2.2 This is the Nearest One Head Gravitational Potential Energy 14.8 Energy Considerations in Planetary and Satellite Motion

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1 2.2 This is the Neaest One Head 423 P U Z Z L E R Moe than 300 yeas ago, Isaac Newton ealized that the sae gavitational foce that causes apples to fall to the Eath also holds the Moon in its obit. In ecent yeas, scientists have used the Hubble Space Telescope to collect evidence of the gavitational foce acting even fathe away, such as at this potoplanetay disk in the constellation Tauus. What popeties of an object such as a potoplanet o the Moon deteine the stength of its gavitational attaction to anothe object? (Left, Lay West/FPG Intenational; ight, Coutesy of NASA) web Fo oe infoation about the Hubble, visit the Space Telescope Science Institute at c h a p t e The Law of Gavity Chapte Outline 14.1 Newton s Law of Univesal Gavitation 14.2 Measuing the Gavitational Constant 14.3 Fee-Fall Acceleation and the Gavitational Foce 14.4 Keple s Laws 14.5 The Law of Gavity and the Motion of Planets 14.6 The Gavitational Field 14.7 Gavitational Potential Enegy 14.8 Enegy Consideations in Planetay and Satellite Motion 14.9 (Optional) The Gavitational Foce Between an Extended Object and a Paticle (Optional) The Gavitational Foce Between a Paticle and a Spheical Mass 423

2 424 CHAPTER 14 The Law of Gavity Befoe 1687, a lage aount of data had been collected on the otions of the Moon and the planets, but a clea undestanding of the foces causing these otions was not available. In that yea, Isaac Newton povided the key that unlocked the secets of the heavens. He knew, fo his fist law, that a net foce had to be acting on the Moon because without such a foce the Moon would ove in a staight-line path athe than in its alost cicula obit. Newton easoned that this foce was the gavitational attaction exeted by the Eath on the Moon. He ealized that the foces involved in the Eath Moon attaction and in the Sun planet attaction wee not soething special to those systes, but athe wee paticula cases of a geneal and univesal attaction between objects. In othe wods, Newton saw that the sae foce of attaction that causes the Moon to follow its path aound the Eath also causes an apple to fall fo a tee. As he put it, I deduced that the foces which keep the planets in thei obs ust be ecipocally as the squaes of thei distances fo the centes about which they evolve; and theeby copaed the foce equisite to keep the Moon in he ob with the foce of gavity at the suface of the Eath; and found the answe petty nealy. In this chapte we study the law of gavity. We place ephasis on descibing the otion of the planets because astonoical data povide an ipotant test of the validity of the law of gavity. We show that the laws of planetay otion developed by Johannes Keple follow fo the law of gavity and the concept of consevation of angula oentu. We then deive a geneal expession fo gavitational potential enegy and exaine the enegetics of planetay and satellite otion. We close by showing how the law of gavity is also used to deteine the foce between a paticle and an extended object NEWTON S LAW OF UNIVERSAL GRAVITATION You ay have head the legend that Newton was stuck on the head by a falling apple while napping unde a tee. This alleged accident supposedly popted hi to iagine that pehaps all bodies in the Univese wee attacted to each othe in the sae way the apple was attacted to the Eath. Newton analyzed astonoical data on the otion of the Moon aound the Eath. Fo that analysis, he ade the bold assetion that the foce law govening the otion of planets was the sae as the foce law that attacted a falling apple to the Eath. This was the fist tie that eathly and heavenly otions wee unified. We shall look at the atheatical details of Newton s analysis in Section In 1687 Newton published his wok on the law of gavity in his teatise Matheatical Pinciples of Natual Philosophy. Newton s law of univesal gavitation states that evey paticle in the Univese attacts evey othe paticle with a foce that is diectly popotional to the poduct of thei asses and invesely popotional to the squae of the distance between the. If the paticles have asses 1 and 2 and ae sepaated by a distance, the agnitude of this gavitational foce is The law of gavity F g G (14.1)

3 14.1 Newton s Law of Univesal Gavitation 425 whee G is a constant, called the univesal gavitational constant, that has been easued expeientally. As noted in Exaple 6.6, its value in SI units is G N 2 /kg 2 (14.2) The fo of the foce law given by Equation 14.1 is often efeed to as an invese-squae law because the agnitude of the foce vaies as the invese squae of the sepaation of the paticles. 1 We shall see othe exaples of this type of foce law in subsequent chaptes. We can expess this foce in vecto fo by defining a unit vecto ˆ12 (Fig. 14.1). Because this unit vecto is diected fo paticle 1 to paticle 2, the foce exeted by paticle 1 on paticle 2 is F 12 G ˆ12 (14.3) whee the inus sign indicates that paticle 2 is attacted to paticle 1, and hence the foce ust be diected towad paticle 1. By Newton s thid law, the foce exeted by paticle 2 on paticle 1, designated F 21, is equal in agnitude to F 12 and in the opposite diection. That is, these foces fo an action eaction pai, and F 21 F 12. Seveal featues of Equation 14.3 deseve ention. The gavitational foce is a field foce that always exists between two paticles, egadless of the ediu that sepaates the. Because the foce vaies as the invese squae of the distance between the paticles, it deceases apidly with inceasing sepaation. We can elate this fact to the geoety of the situation by noting that the intensity of light eanating fo a point souce dops off in the sae 1/ 2 anne, as shown in Figue Anothe ipotant point about Equation 14.3 is that the gavitational foce exeted by a finite-size, spheically syetic ass distibution on a paticle outside the distibution is the sae as if the entie ass of the distibution wee concentated at the cente. Fo exaple, the foce exeted by the 1 F 21 ˆ 12 Figue 14.1 Popeties of the gavitational foce QuickLab F 12 2 The gavitational foce between two paticles is attactive. The unit vecto ˆ12 is diected fo paticle 1 to paticle 2. Note that F 21 F 12. Inflate a balloon just enough to fo a sall sphee. Measue its diaete. Use a ake to colo in a 1-c squae on its suface. Now continue inflating the balloon until it eaches twice the oiginal diaete. Measue the size of the squae you have dawn. Also note how the colo of the aked aea has changed. Have you veified what is shown in Figue 14.2? 2 Figue 14.2 Light adiating fo a point souce dops off as 1/ 2, a elationship that atches the way the gavitational foce depends on distance. When the distance fo the light souce is doubled, the light has to cove fou ties the aea and thus is one fouth as bight. 1 An invese elationship between two quantities x and y is one in which y k/x, whee k is a constant. A diect popotion between x and y exists when y kx.

4 426 CHAPTER 14 The Law of Gavity Eath on a paticle of ass nea the Eath s suface has the agnitude F g G M E (14.4) R 2 E whee M E is the Eath s ass and R E its adius. This foce is diected towad the cente of the Eath. We have evidence of the fact that the gavitational foce acting on an object is diectly popotional to its ass fo ou obsevations of falling objects, discussed in Chapte 2. All objects, egadless of ass, fall in the absence of ai esistance at the sae acceleation g nea the suface of the Eath. Accoding to Newton s second law, this acceleation is given by g F g /, whee is the ass of the falling object. If this atio is to be the sae fo all falling objects, then F g ust be diectly popotional to, so that the ass cancels in the atio. If we conside the oe geneal situation of a gavitational foce between any two objects with ass, such as two planets, this sae aguent can be applied to show that the gavitational foce is popotional to one of the asses. We can choose eithe of the asses in the aguent, howeve; thus, the gavitational foce ust be diectly popotional to both asses, as can be seen in Equation MEASURING THE GRAVITATIONAL CONSTANT The univesal gavitational constant G was easued in an ipotant expeient by Heny Cavendish ( ) in The Cavendish appaatus consists of two sall sphees, each of ass, fixed to the ends of a light hoizontal od suspended by a fine fibe o thin etal wie, as illustated in Figue When two lage sphees, each of ass M, ae placed nea the salle ones, the attactive foce between salle and lage sphees causes the od to otate and twist the wie suspension to a new equilibiu oientation. The angle of otation is easued by the deflection of a light bea eflected fo a io attached to the vetical suspension. The deflection of the light is an effective technique fo aplifying the otion. The expeient is caefully epeated with diffeent asses at vaious sepaations. In addition to poviding a value fo G, the esults show expeientally that the foce is attactive, popotional to the poduct M, and invesely popotional to the squae of the distance. Mio Light souce M Figue 14.3 Scheatic diaga of the Cavendish appaatus fo easuing G. As the sall sphees of ass ae attacted to the lage sphees of ass M, the od between the two sall sphees otates though a sall angle. A light bea eflected fo a io on the otating appaatus easues the angle of otation. The dashed line epesents the oiginal position of the od. EXAMPLE 14.1 Billiads, Anyone? Thee kg billiad balls ae placed on a table at the cones of a ight tiangle, as shown in Figue Calculate the gavitational foce on the cue ball (designated 1 ) esulting fo the othe two balls. Solution Fist we calculate sepaately the individual foces on the cue ball due to the othe two balls, and then we find the vecto su to get the esultant foce. We can see gaphically that this foce should point upwad and towad the

