9 781292 021034

ISBN 978-1-29202-103-4

Essential University PhysicsVolume 1

Richard WolfsonSecond Edition

Essential University Physics Volum

e 1 Wolfson 2e

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ISBN 10: 1-292-02103-9ISBN 13: 978-1-292-02103-4

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CHAPTER SUMMARY

Big Picture

Applications

Key Concepts and Equations

The big idea here is universal gravitation—an attractive force that acts between all matterwith a strength that depends directly on the product of two interacting masses and inverselyon the square of the distance between them. Gravitation is responsible for the familiarbehavior of falling objects and also for the orbits of planets and satellites. Depending onenergy, orbits may be closed (elliptical/circular) or open (hyperbolic/parabolic).

Mathematically, Newton’s law of universal gravitation describes the attractive force F between twomasses and located a distance r apart:

F 5Gm1 m2

r2 1universal gravitation2m2m1

Because the strength of gravity varies with distance, potential-energy changes over large distancesaren’t just a product of force and distance. Integration shows that the potential energy change involved in moving a mass m originally a distance from the center of a mass M to a distance is

With gravity, it’s convenient to choose the zero of potential energy at infinity; then

for the potential energy of a mass m located a distance r from the center of a mass M.

U 5 2

GMm

r 1potential energy, U 5 0 at infinity2

DU 5 GMma 1r1

21r2b 1change in potential energy2

r2r1

DU

A total energy—kinetic plus potential—of zero marks the dividingline between closed and open orbits. An object located a distance rfrom a gravitating mass M must have at least the escape speed toachieve an open orbit and escape M’s vicinity forever:

vesc 5A

2GM

r

Circular orbits are readily analyzed using Newton’s laws and conceptsfrom circular motion. A circular orbit of radius r about a mass M has aperiod given by

Its kinetic and potential energies are related by Totalenergy is negative, as appropriate for a closed orbit, and the objectactually moves faster the lower its total energy.

U 5 22K.

T2 54p2

r3

GM

The gravitational field conceptprovides a way to describe gravitythat avoids the troublesome action-at-a-distance. A gravitating masscreates a field in the space aroundit, and a second mass responds tothe field in its immediate vicinity.

A special orbit is the geosynchronous orbit, parallel to Earth’s equa-tor at an altitude of about 36,000 km. Here the orbital period is 24 h, soa satellite in geosynchronous orbit appears from Earth’s surface tobe fixed in the sky. TV, communications, and weather satellites usegeosynchronous orbits.

Distance, r

EarthSatellite

U 5 0

rPote

ntia

l ene

rgy,

U

It would take thismuch energy forthe satellite toescape infinitelyfar from Earth.

The satellite’s potentialenergy is negative.

Gravity governsboth the fallingapple and theorbiting Moon.

Open(hyperbola)

Borderline(parabola)

Closed(ellipse/circle)

r

m1 m2Fr

Fr

Gravitational field

Force arises from fieldat Moon’s location.

Fr

This equation applies to point masses of negligible size and to spherically symmetric masses of any size. It’s an excellent approximation for anyobjects whose size is much smaller than their separation. In all cases, r is measured from the centers of the gravitating objects.

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Answers to Chapter Questions

Answer to Chapter Opening Question

The satellite orbits Earth in 24 hours, so from Earth’s surface itappears at a fixed position in the sky.

Answers to GOT IT? Questions

1. (d) Quadrupled. If the original distance were r, the original forcewould be proportional to At half that distance, the force isproportional to 1/1r/222 5 4/r2.

1/r2.

Gravity

2.3. (a), (c), and (d). Since B has higher total energy, it must have

lower kinetic energy and is therefore moving slower. B is fartherfrom the gravitating body, so its potential energy is higher—stillnegative, but less so than A’s. For circular orbits, the ratio of po-tential energy to total energy is always the same—namely,U 5 2E.

Hyperbolic . parabolic . elliptical . circular.

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For Thought and Discussion1. What do Newton’s apple and the Moon have in common?2. Explain the difference between G and g.3. When you stand on Earth, the distance between you and Earth is

zero. So why isn’t the gravitational force infinite?4. The force of gravity on an object is proportional to the object’s

mass, yet all objects fall with the same gravitational acceleration.Why?

5. A friend who knows nothing about physics asks what keeps anorbiting satellite from falling to Earth. Give an answer that willsatisfy your friend.

6. Could you put a satellite in an orbit that keeps it stationary overthe south pole? Explain.

7. Why are satellites generally launched eastward and from low latitudes? (Hint: Think about Earth’s rotation.)

8. Given Earth’s mass, the Moon’s distance and orbital period, andthe value of G, could you calculate the Moon’s mass? If yes,how? If no, why not?

