum speed of 125000 $\mathrm{km} / \mathrm{h}$ on its way to photograph Jupiter. Beyond what distance from the Sun is this speed sufficient to escape the solar system?

Donald Albin

Numerade Educator

Problem 39

A satellite of mass 200 kg is placed into Earth orbit at a height of 200 km above the surface. (a) Assuming a circular orbit, how long does the satellite take to complete one orbit? (b) What is the satellite’s speed? (c) Starting from the satellite on the Earth’s surface, what is the minimum energy input necessary to place this satellite in orbit? Ignore air resistance but include the effect of the planet’s daily rotation.

Donald Albin

Numerade Educator

Problem 40

A satellite of mass $m,$ originally on the surface of the Earth, is placed into Earth orbit at an altitude $h .$ (a) Assuming a circular orbit, how long does the satellite take to complete one orbit? (b) What is the satellite's speed? (c) What is the minimum energy input necessary to place this satellite in orbit? Ignore air resistance but include the effect of the planet's daily rotation. Represent the mass and radius of the Earth as $M_{E}$ and $R_{E},$ respectively.

Donald Albin

Numerade Educator

Problem 41

Ganymede is the largest of Jupiter’s moons. Consider a rocket on the surface of Ganymede, at the point farthest from the planet (Fig. P13.41). Model the rocket as a particle. (a) Does the presence of Ganymede make Jupiter exert a larger, smaller, or same size force on the rocket compared with the force it would exert if Ganymede were not interposed? (b) Determine the escape speed for the rocket from the planet–satellite system. The radius of Ganymede is $2.64 \times 10^{6} \mathrm{m},$ and its mass is $1.495 \times 10^{23} \mathrm{kg} .$ The distance between Jupiter and Ganymede is $1.071 \times 10^{9} \mathrm{m},$ and the mass of Jupiter is $1.90 \times 10^{27} \mathrm{kg}$ . Ignore the motion of Jupiter and Ganymede as they revolve about their center of mass.

Donald Albin

Numerade Educator

Problem 42

A satellite moves around the Earth in a circular orbit of radius $r$ (a) What is the speed $v_{i}$ of the satellite? (b) Suddenly, an explosion breaks the satellite into two pieces, with masses $m$ and 4$m$ . Immediately after the explosion, the smaller piece of mass $m$ is stationary with respect to the Earth and falls directly toward the Earth. What is the speed $v$ of the larger piece immediately the explosion? (c) Because of the increase in its speed, this larger piece now moves in a new elliptical orbit. Find its distance away from the center of the Earth when it reaches the other end of the ellipse.

Donald Albin

Numerade Educator

Problem 43

A cylindrical habitat in space 6.00 km in diameter and 30.0 km long has been proposed (by G. K. O’Neill, 1974). Such a habitat would have cities, land, and lakes on the inside surface and air and clouds in the center. They would all be held in place by rotation of the cylinder about its long axis. How fast would the cylinder have to rotate to imitate the Earth’s gravitational field at the walls of the cylinder?

Donald Albin

Numerade Educator

Problem 44

A rocket is fired straight up through the atmosphere from the South Pole, burning out at an altitude of 250 km when traveling at 6.00 km/s. (a) What maximum distance from the Earth’s surface does it travel before falling back to the Earth? (b) Would its maximum distance from the surface be larger if the same rocket were fired with the same fuel load from a launch site on the equator? Why or why not?

Donald Albin

Numerade Educator

Problem 45

Let $\Delta g_{M}$ represent the difference in the gravitational fields produced by the Moon at the points on the Earth's surface nearest to and farthest from the Moon. Find the fraction $\Delta g_{M} / g,$ where $g$ is the Earth's gravitational field. (This difference is responsible for the occurrence of the lunar tides on the Earth.)

Donald Albin

Numerade Educator

Problem 46

Why is the following situation impossible? A spacecraft is launched into a circular orbit around the Earth and circles the Earth once an hour.

Donald Albin

Numerade Educator

Problem 47

(a) A space vehicle is launched vertically upward from the Earth’s surface with an initial speed of 8.76 km/s, which is less than the escape speed of 11.2 km/s. What maximum height does it attain? (b) A meteoroid falls toward the Earth. It is essentially at rest with respect to the Earth when it is at a height of $2.51 \times 10^{7} \mathrm{m}$ above the Earth's surface. With what speed does the meteorite (a meteoroid that survives to impact the Earth's surface) strike the Earth?