5 14.3 Fee-Fall Acceleation and the Gavitational Foce 427 ight. We locate ou coodinate axes as shown in Figue 14.4, placing ou oigin at the position of the cue ball. The foce exeted by 2 on the cue ball is diected upwad and is given by N2 (0.300 kg)(0.300 kg) kg 2 (0.400 ) 2 j j N F 21 G This esult shows that the gavitational foces between eveyday objects have exteely sall agnitudes. The foce ex j 21 eted by 3 on the cue ball is diected to the ight: F 31 G i N2 (0.300 kg)(0.300 kg) kg 2 (0.300 ) 2 i i N Theefoe, the esultant foce on the cue ball is F F 21 F 31 (3.75j 6.67i) N x y F 21 F F and the agnitude of this foce is F F 2 21 F 2 31 (3.75) 2 (6.67) N Figue 14.4 The esultant gavitational foce acting on the cue ball is the vecto su F 21 F 31. Execise Find the diection of F. Answe 29.3 counteclockwise fo the positive x axis FREE-FALL ACCELERATION AND THE GRAVITATIONAL FORCE In Chapte 5, when defining g as the weight of an object of ass, we efeed to g as the agnitude of the fee-fall acceleation. Now we ae in a position to obtain a oe fundaental desciption of g. Because the foce acting on a feely falling object of ass nea the Eath s suface is given by Equation 14.4, we can equate g to this foce to obtain g G M E R 2 E g G M E (14.5) R 2 E Now conside an object of ass located a distance h above the Eath s suface o a distance fo the Eath s cente, whee R E h. The agnitude of the gavitational foce acting on this object is F g G M E 2 G M E (R E h) 2 The gavitational foce acting on the object at this position is also F g g, whee g is the value of the fee-fall acceleation at the altitude h. Substituting this expes- Fee-fall acceleation nea the Eath s suface

6 428 CHAPTER 14 The Law of Gavity sion fo F g into the last equation shows that g is Vaiation of g with altitude g GM E 2 GM E (R E h) 2 (14.6) Thus, it follows that g deceases with inceasing altitude. Because the weight of a body is g, we see that as :, its weight appoaches zeo. EXAMPLE 14.2 Vaiation of g with Altitude h The Intenational Space Station is designed to opeate at an altitude of 350 k. When copleted, it will have a weight (easued at the Eath s suface) of N. What is its weight when in obit? Solution Because the station is above the suface of the Eath, we expect its weight in obit to be less than its weight on Eath, N. Using Equation 14.6 with h 350 k, we obtain g GM E (R E h) 2 ( N 2 /kg 2 )( kg) ( ) /s 2 Because g/g 8.83/ , we conclude that the weight of the station at an altitude of 350 k is 90.1% of the value at the Eath s suface. So the station s weight in obit is (0.901)( N) N Values of g at othe altitudes ae listed in Table TABLE 14.1 web The official web site fo the Intenational Space Station is Fee-Fall Acceleation g at Vaious Altitudes Above the Eath s Suface Altitude h (k) g (/s 2 ) EXAMPLE 14.3 The Density of the Eath Using the fact that g 9.80 /s 2 at the Eath s suface, find the aveage density of the Eath. Solution Using g 9.80 /s 2 and R E , we find fo Equation 14.5 that M E kg. Fo this esult, and using the definition of density fo Chapte 1, we obtain V 4 3 3R kg 4 E 3( ) 3 Because this value is about twice the density of ost ocks at the Eath s suface, we conclude that the inne coe of the Eath has a density uch highe than the aveage value. It is ost aazing that the Cavendish expeient, which deteines G (and can be done on a tabletop), cobined with siple fee-fall easueents of g, povides infoation about the coe of the Eath kg/ 3

7 14.4 Keple s Laws 429 Astonauts F. Stoy Musgave and Jeffey A. Hoffan, along with the Hubble Space Telescope and the space shuttle Endeavo, ae all falling aound the Eath KEPLER S LAWS People have obseved the oveents of the planets, stas, and othe celestial bodies fo thousands of yeas. In ealy histoy, scientists egaded the Eath as the cente of the Univese. This so-called geocentic odel was elaboated and foalized by the Geek astonoe Claudius Ptoley (c. 100 c. 170) in the second centuy A.D. and was accepted fo the next yeas. In 1543 the Polish astonoe Nicolaus Copenicus ( ) suggested that the Eath and the othe planets evolved in cicula obits aound the Sun (the heliocentic odel). The Danish astonoe Tycho Bahe ( ) wanted to deteine how the heavens wee constucted, and thus he developed a poga to deteine the positions of both stas and planets. It is inteesting to note that those obsevations of the planets and 777 stas visible to the naked eye wee caied out with only a lage sextant and a copass. (The telescope had not yet been invented.) The Gean astonoe Johannes Keple was Bahe s assistant fo a shot while befoe Bahe s death, wheeupon he acquied his ento s astonoical data and spent 16 yeas tying to deduce a atheatical odel fo the otion of the planets. Such data ae difficult to sot out because the Eath is also in otion aound the Sun. Afte any laboious calculations, Keple found that Bahe s data on the evolution of Mas aound the Sun povided the answe. Johannes Keple Gean astonoe ( ) The Gean astonoe Johannes Keple is best known fo developing the laws of planetay otion based on the caeful obsevations of Tycho Bahe. (At Resouce) Fo oe infoation about Johannes Keple, visit ou Web site at

8 430 CHAPTER 14 The Law of Gavity Keple s analysis fist showed that the concept of cicula obits aound the Sun had to be abandoned. He eventually discoveed that the obit of Mas could be accuately descibed by an ellipse. Figue 14.5 shows the geoetic desciption of an ellipse. The longest diension is called the ajo axis and is of length 2a, whee a is the seiajo axis. The shotest diension is the ino axis, of length 2b, whee b is the seiino axis. On eithe side of the cente is a focal point, a distance c fo the cente, whee a 2 b 2 c 2. The Sun is located at one of the focal points of Mas s obit. Keple genealized his analysis to include the otions of all planets. The coplete analysis is suaized in thee stateents known as Keple s laws: Keple s laws 1. All planets ove in elliptical obits with the Sun at one focal point. 2. The adius vecto dawn fo the Sun to a planet sweeps out equal aeas in equal tie intevals. 3. The squae of the obital peiod of any planet is popotional to the cube of the seiajo axis of the elliptical obit. F 1 Figue 14.5 c a F 2 Plot of an ellipse. The seiajo axis has a length a, and the seiino axis has a length b. The focal points ae located at a distance c fo the cente, whee a 2 b 2 c 2. b Most of the planetay obits ae close to cicula in shape; fo exaple, the seiajo and seiino axes of the obit of Mas diffe by only 0.4%. Mecuy and Pluto have the ost elliptical obits of the nine planets. In addition to the planets, thee ae any asteoids and coets obiting the Sun that obey Keple s laws. Coet Halley is such an object; it becoes visible when it is close to the Sun evey 76 yeas. Its obit is vey elliptical, with a seiino axis 76% salle than its seiajo axis. Although we do not pove it hee, Keple s fist law is a diect consequence of the fact that the gavitational foce vaies as 1/ 2. That is, unde an invese-squae gavitational-foce law, the obit of a planet can be shown atheatically to be an ellipse with the Sun at one focal point. Indeed, half a centuy afte Keple developed his laws, Newton deonstated that these laws ae a consequence of the gavitational foce that exists between any two asses. Newton s law of univesal gavitation, togethe with his developent of the laws of otion, povides the basis fo a full atheatical solution to the otion of planets and satellites THE LAW OF GRAVITY AND THE MOTION OF PLANETS In foulating his law of gavity, Newton used the following easoning, which suppots the assuption that the gavitational foce is popotional to the invese squae of the sepaation between the two inteacting bodies. He copaed the acceleation of the Moon in its obit with the acceleation of an object falling nea the Eath s suface, such as the legenday apple (Fig. 14.6). Assuing that both acceleations had the sae cause naely, the gavitational attaction of the Eath Newton used the invese-squae law to eason that the acceleation of the Moon towad the Eath (centipetal acceleation) should be popotional to 1/ 2 M, whee M is the distance between the centes of the Eath and the Moon. Futheoe, the acceleation of the apple towad the Eath should be popotional to 1/R 2 E, whee R E is the adius of the Eath, o the distance between the centes of the Eath and the apple. Using the values M and

9 14.5 The Law of Gavity and the Motion of Planets 431 Moon R E M a M v g Figue 14.6 As it evolves aound the Eath, the Moon expeiences a centipetal acceleation a M diected towad Eath the Eath. An object nea the Eath s suface, such as the apple shown hee, expeiences an acceleation g. (Diensions ae not to scale.) R E , Newton pedicted that the atio of the Moon s acceleation a M to the apple s acceleation g would be a M g (1/ M) 2 (1/R E ) 2 R E M Theefoe, the centipetal acceleation of the Moon is a M ( )(9.80 /s 2 ) /s 2 Newton also calculated the centipetal acceleation of the Moon fo a knowledge of its ean distance fo the Eath and its obital peiod, T days s. In a tie T, the Moon tavels a distance 2 M, which equals the cicufeence of its obit. Theefoe, its obital speed is 2 M /T and its centipetal acceleation is a M v 2 (2 M /T) M M M T ( ) ( s) 2 Acceleation of the Moon /s /s In othe wods, because the Moon is oughly 60 Eath adii away, the gavitational acceleation at that distance should be about 1/60 2 of its value at the Eath s suface. This is just the acceleation needed to account fo the cicula otion of the Moon aound the Eath. The nealy pefect ageeent between this value and the value Newton obtained using g povides stong evidence of the invese-squae natue of the gavitational foce law. Although these esults ust have been vey encouaging to Newton, he was deeply toubled by an assuption he ade in the analysis. To evaluate the acceleation of an object at the Eath s suface, Newton teated the Eath as if its ass wee all concentated at its cente. That is, he assued that the Eath acted as a paticle as fa as its influence on an exteio object was concened. Seveal yeas late, in 1687, on the basis of his pioneeing wok in the developent of calculus, Newton poved that this assuption was valid and was a natual consequence of the law of univesal gavitation.