9. How should a satellite be launched so that its orbit takes it overevery point on the (rotating) Earth?

10. Does the gravitational force of the Sun do work on a planet in acircular orbit? In an elliptical orbit? Explain.

Exercises and ProblemsExercises

Section 2 Universal Gravitation

11. Space explorers land on a planet that has the same mass as Earth,but find they weigh twice as much as they would on Earth. What’sthe planet’s radius?

12. Use data for the Moon’s orbit from Appendix: Astrophysical Datato compute the Moon’s acceleration in its circular orbit, and ver-ify that the result is consistent with Newton’s law of gravitation.

13. To what fraction of its current radius would Earth have to shrink(with no change in mass) for the gravitational acceleration at itssurface to triple?

14. Calculate the gravitational acceleration at the surface of (a) Mer-cury and (b) Saturn’s moon Titan.

15. Two identical lead spheres with their centers 14 cm apart attracteach other with a force. Find their mass.

16. What’s the approximate value of the gravitational force betweena 67-kg astronaut and a 73,000-kg spacecraft when they’re 84 mapart?

17. A sensitive gravimeter is carried to the top of Chicago’s Willis(formerly Sears) Tower, where its reading for the acceleration ofgravity is lower than at street level. Find the build-ing’s height.

1.36 mm/s2

0.25-mN

Section 3 Orbital Motion

18. At what altitude will a satellite complete a circular orbit of Earthin 2.0 h?

19. Find the speed of a satellite in geosynchronous orbit.20. Mars’s orbit has a diameter 1.52 times that of Earth’s orbit. How

long does it take Mars to orbit the Sun?21. Calculate the orbital period for Jupiter’s moon Io, which orbits

from the planet’s center.22. An astronaut hits a golf ball horizontally from the top of a lunar

mountain so fast that it goes into circular orbit. What’s its orbitalperiod?

23. The Mars Reconnaissance Orbiter circles the red planet with a112-min period. What’s the spacecraft’s altitude?

Section 4 Gravitational Energy

24. Earth’s distance from the Sun varies from 147 Gm at perihelionto 152 Gm at aphelion because its orbit isn’t quite circular. Findthe change in potential energy as Earth goes from perihelion toaphelion.

25. So-called suborbital missions take scientific instruments intospace for brief periods without the expense of getting into orbit;their trajectories are often simple “up and down” vertical paths.How much energy does it take to launch a 230-kg instrument ona vertical trajectory that peaks at 1800 km altitude?

26. A rocket is launched vertically upward from Earth’s surface at5.1 km/s. What’s its maximum altitude?

27. What vertical launch speed is necessary to get a rocket to an alti-tude of 1100 km?

28. Find the energy necessary to put 1 kg, initially at rest on Earth’ssurface, into geosynchronous orbit.

29. What’s the total mechanical energy associated with Earth’s orbital motion?

30. The escape speed from a planet of mass is 7.1 km/s. Find the planet’s radius.

31. Determine escape speeds from (a) Jupiter’s moon Callisto and(b) a neutron star, with the Sun’s mass crammed into a sphere ofradius 6.0 km.

32. To what radius would Earth have to shrink, with no change inmass, for escape speed at its surface to be 30 km/s?

Problems

33. The gravitational acceleration at a planet’s surface is Find the acceleration at an altitude equal to half the planet’sradius.

34. One of the longest-standing athletic records is Cuban JavierSotomayor’s 2.45-m high jump. How high could Sotomayorjump on (a) Mars and (b) Earth’s Moon?

35. You’re the navigator on a spaceship studying an unexploredplanet. Your ship has just gone into a circular orbit around theplanet, and you determine that the gravitational acceleration at

22.5 m/s2.

2.9 3 1024 kg

4.223105 km

For homework assigned on MasteringPhysics, go to www.masteringphysics.com Biology and/or medical-related problems C Computer problemsBIO

BIO

GravityProblem Set

From Chapter 8 of Essential University Physics, Second Edition, Richard Wolfson. Copyright © 2012 by Pearson Education, Inc. Published byPearson Addison-Wesley. All rights reserved.

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your orbital altitude is half what it would be at the surface. Whatdo you report for your altitude, in terms of the planet’s radius?

36. If you’re standing on the ground 15 m directly below the centerof a spherical water tank containing of water, by whatfraction is your weight reduced due to the water’s gravitationalattraction?

37. Given the Moon’s orbital radius of 384,400 km and period of27.3 days, calculate its acceleration in its circular orbit, andcompare with the acceleration of gravity at Earth’s surface.Show that the Moon’s acceleration is lower by the ratio of thesquare of Earth’s radius to the square of the Moon’s orbitalradius, thus confirming the inverse-square law for the gravita-tional force.