Donald Albin

Numerade Educator

Problem 48

(a) A space vehicle is launched vertically upward from the Earth's surface with an initial speed of $v_{i}$ that is comparable to but less than the escape speed $v_{\text { esc }}$ . What maximum height does it attain? (b) A meteoroid falls toward the Earth. It is essentially at rest with respect to the Earth when it is at a height $h$ above the Earth's surface. With what speed does the meteorite (a meteoroid that survives to impact the Earth’s surface) strike the Earth? (c) What If? Assume a baseball is tossed up with an initial speed that is very small compared to the escape speed. Show that the result from part (a) is consistent with Equation 4.12.

Donald Albin

Numerade Educator

Problem 49

Assume you are agile enough to run across a horizontal surface at 8.50 m/s, independently of the value of the gravitational field. What would be (a) the radius and (b) the mass of an airless spherical asteroid of uniform density $1.10 \times 10^{3} \mathrm{kg} / \mathrm{m}^{3}$ on which you could launch yourself into orbit by running? (c) What would be your period? (d) Would your running significantly affect the rotation of the asteroid? Explain.

Donald Albin

Numerade Educator

Problem 50

A sleeping area for a long space voyage consists of two cabins each connected by a cable to a central hub as shown in Figure P13.50. The cabins are set spinning around the hub axis, which is connected to the rest of the spacecraft to generate artificial gravity in the cabins. A space traveler lies in a bed parallel to the outer wall as shown in Figure P13.50. (a) With $r=10.0 \mathrm{m},$ what would the angular speed of the 60.0 -kg traveler need to be if he is to experience half his normal Earth weight? (b) If the astronaut stands up perpendicular to the bed, without holding on to anything with his hands, will his head be moving at a faster, a slower, or the same tangential speed as his feet? Why? (c) Why is the action in part (b) dangerous?

Donald Albin

Numerade Educator

Problem 51

Two hypothetical planets of masses $m_{1}$ and $m_{2}$ and radii $r_{1}$ and $r_{2},$ respectively, are nearly at rest when they are an infinite distance apart. Because of their gravitational attraction, they head toward each other on a collision course. (a) When their center-to-center separation is $d$ , find expressions for the speed of each planet and for their relative speed. (b) Find the kinetic energy of each planet just before they collide, taking $m_{1}=2.00 \times 10^{24} \mathrm{kg}, m_{2}=$ 8.00 $\times 10^{24} \mathrm{kg}, r_{1}=3.00 \times 10^{6} \mathrm{m},$ and $r_{2}=5.00 \times 10^{6} \mathrm{m} .$ Note: Both the energy and momentum of the isolated two-planet system are constant.

Donald Albin

Numerade Educator

Problem 52

Two spheres having masses M and 2M and radii R and 3R, respectively, are simultaneously released from rest when the distance between their centers is 12R. Assume the two spheres interact only with each other and we wish to find the speeds with which they collide. (a) What two isolated system models are appropriate for this system? (b) Write an equation from one of the models and solve it for $\overrightarrow{\mathbf{v}}_{1},$ the velocity of the sphere of mass $M$ at any time after release in terms of $\overrightarrow{\mathbf{v}}_{2},$ the velocity of $2 M .$ (c) Write an equation from the other model and solve it for speed $v_{1}$ in terms of speed $v_{2}$ when the spheres collide. (d) Combine the two equations to find the two speeds $v_{1}$ and $v_{2}$ when the spheres collide.

Donald Albin

Numerade Educator

Problem 53

A ring of matter is a familiar structure in planetary and stellar astronomy. Examples include Saturn’s rings and a ring nebula. Consider a uniform ring of mass $2.36 \times$ $10^{20} \mathrm{kg}$ and radius $1.00 \times 10^{8} \mathrm{m} .$ An object of mass 1000 $\mathrm{kg}$ is placed at a point $A$ on the axis of the ring, $2.00 \times 10^{8} \mathrm{m}$ from the center of the ring (Fig. Pl3.53). When the object is released, the attraction of the ring makes the object move along the axis toward the center of the ring (point B). (a) Calculate the gravitational potential energy of the object–ring system when the object is at A. (b) Calculate the gravitational potential energy of the system when the object is at B. (c) Calculate the speed of the object as it passes through B.