10 432 CHAPTER 14 The Law of Gavity Figue 14.7 Keple s thid law M S M p A planet of ass M p oving in a cicula obit aound the Sun. The obits of all planets except Mecuy and Pluto ae nealy cicula. v Keple s Thid Law It is infoative to show that Keple s thid law can be pedicted fo the invesesquae law fo cicula obits. 2 Conside a planet of ass M p oving aound the Sun of ass M S in a cicula obit, as shown in Figue Because the gavitational foce exeted by the Sun on the planet is a adially diected foce that keeps the planet oving in a cicle, we can apply Newton s second law (F a) to the planet: GM S M p 2 M pv 2 Because the obital speed v of the planet is siply 2/T, whee T is its peiod of evolution, the peceding expession becoes GM S 2 (2/T) 2 T GM S 3 K S 3 (14.7) whee K S is a constant given by K S 4 2 GM S s 2 / 3 Equation 14.7 is Keple s thid law. It can be shown that the law is also valid fo elliptical obits if we eplace with the length of the seiajo axis a. Note that the constant of popotionality K S is independent of the ass of the planet. Theefoe, Equation 14.7 is valid fo any planet. 3 Table 14.2 contains a collection of useful planetay data. The last colun veifies that T 2 / 3 is a constant. The sall vaiations in the values in this colun eflect uncetainties in the easued values of the peiods and seiajo axes of the planets. If we wee to conside the obit aound the Eath of a satellite such as the Moon, then the popotionality constant would have a diffeent value, with the Sun s ass eplaced by the Eath s ass. EXAMPLE 14.4 The Mass of the Sun Calculate the ass of the Sun using the fact that the peiod of the Eath s obit aound the Sun is s and its distance fo the Sun is Solution M S GT 2 Using Equation 14.7, we find that 4 2 ( ) 3 ( N 2 /kg 2 )( s) kg In Exaple 14.3, an undestanding of gavitational foces enabled us to find out soething about the density of the Eath s coe, and now we have used this undestanding to deteine the ass of the Sun. 2 The obits of all planets except Mecuy and Pluto ae vey close to being cicula; hence, we do not intoduce uch eo with this assuption. Fo exaple, the atio of the seiino axis to the seiajo axis fo the Eath s obit is b/a Equation 14.7 is indeed a popotion because the atio of the two quantities T 2 and 3 is a constant. The vaiables in a popotion ae not equied to be liited to the fist powe only.

11 14.5 The Law of Gavity and the Motion of Planets 433 TABLE 14.2 Useful Planetay Data Mean Peiod of Radius Revolution Mean Distance Body Mass (kg) () (s) fo Sun () T 2 3 (s2 / 3 ) Mecuy Venus Eath Mas Jupite Satun Uanus Neptune Pluto Moon Sun Keple s Second Law and Consevation of Angula Moentu Conside a planet of ass M p oving aound the Sun in an elliptical obit (Fig. 14.8). The gavitational foce acting on the planet is always along the adius vecto, diected towad the Sun, as shown in Figue 14.9a. When a foce is diected towad o away fo a fixed point and is a function of only, it is called a cental foce. The toque acting on the planet due to this foce is clealy zeo; that is, because F is paallel to, F F ˆ 0 (You ay want to evisit Section 11.2 to efesh you eoy on the vecto poduct.) Recall fo Equation 11.19, howeve, that toque equals the tie ate of change of angula oentu: Theefoe, because the gavitational d L/dt. D C Sun S Figue 14.8 Keple s second law is called the law of equal aeas. When the tie inteval equied fo a planet to tavel fo A to B is equal to the tie inteval equied fo it to go fo C to D, the two aeas swept out by the planet s adius vecto ae equal. Note that in ode fo this to be tue, the planet ust be oving faste between C and D than between A and B. A B Sepaate views of Jupite and of Peiodic Coet Shoeake Levy 9 both taken with the Hubble Space Telescope about two onths befoe Jupite and the coet collided in July 1994 wee put togethe with the use of a copute. Thei elative sizes and distances wee alteed. The black spot on Jupite is the shadow of its oon Io.

12 434 CHAPTER 14 The Law of Gavity Sun M S Sun F g (a) M p v d = vdt foce exeted by the Sun on a planet esults in no toque on the planet, the angula oentu L of the planet is constant: L p M p v M p v constant (14.8) Because L eains constant, the planet s otion at any instant is esticted to the plane foed by and v. We can elate this esult to the following geoetic consideation. The adius vecto in Figue 14.9b sweeps out an aea da in a tie dt. This aea equals onehalf the aea d of the paalleloga foed by the vectos and d (see Section 11.2). Because the displaceent of the planet in a tie dt is d vdt, we can say that da da 1 2 d 1 2 v dt L 2M p dt Figue 14.9 (b) (a) The gavitational foce acting on a planet is diected towad the Sun, along the adius vecto. (b) As a planet obits the Sun, the aea swept out by the adius vecto in a tie dt is equal to one-half the aea of the paalleloga foed by the vectos and d vdt. whee L and M p ae both constants. Thus, we conclude that da dt L constant 2M p (14.9) the adius vecto fo the Sun to a planet sweeps out equal aeas in equal tie intevals. It is ipotant to ecognize that this esult, which is Keple s second law, is a consequence of the fact that the foce of gavity is a cental foce, which in tun iplies that angula oentu is constant. Theefoe, Keple s second law applies to any situation involving a cental foce, whethe invese-squae o not. EXAMPLE 14.5 Motion in an Elliptical Obit A satellite of ass oves in an elliptical obit aound the Eath (Fig ). The iniu distance of the satellite fo the Eath is called the peigee (indicated by p in Fig. Figue v a p a p a As a satellite oves aound the Eath in an elliptical obit, its angula oentu is constant. Theefoe, v a a v p p, whee the subscipts a and p epesent apogee and peigee, espectively. v p 14.10), and the axiu distance is called the apogee (indicated by a). If the speed of the satellite at p is v p, what is its speed at a? Solution As the satellite oves fo peigee towad apogee, it is oving fathe fo the Eath. Thus, a coponent of the gavitational foce exeted by the Eath on the satellite is opposite the velocity vecto. Negative wok is done on the satellite, which causes it to slow down, accoding to the wok kinetic enegy theoe. As a esult, we expect the speed at apogee to be lowe than the speed at peigee. The angula oentu of the satellite elative to the Eath is v v. At the points a and p, v is pependicula to. Theefoe, the agnitude of the angula oentu at these positions is L a v a a and L p v p p. Because angula oentu is constant, we see that v a a v p p p v a v p a

13 14.6 The Gavitational Field 435 Quick Quiz 14.1 How would you explain the fact that Satun and Jupite have peiods uch geate than one yea? 14.6 THE GRAVITATIONAL FIELD When Newton published his theoy of univesal gavitation, it was consideed a success because it satisfactoily explained the otion of the planets. Since 1687 the sae theoy has been used to account fo the otions of coets, the deflection of a Cavendish balance, the obits of binay stas, and the otation of galaxies. Nevetheless, both Newton s contepoaies and his successos found it difficult to accept the concept of a foce that acts though a distance, as entioned in Section 5.1. They asked how it was possible fo two objects to inteact when they wee not in contact with each othe. Newton hiself could not answe that question. An appoach to descibing inteactions between objects that ae not in contact cae well afte Newton s death, and it enables us to look at the gavitational inteaction in a diffeent way. As descibed in Section 5.1, this altenative appoach uses the concept of a gavitational field that exists at evey point in space. When a paticle of ass is placed at a point whee the gavitational field is g, the paticle expeiences a foce F g g. In othe wods, the field exets a foce on the paticle. Hence, the gavitational field g is defined as g F g (14.10) Gavitational field That is, the gavitational field at a point in space equals the gavitational foce expeienced by a test paticle placed at that point divided by the ass of the test paticle. Notice that the pesence of the test paticle is not necessay fo the field to exist the Eath ceates the gavitational field. We call the object ceating the field the souce paticle (although the Eath is clealy not a paticle; we shall discuss shotly the fact that we can appoxiate the Eath as a paticle fo the pupose of finding the gavitational field that it ceates). We can detect the pesence of the field and easue its stength by placing a test paticle in the field and noting the foce exeted on it. Although the gavitational foce is inheently an inteaction between two objects, the concept of a gavitational field allows us to facto out the ass of one of the objects. In essence, we ae descibing the effect that any object (in this case, the Eath) has on the epty space aound itself in tes of the foce that would be pesent if a second object wee soewhee in that space. 4 As an exaple of how the field concept woks, conside an object of ass nea the Eath s suface. Because the gavitational foce acting on the object has a agnitude GM E / 2 (see Eq. 14.4), the field g at a distance fo the cente of the Eath is g F g GM E 2 ˆ (14.11) whee ˆ is a unit vecto pointing adially outwad fo the Eath and the inus 4 We shall etun to this idea of ass affecting the space aound it when we discuss Einstein s theoy of gavitation in Chapte 39.