38. The equation relates force to the derivative of

potential energy. Use this fact to differentiate Equation 6,

, for gravitational potential energy, and show

that you recover Newton’s law of gravitation.

39. During the Apollo Moon landings, one astronaut remained withthe command module in lunar orbit, about 130 km above the sur-face. For half of each orbit, this astronaut was completely cut offfrom the rest of humanity as the spacecraft rounded the far sideof the Moon. How long did this period last?

40. A white dwarf is a collapsed star with roughly the Sun’s masscompressed into the size of Earth. What would be (a) the orbitalspeed and (b) the orbital period for a spaceship in orbit just abovethe surface of a white dwarf?

41. Given that our Sun orbits the galaxy with a period of 200 My at from the galactic center, estimate the galaxy’smass. Assume (incorrectly) that the galaxy is essentially spher-ical and that most of its mass lies interior to the Sun’s orbit.

42. You’re preparing an exhibit for the Golf Hall of Fame, and yourealize that the longest golf shot in history was Astronaut AlanShepard’s lunar drive. Shepard, swinging single-handed with agolf club attached to a lunar sample scoop, claimed his ball went“miles and miles.” The record for a single-handed golf shot onEarth is 257 m. Could Shepard’s ball really have gone “miles andmiles”? Assume the ball’s initial speed is independent of gravita-tional acceleration.

43. Exact solutions for gravitational problems involving more thantwo bodies are notoriously difficult. One solvable problem in-volves a configuration of three equal-mass objects spaced in anequilateral triangle. Forces due to their mutual gravitation causethe configuration to rotate. Suppose three identical stars, each of mass M, form a triangle of side L. Find an expression for theperiod of their orbital motion.

44. Satellites A and B are in circular orbits, with A twice as far fromEarth’s center as B. How do their orbital periods compare?

45. The asteroid Pasachoff orbits the Sun with period 1417 days.Find the semimajor axis of its orbit from Kepler’s third law. UseEarth’s orbital radius and period, respectively, as your units ofdistance and time.

46. We still don’t have a permanent solution for the disposal of ra-dioactive waste. As a nuclear waste specialist with the Depart-ment of Energy, you’re asked to evaluate a proposal to shootwaste canisters into the Sun. You need to report the speed atwhich a canister, dropped from rest in the vicinity of Earth’s orbit, would hit the Sun. What’s your answer?

2.6 31020 m

U1r2 5 2

GMm

r

Fx 5 2dU

dx

4 3 106 kg

Gravity: Problem Set

8500 km

FIGURE 15 Problem 52

47. At perihelion in February 1986, Comet Halley was from the Sun and was moving at 54.6 km/s. What was Halley’sspeed when it crossed Neptune’s orbit in 2006?

48. Neglecting air resistance, to what height would you have to fire arocket for the constant-acceleration equations to give a height inerror by 1%? Would those equations overestimate or underesti-mate the height?

49. Show that an object released from rest very far from Earthreaches Earth’s surface at essentially escape speed.

50. By what factor must an object’s speed in circular orbit be in-creased to reach escape speed from its orbital altitude?

51. You’re in charge of tracking celestial objects that might pose adanger to Earth. Astronomers have discovered a new comet that’smoving at 45 km/s as it crosses Earth’s orbit. Determine whetherthe comet will again return to Earth’s vicinity.

52. Two meteoroids are 250,000 km from Earth’s center and mov-ing at 2.1 km/s. One is headed straight for Earth, while theother is on a path that will come within 8500 km of Earth’scenter (Fig. 15). Find the speed (a) of the first meteoroid whenit strikes Earth and (b) of the second meteoroid at its closestapproach. (c) Will the second meteoroid ever return to Earth’svicinity?

8.79 3107 km

53. Neglecting Earth’s rotation, show that the energy needed tolaunch a satellite of mass m into circular orbit at altitude h is

54. A projectile is launched vertically upward from a planet of massM and radius R; its initial speed is twice the escape speed. Derivean expression for its speed as a function of the distance r fromthe planet’s center.

55. A spacecraft is in circular orbit 5500 km above Earth’s surface.How much will its altitude decrease if it moves to a new circularorbit where (a) its orbital speed is 10% higher or (b) its orbitalperiod is 10% shorter?

56. Two meteoroids are 160,000 km from Earth’s center and headingstraight toward Earth, one at 10 km/s, the other at 20 km/s. Atwhat speeds will they strike Earth?

57. Two rockets are launched from Earth’s surface, one at 12 km/sand the other at 18 km/s. How fast is each moving when itcrosses the Moon’s orbit?