Donald Albin

Numerade Educator

Problem 54

(a) Show that the rate of change of the free-fall acceleration with vertical position near the Earth’s surface is
$$\frac{d g}{d r}=-\frac{2 G M_{E}}{R_{E}^{3}}$$
This rate of change with position is called a gradient. (b) Assuming $h$ is small in comparison to the radius of the Earth, show that the difference in free-fall acceleration between two points separated by vertical distance $h$ is
$$|\Delta g|=\frac{2 G M_{E} h}{R_{E}^{3}}$$
(c) Evaluate this difference for $h=6.00 \mathrm{m},$ a typical height for a two-story building.

Donald Albin

Numerade Educator

Problem 55

As an astronaut, you observe a small planet to be spherical. After landing on the planet, you set off, walking always straight ahead, and find yourself returning to your spacecraft from the opposite side after completing a lap of 25.0 km. You hold a hammer and a falcon feather at a height of 1.40 m, release them, and observe that they fall together to the surface in 29.2 s. Determine the mass of the planet.

Donald Albin

Numerade Educator

Problem 56

A certain quaternary star system consists of three stars, each of mass m, moving in the same circular orbit of radius r about a central star of mass M. The stars orbit in the same sense and are positioned one-third of a revolution apart from one another. Show that the period of each of the three stars is given by
$$T=2 \pi \sqrt{\frac{r^{3}}{G(M+m / \sqrt{3})}}$$

Donald Albin

Numerade Educator

Problem 57

Studies of the relationship of the Sun to our galaxy—the Milky Way—have revealed that the Sun is located near the outer edge of the galactic disc, about 30000 ly $(1 \text { ly }=$ $9.46 \times 10^{15} \mathrm{m} )$ from the center. The Sun has an orbital speed of approximately 250 $\mathrm{km} / \mathrm{s}$ around the galactic center. (a) What is the period of the Sun's galactic motion? (b) What is the order of magnitude of the mass of the Milky Way galaxy? (c) Suppose the galaxy is made mostly of stars of which the Sun is typical. What is the order of magnitude of the number of stars in the Milky Way?

Donald Albin

Numerade Educator

Problem 58

Two identical hard spheres, each of mass m and radius r, are released from rest in otherwise empty space with their centers separated by the distance R. They are allowed to collide under the influence of their gravitational attraction. (a) Show that the magnitude of the impulse received by each sphere before they make contact is given by $\left[G m^{3}(1 / 2 r-1 / R)\right]^{1 / 2} .$ (b) What If? Find the
magnitude of the impulse each receives during their contact if they collide elastically.

Donald Albin

Numerade Educator

Problem 59

The maximum distance from the Earth to the Sun (at aphelion) is $1.521 \times 10^{11} \mathrm{m},$ and the distance of closest approach (at perihelion) is $1.471 \times 10^{11} \mathrm{m} .$ The Earth's orbital speed at perihelion is $3.027 \times 10^{4} \mathrm{m} / \mathrm{s}$ . Determine (a) the Earth's orbital speed at aphelion and the kinetic and potential energies of the Earth-Sun system (b) at perihelion, and $(\mathrm{c})$ at aphelion. (d) Is the total energy of the system constant? Explain. Ignore the effect of the Moon and other planets.

Donald Albin

Numerade Educator

Problem 60

Many people assume air resistance acting on a moving object will always make the object slow down. It can, however, actually be responsible for making the object speed up. Consider a 100-kg Earth satellite in a circular orbit at an altitude of 200 km. A small force of air resistance makes the satellite drop into a circular orbit with an altitude of 100 km. (a) Calculate the satellite’s initial speed. (b) Calculate its final speed in this process. (c) Calculate the initial energy of the satellite–Earth system. (d) Calculate the final energy of the system. (e) Show that the system has lost mechanical energy and find the amount of the loss due to friction. (f) What force makes the satellite’s speed increase? Hint: You will find a free-body diagram useful in explaining your answer.

Donald Albin

Numerade Educator

Problem 61

X-ray pulses from Cygnus X-1, the first black hole to be identified and a celestial x-ray source, have been recorded during high-altitude rocket flights. The signals can be interpreted as originating when a blob of ionized matter orbits a black hole with a period of 5.0 ms. If the blob is in a circular orbit about a black hole whose mass is $20 M_{\mathrm{Sun}},$ what is the orbit radius?