14 436 CHAPTER 14 The Law of Gavity Figue (a) (a) The gavitational field vectos in the vicinity of a unifo spheical ass such as the Eath vay in both diection and agnitude. The vectos point in the diection of the acceleation a paticle would expeience if it wee placed in the field. The agnitude of the field vecto at any location is the agnitude of the fee-fall acceleation at that location. (b) The gavitational field vectos in a sall egion nea the Eath s suface ae unifo in both diection and agnitude. (b) sign indicates that the field points towad the cente of the Eath, as illustated in Figue 14.11a. Note that the field vectos at diffeent points suounding the Eath vay in both diection and agnitude. In a sall egion nea the Eath s suface, the downwad field g is appoxiately constant and unifo, as indicated in Figue 14.11b. Equation is valid at all points outside the Eath s suface, assuing that the Eath is spheical. At the Eath s suface, whee R E, g has a agnitude of 9.80 N/kg GRAVITATIONAL POTENTIAL ENERGY In Chapte 8 we intoduced the concept of gavitational potential enegy, which is the enegy associated with the position of a paticle. We ephasized that the gavitational potential enegy function U gy is valid only when the paticle is nea the Eath s suface, whee the gavitational foce is constant. Because the gavitational foce between two paticles vaies as 1/ 2, we expect that a oe geneal potential enegy function one that is valid without the estiction of having to be nea the Eath s suface will be significantly diffeent fo U gy. Befoe we calculate this geneal fo fo the gavitational potential enegy function, let us fist veify that the gavitational foce is consevative. (Recall fo Section 8.2 that a foce is consevative if the wok it does on an object oving between any two points is independent of the path taken by the object.) To do this, we fist note that the gavitational foce is a cental foce. By definition, a cental foce is any foce that is diected along a adial line to a fixed cente and has a agnitude that depends only on the adial coodinate. Hence, a cental foce can be epesented by F()ˆ, whee ˆ is a unit vecto diected fo the oigin to the paticle, as shown in Figue Conside a cental foce acting on a paticle oving along the geneal path P to Q in Figue The path fo P to Q can be appoxiated by a seies of

15 14.7 Gavitational Potential Enegy 437 steps accoding to the following pocedue. In Figue 14.12, we daw seveal thin wedges, which ae shown as dashed lines. The oute bounday of ou set of wedges is a path consisting of shot adial line segents and acs (gay in the figue). We select the length of the adial diension of each wedge such that the shot ac at the wedge s wide end intesects the actual path of the paticle. Then we can appoxiate the actual path with a seies of zigzag oveents that altenate between oving along an ac and oving along a adial line. By definition, a cental foce is always diected along one of the adial segents; theefoe, the wok done by F along any adial segent is dw F d F() d You should ecall that, by definition, the wok done by a foce that is pependicula to the displaceent is zeo. Hence, the wok done in oving along any ac is zeo because F is pependicula to the displaceent along these segents. Theefoe, the total wok done by F is the su of the contibutions along the adial segents: W f i F() d whee the subscipts i and f efe to the initial and final positions. Because the integand is a function only of the adial position, this integal depends only on the initial and final values of. Thus, the wok done is the sae ove any path fo P to Q. Because the wok done is independent of the path and depends only on the end points, we conclude that any cental foce is consevative. We ae now assued that a potential enegy function can be obtained once the fo of the cental foce is specified. Recall fo Equation 8.2 that the change in the gavitational potential enegy associated with a given displaceent is defined as the negative of the wok done by the gavitational foce duing that displaceent: U U f U i f (14.12) We can use this esult to evaluate the gavitational potential enegy function. Conside a paticle of ass oving between two points P and Q above the Eath s suface (Fig ). The paticle is subject to the gavitational foce given by Equation We can expess this foce as F() GM E 2 i F() d whee the negative sign indicates that the foce is attactive. Substituting this expession fo F() into Equation 14.12, we can copute the change in the gavita- O Wok done by a cental foce ˆ i P ˆ F Figue f Radial segent Q Ac A paticle oves fo P to Q while acted on by a cental foce F, which is diected adially. The path is boken into a seies of adial segents and acs. Because the wok done along the acs is zeo, the wok done is independent of the path and depends only on f and i. P F g i R E M E f F g Q Figue As a paticle of ass oves fo P to Q above the Eath s suface, the gavitational potential enegy changes accoding to Equation

16 438 CHAPTER 14 The Law of Gavity tional potential enegy function: U f U i GM E f i d 2 GM E 1 f i Change in gavitational potential enegy U f U i GM E 1 f 1 i (14.13) Gavitational potential enegy of the Eath paticle syste fo R E M E Eath As always, the choice of a efeence point fo the potential enegy is copletely abitay. It is custoay to choose the efeence point whee the foce is zeo. Taking U i 0 at i, we obtain the ipotant esult U GM E (14.14) This expession applies to the Eath paticle syste whee the two asses ae sepaated by a distance, povided that R E. The esult is not valid fo paticles inside the Eath, whee R E. (The situation in which R E is teated in Section ) Because of ou choice of U i, the function U is always negative (Fig ). Although Equation was deived fo the paticle Eath syste, it can be applied to any two paticles. That is, the gavitational potential enegy associated with any pai of paticles of asses 1 and 2 sepaated by a distance is U U G 1 2 (14.15) O GM E R E Figue R E Gaph of the gavitational potential enegy U vesus fo a paticle above the Eath s suface. The potential enegy goes to zeo as appoaches infinity Figue paticles Thee inteacting This expession shows that the gavitational potential enegy fo any pai of paticles vaies as 1/, wheeas the foce between the vaies as 1/ 2. Futheoe, the potential enegy is negative because the foce is attactive and we have taken the potential enegy as zeo when the paticle sepaation is infinite. Because the foce between the paticles is attactive, we know that an extenal agent ust do positive wok to incease the sepaation between the. The wok done by the extenal agent poduces an incease in the potential enegy as the two paticles ae sepaated. That is, U becoes less negative as inceases. When two paticles ae at est and sepaated by a distance, an extenal agent has to supply an enegy at least equal to G 1 2 / in ode to sepaate the paticles to an infinite distance. It is theefoe convenient to think of the absolute value of the potential enegy as the binding enegy of the syste. If the extenal agent supplies an enegy geate than the binding enegy, the excess enegy of the syste will be in the fo of kinetic enegy when the paticles ae at an infinite sepaation. We can extend this concept to thee o oe paticles. In this case, the total potential enegy of the syste is the su ove all pais of paticles. 5 Each pai contibutes a te of the fo given by Equation Fo exaple, if the syste contains thee paticles, as in Figue 14.15, we find that U total U 12 U 13 U 23 G (14.16) The absolute value of U total epesents the wok needed to sepaate the paticles by an infinite distance The fact that potential enegy tes can be added fo all pais of paticles stes fo the expeiental fact that gavitational foces obey the supeposition pinciple.

17 14.8 Enegy Consideations in Planetay and Satellite Motion 439 EXAMPLE 14.6 The Change in Potential Enegy A paticle of ass is displaced though a sall vetical distance y nea the Eath s suface. Show that in this situation the geneal expession fo the change in gavitational potential enegy given by Equation educes to the failia elationship U g y. Solution We can expess Equation in the fo U GM E 1 f 1 i GM E f i i f If both the initial and final positions of the paticle ae close to the Eath s suface, then and i f R 2 f i y E. (Recall that is easued fo the cente of the Eath.) Theefoe, the change in potential enegy becoes U GM E R 2 y g y E whee we have used the fact that g GM E /R E (Eq. 14.5). Keep in ind that the efeence point is abitay because it is the change in potential enegy that is eaningful ENERGY CONSIDERATIONS IN PLANETARY AND SATELLITE MOTION Conside a body of ass oving with a speed v in the vicinity of a assive body of ass M, whee M W. The syste ight be a planet oving aound the Sun, a satellite in obit aound the Eath, o a coet aking a one-tie flyby of the Sun. If we assue that the body of ass M is at est in an inetial efeence fae, then the total echanical enegy E of the two-body syste when the bodies ae sepaated by a distance is the su of the kinetic enegy of the body of ass and the potential enegy of the syste, given by Equation 14.15: 6 E K U v M E 1 2 v 2 GM (14.17) This equation shows that E ay be positive, negative, o zeo, depending on the value of v. Howeve, fo a bound syste, 7 such as the Eath Sun syste, E is necessaily less than zeo because we have chosen the convention that U : 0 as :. We can easily establish that E 0 fo the syste consisting of a body of ass oving in a cicula obit about a body of ass M W (Fig ). Newton s second law applied to the body of ass gives GM 2 a v 2 Figue A body of ass oving in a cicula obit about a uch lage body of ass M. 6 You ight ecognize that we have ignoed the acceleation and kinetic enegy of the lage body. To see that this siplification is easonable, conside an object of ass falling towad the Eath. Because the cente of ass of the object Eath syste is effectively stationay, it follows that v M E v E. Thus, the Eath acquies a kinetic enegy equal to 1 2 M Ev 2 E v 2 K M E M E whee K is the kinetic enegy of the object. Because M E W, this esult shows that the kinetic enegy of the Eath is negligible. 7 Of the thee exaples povided at the beginning of this section, the planet oving aound the Sun and a satellite in obit aound the Eath ae bound systes the Eath will always stay nea the Sun, and the satellite will always stay nea the Eath. The one-tie coet flyby epesents an unbound syste the coet inteacts once with the Sun but is not bound to it. Thus, in theoy the coet can ove infinitely fa away fo the Sun.