58. A satellite is in an elliptical orbit at altitudes ranging from 230 to890 km. At its highest point, it’s moving at 7.23 km/s. How fastis it moving at its lowest point?

59. A missile’s trajectory takes it to a maximum altitude of 1200 km.If its launch speed is 6.1 km/s, how fast is it moving at the peakof its trajectory?

60. A 720-kg spacecraft has total energy TJ and is in circularorbit around the Sun. Find (a) its orbital radius, (b) its kinetic en-ergy, and (c) its speed.

20.53

aGMEm

REb a RE 1 2h

21RE 1 h2 b .

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61. Mercury’s orbital speed varies from 38.8 km/s at aphelion to 59.0 km/s at perihelion. If the planet is from theSun’s center at aphelion, how far is it at perihelion?

62. Show that the form follows from Equation 5

when [Hint: Write and apply the binomialapproximation.]

63. Two satellites are in geosynchronous orbit but in diametricallyopposite positions (Fig. 16). In order to catch up with the other,one satellite descends into a lower circular orbit (see Conceptual

r2 5 r1 1 Drr1 . r2.

5 GMma 1r1

21r2b

DU12 5 3

r2

r1

GMm

r2 dr 5 GMm 3

r2

r1

r22 dr 5 GMm

r21

21`r2

r1

DU 5 mg Dr

6.99 31010 m

Gravity: Problem Set

simplifying assumption that asteroids are spherical, with averagedensity For safety, make sure even a jumper capableof 3 m on Earth will return to the surface. What do you report forthe minimum asteroid diameter?

67. The Olympic Committee is keeping you busy! You’re now askedto consider a proposal for lunar hockey. The record speed for ahockey puck is 168 km/h. Is there any danger that hockey puckswill go into lunar orbit?

68. Tidal forces are proportional to the variation in gravity with posi-

tion. By differentiating Equation 1, , estimate the

ratio of the tidal forces due to the Sun and the Moon. Compareyour answer with the ratio of the gravitational forces that the Sunand Moon exert on Earth.

69. Spacecraft that study the Sun are often placed at the so-called L1Lagrange point, located sunward of Earth on the Sun-Earth line.L1 is the point where Earth’s and Sun’s gravity together producean orbital period of one year, so that a spacecraft at L1 stays fixedrelative to Earth as both planet and spacecraft orbit the Sun. Thisplacement ensures an uninterrupted view of the Sun, without be-ing periodically eclipsed by Earth as would occur in Earth orbit.Find L1’s location relative to Earth. (Hint: This problem callsfor numerical methods or solving a higher-order polynomialequation.)

Passage Problems

The Global Positioning System (GPS) uses a “constellation” ofsome 30 satellites to provide accurate positioning for any point onEarth (Fig. 18). GPS receivers time radio signals traveling at thespeed of light from three of the satellites to find the receiver’s posi-tion. Signals from one or more additional satellites provide correc-tions, eliminating the need for high-accuracy clocks in individual

F 5Gm1 m2

r2

2500 kg/m3.

FIGURE 17 Problem 64

Example 1 from the chapter, “Gravity” for a description of thismaneuver). How far should it descend if it’s to catch up in 10 or-bits? Neglect rocket firing times and time spent moving betweenthe two circular orbits.

64. We derived Equation 4 on the assumption that the

massive gravitating center remains fixed. Now consider two ob-jects with equal mass M orbiting each other, as shown in Fig. 17.Show that the orbital period is given by whered is the distance between the objects.

T2 5 2p2d3/GM,

T2 54p2

r3

GM

20,200 km

FIGURE 18 GPS satellites (Passage Problems 70–73)

BIO

C

FIGURE 16 Problem 63

65. Tidal effects in the Earth-Moon system cause the Moon’sorbital period to increase at a current rate of about 35 ms percentury. Assuming the Moon’s orbit is circular, to what rate ofchange in the Earth-Moon distance does this correspond?(Hint: Differentiate Kepler’s third law, Equation 4.)

66. As a member of the 2040 Olympic committee, you’re consider-ing a new sport: asteroid jumping. On Earth, world-class highjumpers routinely clear 2 m. Your job is to make sure athletesjumping from asteroids will return to the asteroid. Make the

GPS receivers. GPS satellites are in circular orbits at 20,200 kmaltitude.

70. What’s the approximate orbital period of GPS satellites?a. 90 minb. 8 hc. 12 hd. 24 he. 1 week

71. What’s the approximate speed of GPS satellites?a. 9.8 m/sb. 500 m/sc. 1.7 km/sd. 4 km/se. 12 km/s

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