Donald Albin

Numerade Educator

Problem 62

Show that the minimum period for a satellite in orbit around a spherical planet of uniform density $\rho$ is
$$T_{\min }=\sqrt{\frac{3 \pi}{G \rho}}$$
independent of the planet’s radius.

Donald Albin

Numerade Educator

Problem 63

Astronomers detect a distant meteoroid moving along a straight line that, if extended, would pass at a distance 3$R_{E}$ from the center of the Earth, where $R_{E}$ is the Earth's radius. What minimum speed must the meteoroid have if it is not to collide with the Earth?

Donald Albin

Numerade Educator

Problem 64

Two stars of masses $M$ and $m,$ separated by a distance $d,$ revolve in circular orbits about their center of mass (Fig. Pl3.64). Show that each star has a period given by $$T^{2}=\frac{4 \pi^{2} d^{3}}{G(M+m)}$$

Donald Albin

Numerade Educator

Problem 65

Two identical particles, each of mass 1000 $\mathrm{kg}$ , are coasting in free space along the same path, one in front of the other by 20.0 $\mathrm{m}$ . At the instant their separation distance has this value, each particle has precisely the same velocity of 800 $\mathrm{i} \mathrm{m} / \mathrm{s}$ . What are their precise velocities when they are 2.00 $\mathrm{m}$ apart?

Donald Albin

Numerade Educator

Problem 66

Consider an object of mass $m,$ not necessarily small compared with the mass of the Earth, released at a distance of $1.20 \times 10^{7} \mathrm{m}$ from the center of the Earth. Assume the Earth and the object behave as a pair of particles, isolated from the rest of the Universe. (a) Find the magnitude of the acceleration $a_{\text { rel }}$ with which each starts to move relative to the other as a function of $m .$ Evaluate the acceleration (b) for $m=5.00 \mathrm{kg},(\mathrm{c})$ for $m=2000 \mathrm{kg}$ , and $(\mathrm{d})$ for $m=$ $2.00 \times 10^{24} \mathrm{kg} .$ (e) Describe the pattern of variation of $a_{\mathrm{rel}}$ with $m .$

Donald Albin

Numerade Educator

Problem 67

As thermonuclear fusion proceeds in its core, the Sun loses mass at a rate of $3.64 \times 10^{9} \mathrm{kg} / \mathrm{s}$ . During the 5000-yr period of recorded history, by how much has the length of the year changed due to the loss of mass from the Sun? Suggestions: Assume the Earth's orbit is circular. No external torque acts on the Earth-Sun system, so the angular momentum of the Earth is constant.

Donald Albin

Numerade Educator

Problem 68

The Solar and Heliospheric Observatory (SOHO) spacecraft has a special orbit, located between the Earth and the Sun along the line joining them, and it is always close enough to the Earth to transmit data easily. Both objects exert gravitational forces on the observatory. It moves around the Sun in a near-circular orbit that is smaller than the Earth’s circular orbit. Its period, however, is not less than 1 yr but just equal to 1 yr. Show that its distance from the Earth must be $1.48 \times 10^{9} \mathrm{m} .$ In $1772,$ Joseph Louis Lagrange determined theoretically the special location allowing this orbit. Suggestions: Use data that are precise to four digits. The mass of the Earth is $5.974 \times 10^{24} \mathrm{kg} .$ You will not be able to easily solve the equation you generate; instead, use a computer to verify that $1.48 \times 10^{9} \mathrm{m}$ is the correct value.

Donald Albin

Numerade Educator

Problem 69

The oldest artificial satellite still in orbit is Vanguard I, launched March 3, 1958. Its mass is 1.60 kg. Neglecting atmospheric drag, the satellite would still be in its initial orbit, with a minimum distance from the center of the Earth of 7.02 Mm and a speed at this perigee point of 8.23 km/s. For this orbit, find (a) the total energy of the satellite–Earth system and (b) the magnitude of the angular momentum of the satellite. (c) At apogee, find the satellite’s speed and its distance from the center of the Earth. (d) Find the semimajor axis of its orbit. (e) Determine its period.

Donald Albin

Numerade Educator