18 440 CHAPTER 14 The Law of Gavity Total enegy fo cicula obits Multiplying both sides by and dividing by 2 gives 1 2 v 2 GM 2 Substituting this into Equation 14.17, we obtain E GM 2 E GM 2 GM (14.18) (14.19) This esult clealy shows that the total echanical enegy is negative in the case of cicula obits. Note that the kinetic enegy is positive and equal to one-half the absolute value of the potential enegy. The absolute value of E is also equal to the binding enegy of the syste, because this aount of enegy ust be povided to the syste to ove the two asses infinitely fa apat. The total echanical enegy is also negative in the case of elliptical obits. The expession fo E fo elliptical obits is the sae as Equation with eplaced by the seiajo axis length a. Futheoe, the total enegy is constant if we assue that the syste is isolated. Theefoe, as the body of ass oves fo P to Q in Figue 14.13, the total enegy eains constant and Equation gives E 1 2 v i 2 GM i 1 2 v f 2 GM f (14.20) Cobining this stateent of enegy consevation with ou ealie discussion of consevation of angula oentu, we see that both the total enegy and the total angula oentu of a gavitationally bound, two-body syste ae constants of the otion. EXAMPLE 14.7 Changing the Obit of a Satellite The space shuttle eleases a 470-kg counications satellite while in an obit that is 280 k above the suface of the Eath. A ocket engine on the satellite boosts it into a geosynchonous obit, which is an obit in which the satellite stays diectly ove a single location on the Eath. How uch enegy did the engine have to povide? Solution Fist we ust deteine the adius of a geosynchonous obit. Then we can calculate the change in enegy needed to boost the satellite into obit. The peiod of the obit T ust be one day ( s), so that the satellite tavels once aound the Eath in the sae tie that the Eath spins once on its axis. Knowing the peiod, we can then apply Keple s thid law (Eq. 14.7) to find the adius, once we eplace K S with K E 4 2 /GM E s 2 / 3 : T 2 K E 3 3 T 2 3 ( s) K E s 2 / R f This is a little oe than i above the Eath s suface. We ust also deteine the initial adius (not the altitude above the Eath s suface) of the satellite s obit when it was still in the shuttle s cago bay. This is siply Now, applying Equation 14.19, we obtain, fo the total initial and final enegies, The enegy equied fo the engine to boost the satellite is E engine E f E i GM E 2 1 R f 1 R i ( N 2 /kg 2 )( kg)(470 kg) 2 R E 280 k R i E i GM E 2R i J E f GM E 2R f

19 14.8 Enegy Consideations in Planetay and Satellite Motion 441 This is the enegy equivalent of 89 gal of gasoline. NASA enginees ust account fo the changing ass of the spacecaft as it ejects buned fuel, soething we have not done hee. Would you expect the calculation that includes the effect of this changing ass to yield a geate o lesse aount of enegy equied fo the engine? If we wish to deteine how the enegy is distibuted afte the engine is fied, we find fo Equation that the change in kinetic enegy is K (GM E /2)(1/R f 1/R i ) J (a decease), and the coesponding change in potential enegy is U GM E (1/R f 1/R i ) J (an incease). Thus, the change in echanical enegy of the syste is E K U J, as we aleady calculated. The fiing of the engine esults in an incease in the total echanical enegy of the syste. Because an incease in potential enegy is accopanied by a decease in kinetic enegy, we conclude that the speed of an obiting satellite deceases as its altitude inceases. Escape Speed Suppose an object of ass is pojected vetically upwad fo the Eath s suface with an initial speed v i, as illustated in Figue We can use enegy consideations to find the iniu value of the initial speed needed to allow the object to escape the Eath s gavitational field. Equation gives the total enegy of the object at any point. At the suface of the Eath, v v i and i R E. When the object eaches its axiu altitude, v v f 0 and f ax. Because the total enegy of the syste is constant, substituting these conditions into Equation gives Solving fo v 2 i gives 1 2 v i 2 GM E R E GM E ax v 2 i 2GM E 1 1 R E ax (14.21) Theefoe, if the initial speed is known, this expession can be used to calculate the axiu altitude h because we know that h ax R E We ae now in a position to calculate escape speed, which is the iniu speed the object ust have at the Eath s suface in ode to escape fo the influence of the Eath s gavitational field. Taveling at this iniu speed, the object continues to ove fathe and fathe away fo the Eath as its speed asyptotically appoaches zeo. Letting ax : in Equation and taking v i v esc, we obtain R E v f = 0 v i M E Figue h ax An object of ass pojected upwad fo the Eath s suface with an initial speed v i eaches a axiu altitude h. v esc 2GM E R E (14.22) Escape speed Note that this expession fo v esc is independent of the ass of the object. In othe wods, a spacecaft has the sae escape speed as a olecule. Futheoe, the esult is independent of the diection of the velocity and ignoes ai esistance. If the object is given an initial speed equal to v esc, its total enegy is equal to zeo. This can be seen by noting that when :, the object s kinetic enegy and its potential enegy ae both zeo. If v i is geate than v esc, the total enegy is geate than zeo and the object has soe esidual kinetic enegy as :.

20 442 CHAPTER 14 The Law of Gavity EXAMPLE 14.8 Escape Speed of a Rocket Calculate the escape speed fo the Eath fo a kg spacecaft, and deteine the kinetic enegy it ust have at the Eath s suface in ode to escape the Eath s gavitational field. Solution v esc 2GM E R E Using Equation gives 2( N 2 /kg 2 )( kg) This coesponds to about i/h. The kinetic enegy of the spacecaft is /s K 1 2 v 2 esc 1 2 ( kg)( /s) J This is equivalent to about gal of gasoline. TABLE 14.3 Escape Speeds fo the Sufaces of the Planets, Moon, and Sun Body v esc (k/s) Mecuy 4.3 Venus 10.3 Eath 11.2 Moon 2.3 Mas 5.0 Jupite 60 Satun 36 Uanus 22 Neptune 24 Pluto 1.1 Sun 618 Equations and can be applied to objects pojected fo any planet. That is, in geneal, the escape speed fo the suface of any planet of ass M and adius R is v esc 2GM R Escape speeds fo the planets, the Moon, and the Sun ae povided in Table Note that the values vay fo 1.1 k/s fo Pluto to about 618 k/s fo the Sun. These esults, togethe with soe ideas fo the kinetic theoy of gases (see Chapte 21), explain why soe planets have atosphees and othes do not. As we shall see late, a gas olecule has an aveage kinetic enegy that depends on the tepeatue of the gas. Hence, lighte olecules, such as hydogen and heliu, have a highe aveage speed than heavie species at the sae tepeatue. When the aveage speed of the lighte olecules is not uch less than the escape speed of a planet, a significant faction of the have a chance to escape fo the planet. This echanis also explains why the Eath does not etain hydogen olecules and heliu atos in its atosphee but does etain heavie olecules, such as oxygen and nitogen. On the othe hand, the vey lage escape speed fo Jupite enables that planet to etain hydogen, the piay constituent of its atosphee. Quick Quiz 14.2 If you wee a space pospecto and discoveed gold on an asteoid, it pobably would not be a good idea to jup up and down in exciteent ove you find. Why? Quick Quiz 14.3 Figue is a dawing by Newton showing the path of a stone thown fo a ountaintop. He shows the stone landing fathe and fathe away when thown at highe and highe speeds (at points D, E, F, and G), until finally it is thown all the way aound the Eath. Why didn t Newton show the stone landing at B and A befoe it was going fast enough to coplete an obit?

21 14.9 The Gavitational Foce Between an Extended Object and a Paticle 443 Figue The geate the velocity...with which [a stone] is pojected, the fathe it goes befoe it falls to the Eath. We ay theefoe suppose the velocity to be so inceased, that it would descibe an ac of 1, 2, 5, 10, 100, 1000 iles befoe it aived at the Eath, till at last, exceeding the liits of the Eath, it should pass into space without touching. Si Isaac Newton, Syste of the Wold. Optional Section 14.9 THE GRAVITATIONAL FORCE BETWEEN AN EXTENDED OBJECT AND A PARTICLE We have ephasized that the law of univesal gavitation given by Equation 14.3 is valid only if the inteacting objects ae teated as paticles. In view of this, how can we calculate the foce between a paticle and an object having finite diensions? This is accoplished by teating the extended object as a collection of paticles and aking use of integal calculus. We fist evaluate the potential enegy function, and then calculate the gavitational foce fo that function. We obtain the potential enegy associated with a syste consisting of a paticle of ass and an extended object of ass M by dividing the object into any eleents, each having a ass M i (Fig ). The potential enegy associated with the syste consisting of any one eleent and the paticle is U G M i / i, whee i is the distance fo the paticle to the eleent M i. The total potential enegy of the oveall syste is obtained by taking the su ove all eleents as M i : 0. In this liit, we can expess U in integal fo as U G dm (14.23) Once U has been evaluated, we obtain the foce exeted by the extended object on the paticle by taking the negative deivative of this scala function (see Section 8.6). If the extended object has spheical syety, the function U depends only on, and the foce is given by du/d. We teat this situation in Section In pinciple, one can evaluate U fo any geoety; howeve, the integation can be cubesoe. An altenative appoach to evaluating the gavitational foce between a paticle and an extended object is to pefo a vecto su ove all ass eleents of the object. Using the pocedue outlined in evaluating U and the law of univesal gavitation in the fo shown in Equation 14.3, we obtain, fo the total foce exeted on the paticle F g G dm 2 ˆ (14.24) whee ˆ is a unit vecto diected fo the eleent dm towad the paticle (see Fig ) and the inus sign indicates that the diection of the foce is opposite that of ˆ. This pocedue is not always ecoended because woking with a vecto function is oe difficult than woking with the scala potential enegy function. Howeve, if the geoety is siple, as in the following exaple, the evaluation of F can be staightfowad. M Figue M i A paticle of ass inteacting with an extended object of ass M. The total gavitational foce exeted by the object on the paticle can be obtained by dividing the object into nueous eleents, each having a ass M i, and then taking a vecto su ove the foces exeted by all eleents. Total foce exeted on a paticle by an extended object ˆ i

22 444 CHAPTER 14 The Law of Gavity EXAMPLE 14.9 Gavitational Foce Between a Paticle and a Ba The left end of a hoogeneous ba of length L and ass M is at a distance h fo a paticle of ass (Fig ). Calculate the total gavitational foce exeted by the ba on the paticle. Solution The abitay segent of the ba of length dx has a ass dm. Because the ass pe unit length is constant, it follows that the atio of asses dm/m is equal to the atio y of lengths dx/l, and so dm (M/L) dx. In this poble, the vaiable in Equation is the distance x shown in Figue 14.20, the unit vecto ˆ is ˆ i, and the foce acting on the paticle is to the ight; theefoe, Equation gives us F g G hl F g GM L h Mdx L 1 x hl h 1 x 2 (i) G M hl L i GM h(h L) i h dx x 2 i Figue O h x The gavitational foce exeted by the ba on the paticle is diected to the ight. Note that the ba is not equivalent to a paticle of ass M located at the cente of ass of the ba. L dx x We see that the foce exeted on the paticle is in the positive x diection, which is what we expect because the gavitational foce is attactive. Note that in the liit L : 0, the foce vaies as 1/h 2, which is what we expect fo the foce between two point asses. Futheoe, if h W L, the foce also vaies as 1/h 2. This can be seen by noting that the denoinato of the expession fo F g can be expessed in the fo h 2 (1 L/h), which is appoxiately equal to h 2 when h W L. Thus, when bodies ae sepaated by distances that ae geat elative to thei chaacteistic diensions, they behave like paticles. Optional Section THE GRAVITATIONAL FORCE BETWEEN A PARTICLE AND A SPHERICAL MASS We have aleady stated that a lage sphee attacts a paticle outside it as if the total ass of the sphee wee concentated at its cente. We now descibe the foce acting on a paticle when the extended object is eithe a spheical shell o a solid sphee, and then apply these facts to soe inteesting systes. Spheical Shell Case 1. If a paticle of ass is located outside a spheical shell of ass M at, fo instance, point P in Figue 14.21a, the shell attacts the paticle as though the ass of the shell wee concentated at its cente. We can show this, as Newton did, with integal calculus. Thus, as fa as the gavitational foce acting on a paticle outside the shell is concened, a spheical shell acts no diffeently fo the solid spheical distibutions of ass we have seen. Foce on a paticle due to a spheical shell Case 2. If the paticle is located inside the shell (at point P in Fig b), the gavitational foce acting on it can be shown to be zeo. We can expess these two ipotant esults in the following way: F g GM (14.25a) 2 ˆ fo R F g 0 fo R (14.25b) The gavitational foce as a function of the distance is plotted in Figue 14.21c.

23 14.10 The Gavitational Foce Between a Paticle and a Spheical Mass 445 M Q F QP P F Q P (a) Q M P F Top, P F Botto, P (b) F g O R (c) Figue (a) The nonadial coponents of the gavitational foces exeted on a paticle of ass located at point P outside a spheical shell of ass M cancel out. (b) The spheical shell can be boken into ings. Even though point P is close to the top ing than to the botto ing, the botto ing is lage, and the gavitational foces exeted on the paticle at P by the atte in the two ings cancel each othe. Thus, fo a paticle located at any point P inside the shell, thee is no gavitational foce exeted on the paticle by the ass M of the shell. (c) The agnitude of the gavitational foce vesus the adial distance fo the cente of the shell. The shell does not act as a gavitational shield, which eans that a paticle inside a shell ay expeience foces exeted by bodies outside the shell. Solid Sphee Case 1. If a paticle of ass is located outside a hoogeneous solid sphee of ass M (at point P in Fig ), the sphee attacts the paticle as though the

24 446 CHAPTER 14 The Law of Gavity ass of the sphee wee concentated at its cente. We have used this notion at seveal places in this chapte aleady, and we can ague it fo Equation 14.25a. A solid sphee can be consideed to be a collection of concentic spheical shells. The asses of all of the shells can be intepeted as being concentated at thei coon cente, and the gavitational foce is equivalent to that due to a paticle of ass M located at that cente. Foce on a paticle due to a solid sphee Case 2. If a paticle of ass is located inside a hoogeneous solid sphee of ass M (at point Q in Fig ), the gavitational foce acting on it is due only to the ass M contained within the sphee of adius R, shown in Figue In othe wods, F g GM 2 ˆ fo R F g GM 2 ˆ fo R (14.26a) (14.26b) This also follows fo spheical-shell Case 1 because the pat of the sphee that is P M R M F g Q F g O R Figue The gavitational foce acting on a paticle when it is outside a unifo solid sphee is GM/ 2 and is diected towad the cente of the sphee. The gavitational foce acting on the paticle when it is inside such a sphee is popotional to and goes to zeo at the cente.

25 14.10 The Gavitational Foce Between a Paticle and a Spheical Mass 447 fathe fo the cente than Q can be teated as a seies of concentic spheical shells that do not exet a net foce on the paticle because the paticle is inside the. Because the sphee is assued to have a unifo density, it follows that the atio of asses M/M is equal to the atio of volues V/V, whee V is the total volue of the sphee and V is the volue within the sphee of adius only: M M 4 V V R 3 3 R 3 Solving this equation fo M and substituting the value obtained into Equation 14.26b, we have F g GM R 3 ˆ fo R (14.27) This equation tells us that at the cente of the solid sphee, whee 0, the gavitational foce goes to zeo, as we intuitively expect. The foce as a function of is plotted in Figue Case 3. If a paticle is located inside a solid sphee having a density that is spheically syetic but not unifo, then M in Equation 14.26b is given by an integal of the fo M dv, whee the integation is taken ove the volue contained within the sphee of adius in Figue We can evaluate this integal if the adial vaiation of is given. In this case, we take the volue eleent dv as the volue of a spheical shell of adius and thickness d, and thus dv 4 2 d. Fo exaple, if A, whee A is a constant, it is left to a poble (Poble 63) to show that M A 4. Hence, we see fo Equation 14.26b that F is popotional to 2 in this case and is zeo at the cente. Quick Quiz 14.4 A paticle is pojected though a sall hole into the inteio of a spheical shell. Descibe EXAMPLE A Fee Ride, Thanks to Gavity An object of ass oves in a sooth, staight tunnel dug between two points on the Eath s suface (Fig ). Show that the object oves with siple haonic otion, and find the peiod of its otion. Assue that the Eath s density is unifo. Solution The gavitational foce exeted on the object acts towad the Eath s cente and is given by Equation 14.27: F g GM R 3 ˆ We eceive ou fist indication that this foce should esult in siple haonic otion by copaing it to Hooke s law, fist seen in Section 7.3. Because the gavitational foce on the object is linealy popotional to the displaceent, the object expeiences a Hooke s law foce. The y coponent of the gavitational foce on the object is balanced by the noal foce exeted by the tunnel wall, and the x coponent is F x GM E R 3 cos E Because the x coodinate of the object is x cos, we can wite F x GM E R 3 x E Applying Newton s second law to the otion along the x diection gives F x GM E R 3 x a x E

26 448 CHAPTER 14 The Law of Gavity Figue O y x F g An object oves along a tunnel dug though the Eath. The coponent of the gavitational foce F g along the x axis is the diving foce fo the otion. Note that this coponent always acts towad O. Solving fo a x, we obtain θ If we use the sybol 2 fo the coefficient of x GM E /R 3 E 2 we see that (1) a x 2 x a x GM E R 3 x E an expession that atches the atheatical fo of Equation 13.9, which gives the acceleation of a paticle in siple haonic otion: a 2 x x. Theefoe, Equation (1), x which we have deived fo the acceleation of ou object in the tunnel, is the acceleation equation fo siple haonic otion at angula speed with Thus, the object in the tunnel oves in the sae way as a block hanging fo a sping! The peiod of oscillation is T 2 R 3 2 E GM E s GM E 84.3 in R 3 E 2 ( ) 3 ( N 2 /kg 2 )( kg) This peiod is the sae as that of a satellite taveling in a cicula obit just above the Eath s suface (ignoing any tees, buildings, o othe objects in the way). Note that the esult is independent of the length of the tunnel. A poposal has been ade to opeate a ass-tansit syste between any two cities, using the pinciple descibed in this exaple. A one-way tip would take about 42 in. A oe pecise calculation of the otion ust account fo the fact that the Eath s density is not unifo. Moe ipotant, thee ae any pactical pobles to conside. Fo instance, it would be ipossible to achieve a fictionless tunnel, and so soe auxiliay powe souce would be equied. Can you think of othe pobles? the otion of the paticle inside the shell. SUMMARY Newton s law of univesal gavitation states that the gavitational foce of attaction between any two paticles of asses 1 and 2 sepaated by a distance has the agnitude F g G (14.1) whee G N 2 /kg 2 is the univesal gavitational constant. This equation enables us to calculate the foce of attaction between asses unde a wide vaiety of cicustances. An object at a distance h above the Eath s suface expeiences a gavitational foce of agnitude g, whee g is the fee-fall acceleation at that elevation: g GM E 2 GM E (R E h) 2 (14.6)

27 Suay 449 In this expession, M E is the ass of the Eath and R E is its adius. Thus, the weight of an object deceases as the object oves away fo the Eath s suface. Keple s laws of planetay otion state that 1. All planets ove in elliptical obits with the Sun at one focal point. 2. The adius vecto dawn fo the Sun to a planet sweeps out equal aeas in equal tie intevals. 3. The squae of the obital peiod of any planet is popotional to the cube of the seiajo axis of the elliptical obit. Keple s thid law can be expessed as T GM S 3 (14.7) whee M S is the ass of the Sun and is the obital adius. Fo elliptical obits, Equation 14.7 is valid if is eplaced by the seiajo axis a. Most planets have nealy cicula obits aound the Sun. The gavitational field at a point in space equals the gavitational foce expeienced by any test paticle located at that point divided by the ass of the test paticle: g F g (14.10) The gavitational foce is consevative, and theefoe a potential enegy function can be defined. The gavitational potential enegy associated with two paticles sepaated by a distance is U G 1 2 (14.15) whee U is taken to be zeo as :. The total potential enegy fo a syste of paticles is the su of enegies fo all pais of paticles, with each pai epesented by a te of the fo given by Equation If an isolated syste consists of a paticle of ass oving with a speed v in the vicinity of a assive body of ass M, the total enegy E of the syste is the su of the kinetic and potential enegies: E 1 2 v 2 GM (14.17) The total enegy is a constant of the otion. If the paticle oves in a cicula obit of adius aound the assive body and if M W, the total enegy of the syste is E GM 2 (14.19) The total enegy is negative fo any bound syste. The escape speed fo an object pojected fo the suface of the Eath is v esc 2GM E R E (14.22)

28 450 CHAPTER 14 The Law of Gavity QUESTIONS 1. Use Keple s second law to convince youself that the Eath ust ove faste in its obit duing Decebe, when it is closest to the Sun, than duing June, when it is fathest fo the Sun. 2. The gavitational foce that the Sun exets on the Moon is about twice as geat as the gavitational foce that the Eath exets on the Moon. Why doesn t the Sun pull the Moon away fo the Eath duing a total eclipse of the Sun? 3. If a syste consists of five paticles, how any tes appea in the expession fo the total potential enegy? How any tes appea if the syste consists of N paticles? 4. Is it possible to calculate the potential enegy function associated with a paticle and an extended body without knowing the geoety o ass distibution of the extended body? 5. Does the escape speed of a ocket depend on its ass? Explain. 6. Copae the enegies equied to each the Moon fo a kg spacecaft and a kg satellite. 7. Explain why it takes oe fuel fo a spacecaft to tavel fo the Eath to the Moon than fo the etun tip. Estiate the diffeence. 8. Why don t we put a geosynchonous weathe satellite in obit aound the 45th paallel? Wouldn t this be oe useful fo the United States than such a satellite in obit aound the equato? 9. Is the potential enegy associated with the Eath Moon syste geate than, less than, o equal to the kinetic enegy of the Moon elative to the Eath? 10. Explain why no wok is done on a planet as it oves in a cicula obit aound the Sun, even though a gavita- tional foce is acting on the planet. What is the net wok done on a planet duing each evolution as it oves aound the Sun in an elliptical obit? 11. Explain why the foce exeted on a paticle by a unifo sphee ust be diected towad the cente of the sphee. Would this be the case if the ass distibution of the sphee wee not spheically syetic? 12. Neglecting the density vaiation of the Eath, what would be the peiod of a paticle oving in a sooth hole dug between opposite points on the Eath s suface, passing though its cente? 13. At what position in its elliptical obit is the speed of a planet a axiu? At what position is the speed a iniu? 14. If you wee given the ass and adius of planet X, how would you calculate the fee-fall acceleation on the suface of this planet? 15. If a hole could be dug to the cente of the Eath, do you think that the foce on a ass would still obey Equation 14.1 thee? What do you think the foce on would be at the cente of the Eath? 16. In his 1798 expeient, Cavendish was said to have weighed the Eath. Explain this stateent. 17. The gavitational foce exeted on the Voyage spacecaft by Jupite acceleated it towad escape speed fo the Sun. How is this possible? 18. How would you find the ass of the Moon? 19. The Apollo 13 spaceship developed touble in the oxygen syste about halfway to the Moon. Why did the spaceship continue on aound the Moon and then etun hoe, athe than iediately tun back to Eath? PROBLEMS 1, 2, 3 = staightfowad, inteediate, challenging = full solution available in the Student Solutions Manual and Study Guide WEB = solution posted at = Copute useful in solving poble = Inteactive Physics = paied nueical/sybolic pobles Section 14.1 Newton s Law of Univesal Gavitation Section 14.2 Measuing the Gavitational Constant Section 14.3 Fee-Fall Acceleation and the Gavitational Foce 1. Deteine the ode of agnitude of the gavitational foce that you exet on anothe peson 2 away. In you solution, state the quantities that you easue o estiate and thei values. 2. A 200-kg ass and a 500-kg ass ae sepaated by (a) Find the net gavitational foce exeted by these asses on a 50.0-kg ass placed idway between the. (b) At what position (othe than infinitely e- ote ones) can the 50.0-kg ass be placed so as to expeience a net foce of zeo? 3. Thee equal asses ae located at thee cones of a squae of edge length, as shown in Figue P14.3. Find the gavitational field g at the fouth cone due to these asses. 4. Two objects attact each othe with a gavitational foce of agnitude N when sepaated by 20.0 c. If the total ass of the two objects is 5.00 kg, what is the ass of each? 5. Thee unifo sphees of asses 2.00 kg, 4.00 kg, and 6.00 kg ae placed at the cones of a ight tiangle, as illustated in Figue P14.5. Calculate the esultant gavi-

29 Pobles 451 y O x Figue P14.3 y (0, 3.00) 2.00 kg F 24 ( 4.00, 0) x F 64 O 4.00 kg 6.00 kg 11. A student poposes to easue the gavitational constant G by suspending two spheical asses fo the ceiling of a tall cathedal and easuing the deflection of the cables fo the vetical. Daw a fee-body diaga of one of the asses. If two kg asses ae suspended at the end of long cables, and the cables ae attached to the ceiling apat, what is the sepaation of the asses? 12. On the way to the Moon, the Apollo astonauts eached a point whee the Moon s gavitational pull becae stonge than the Eath s. (a) Deteine the distance of this point fo the cente of the Eath. (b) What is the acceleation due to the Eath s gavity at this point? Section 14.4 Keple s Laws Section 14.5 The Law of Gavity and the Motion of Planets 13. A paticle of ass oves along a staight line with constant speed in the x diection, a distance b fo the x axis (Fig. P14.13). Show that Keple s second law is satisfied by deonstating that the two shaded tiangles in the figue have the sae aea when t 4 t 3 t 2 t 1. Figue P14.5 v 0 y t 1 t 2 t 3 t 4 WEB tational foce on the 4.00-kg ass, assuing that the sphees ae isolated fo the est of the Univese. 6. The fee-fall acceleation on the suface of the Moon is about one-sixth that on the suface of the Eath. If the adius of the Moon is about 0.250R E, find the atio of thei aveage densities, Moon / Eath. 7. Duing a sola eclipse, the Moon, Eath, and Sun all lie on the sae line, with the Moon between the Eath and the Sun. (a) What foce is exeted by the Sun on the Moon? (b) What foce is exeted by the Eath on the Moon? (c) What foce is exeted by the Sun on the Eath? 8. The cente-to-cente distance between the Eath and the Moon is k. The Moon copletes an obit in 27.3 days. (a) Deteine the Moon s obital speed. (b) If gavity wee switched off, the Moon would ove along a staight line tangent to its obit, as descibed by Newton s fist law. In its actual obit in 1.00 s, how fa does the Moon fall below the tangent line and towad the Eath? 9. When a falling eteooid is at a distance above the Eath s suface of 3.00 ties the Eath s adius, what is its acceleation due to the Eath s gavity? 10. Two ocean lines, each with a ass of etic tons, ae oving on paallel couses, 100 apat. What is the agnitude of the acceleation of one of the lines towad the othe due to thei utual gavitational attaction? (Teat the ships as point asses.) b O Figue P A counications satellite in geosynchonous obit eains above a single point on the Eath s equato as the planet otates on its axis. (a) Calculate the adius of its obit. (b) The satellite elays a adio signal fo a tansitte nea the noth pole to a eceive, also nea the noth pole. Taveling at the speed of light, how long is the adio wave in tansit? 15. Plaskett s binay syste consists of two stas that evolve in a cicula obit about a cente of ass idway between the. This eans that the asses of the two stas ae equal (Fig. P14.15). If the obital velocity of each sta is 220 k/s and the obital peiod of each is 14.4 days, find the ass M of each sta. (Fo copaison, the ass of ou Sun is kg.) 16. Plaskett s binay syste consists of two stas that evolve in a cicula obit about a cente of gavity idway between the. This eans that the asses of the two stas ae equal (see Fig. P14.15). If the obital speed of each sta is v and the obital peiod of each is T, find the ass M of each sta. x

30 452 CHAPTER 14 The Law of Gavity 220 k/s CM M 20. Two planets, X and Y, tavel counteclockwise in cicula obits about a sta, as shown in Figue P The adii of thei obits ae in the atio 3:1. At soe tie, they ae aligned as in Figue P14.20a, aking a staight line with the sta. Duing the next five yeas, the angula displaceent of planet X is 90.0, as shown in Figue P14.20b. Whee is planet Y at this tie? X M 220 k/s Figue P14.15 Pobles 15 and 16. Y X Y 17. The Exploe VIII satellite, placed into obit Novebe 3, 1960, to investigate the ionosphee, had the following obit paaetes: peigee, 459 k; apogee, k (both distances above the Eath s suface); and peiod, in. Find the atio v p /v a of the speed at peigee to that at apogee. 18. Coet Halley (Fig. P14.18) appoaches the Sun to within AU, and its obital peiod is 75.6 yeas (AU is the sybol fo astonoical unit, whee 1 AU is the ean Eath Sun distance). How fa fo the Sun will Halley s coet tavel befoe it stats its etun jouney? AU Sun 2a x Figue P14.18 (a) Figue P A synchonous satellite, which always eains above the sae point on a planet s equato, is put in obit aound Jupite so that scientists can study the faous ed spot. Jupite otates once evey 9.84 h. Use the data in Table 14.2 to find the altitude of the satellite. 22. Neuton stas ae exteely dense objects that ae foed fo the enants of supenova explosions. Many otate vey apidly. Suppose that the ass of a cetain spheical neuton sta is twice the ass of the Sun and that its adius is 10.0 k. Deteine the geatest possible angula speed it can have fo the atte at the suface of the sta on its equato to be just held in obit by the gavitational foce. 23. The Sola and Heliospheic Obsevatoy (SOHO) spacecaft has a special obit, chosen so that its view of the Sun is neve eclipsed and it is always close enough to the Eath to tansit data easily. It oves in a neacicle aound the Sun that is salle than the Eath s cicula obit. Its peiod, howeve, is not less than 1 y but is just equal to 1 y. It is always located between the Eath and the Sun along the line joining the. Both objects exet gavitational foces on the obsevatoy. Show that the spacecaft s distance fo the Eath ust be between and In 1772 Joseph Louis Lagange deteined theoetically the special location that allows this obit. The SOHO spacecaft took this position on Febuay 14, (Hint: Use data that ae pecise to fou digits. The ass of the Eath is kg.) (b) WEB 19. Io, a satellite of Jupite, has an obital peiod of 1.77 days and an obital adius of k. Fo these data, deteine the ass of Jupite. Section 14.6 The Gavitational Field 24. A spacecaft in the shape of a long cylinde has a length of 100, and its ass with occupants is kg. It has

31 Pobles stayed too close to a 1.0--adius black hole having a ass 100 ties that of the Sun (Fig. P14.24). The nose of the spacecaft is pointing towad the cente of the black hole, and the distance between the nose and the black hole is 10.0 k. (a) Deteine the total foce on the spacecaft. (b) What is the diffeence in the gavitational fields acting on the occupants in the nose of the ship and on those in the ea of the ship, fathest fo the black hole? 25. Copute the agnitude and diection of the gavitational field at a point P on the pependicula bisecto of two equal asses sepaated by a distance 2a, as shown in Figue P a M M Figue P14.24 Figue P k Black hole 26. Find the gavitational field at a distance along the axis of a thin ing of ass M and adius a. Section 14.7 Gavitational Potential Enegy Note: Assue that U 0 as :. 27. A satellite of the Eath has a ass of 100 kg and is at an altitude of (a) What is the potential enegy of the satellite Eath syste? (b) What is the agnitude of the gavitational foce exeted by the Eath on the satellite? (c) What foce does the satellite exet on the Eath? 28. How uch enegy is equied to ove a kg ass fo the Eath s suface to an altitude twice the Eath s adius? 29. Afte ou Sun exhausts its nuclea fuel, its ultiate fate ay be to collapse to a white-dwaf state, in which it has appoxiately the sae ass it has now but a adius P WEB equal to the adius of the Eath. Calculate (a) the aveage density of the white dwaf, (b) the acceleation due to gavity at its suface, and (c) the gavitational potential enegy associated with a 1.00-kg object at its suface. 30. At the Eath s suface a pojectile is launched staight up at a speed of 10.0 k/s. To what height will it ise? Ignoe ai esistance. 31. A syste consists of thee paticles, each of ass 5.00 g, located at the cones of an equilateal tiangle with sides of 30.0 c. (a) Calculate the potential enegy of the syste. (b) If the paticles ae eleased siultaneously, whee will they collide? 32. How uch wok is done by the Moon s gavitational field as a kg eteo coes in fo oute space and ipacts the Moon s suface? Section 14.8 Enegy Consideations in Planetay and Satellite Motion 33. A 500-kg satellite is in a cicula obit at an altitude of 500 k above the Eath s suface. Because of ai fiction, the satellite is eventually bought to the Eath s suface, and it hits the Eath with a speed of 2.00 k/s. How uch enegy was tansfoed to intenal enegy by eans of fiction? 34. (a) What is the iniu speed, elative to the Sun, that is necessay fo a spacecaft to escape the Sola Syste if it stats at the Eath s obit? (b) Voyage 1 achieved a axiu speed of k/h on its way to photogaph Jupite. Beyond what distance fo the Sun is this speed sufficient fo a spacecaft to escape the Sola Syste? 35. A satellite with a ass of 200 kg is placed in Eath obit at a height of 200 k above the suface. (a) Assuing a cicula obit, how long does the satellite take to coplete one obit? (b) What is the satellite s speed? (c) What is the iniu enegy necessay to place this satellite in obit (assuing no ai fiction)? 36. A satellite of ass is placed in Eath obit at an altitude h. (a) Assuing a cicula obit, how long does the satellite take to coplete one obit? (b) What is the satellite s speed? (c) What is the iniu enegy necessay to place this satellite in obit (assuing no ai fiction)? 37. A spaceship is fied fo the Eath s suface with an initial speed of /s. What will its speed be when it is vey fa fo the Eath? (Neglect fiction.) 38. A kg satellite obits the Eath at a constant altitude of 100 k. How uch enegy ust be added to the syste to ove the satellite into a cicula obit at an altitude of 200 k? 39. A teetop satellite oves in a cicula obit just above the suface of a planet, which is assued to offe no ai esistance. Show that its obital speed v and the escape speed fo the planet ae elated by the expession v esc 2v. 40. The planet Uanus has a ass about 14 ties the Eath s ass, and its adius is equal to about 3.7 Eath

32 454 CHAPTER 14 The Law of Gavity adii. (a) By setting up atios with the coesponding Eath values, find the acceleation due to gavity at the cloud tops of Uanus. (b) Ignoing the otation of the planet, find the iniu escape speed fo Uanus. 41. Deteine the escape velocity fo a ocket on the fa side of Ganyede, the lagest of Jupite s oons. The adius of Ganyede is , and its ass is kg. The ass of Jupite is kg, and the distance between Jupite and Ganyede is Be sue to include the gavitational effect due to Jupite, but you ay ignoe the otions of Jupite and Ganyede as they evolve about thei cente of ass (Fig. P14.41). L d Figue P14.44 R M L Figue P14.45 Ganyede v (Optional) Section The Gavitational Foce Between a Paticle and a Spheical Mass 46. (a) Show that the peiod calculated in Exaple can be witten as Jupite Figue P In Robet Heinlein s The Moon is a Hash Mistess, the colonial inhabitants of the Moon theaten to launch ocks down onto the Eath if they ae not given independence (o at least epesentation). Assuing that a ail gun could launch a ock of ass at twice the luna escape speed, calculate the speed of the ock as it entes the Eath s atosphee. (By luna escape speed we ean the speed equied to escape entiely fo a stationay Moon alone in the Univese.) 43. Deive an expession fo the wok equied to ove an Eath satellite of ass fo a cicula obit of adius 2R E to one of adius 3R E. T 2 R E g whee g is the fee-fall acceleation on the suface of the Eath. (b) What would this peiod be if tunnels wee ade though the Moon? (c) What pactical poble egading these tunnels on Eath would be eoved if they wee built on the Moon? 47. A 500-kg unifo solid sphee has a adius of Find the agnitude of the gavitational foce exeted by the sphee on a 50.0-g paticle located (a) 1.50 fo the cente of the sphee, (b) at the suface of the sphee, and (c) fo the cente of the sphee. 48. A unifo solid sphee of ass 1 and adius R 1 is inside and concentic with a spheical shell of ass 2 and adius R 2 (Fig. P14.48). Find the gavitational foce exeted by the sphees on a paticle of ass located at (a) a, (b) b, and (c) c, whee is easued fo the cente of the sphees. (Optional) Section 14.9 The Gavitational Foce Between an Extended Object and a Paticle 44. Conside two identical unifo ods of length L and ass lying along the sae line and having thei closest points sepaated by a distance d (Fig. P14.44). Show that the utual gavitational foce between these ods has a agnitude F G2 L 2 ln (L d)2 d(2l d) 45. A unifo od of ass M is in the shape of a seicicle of adius R (Fig. P14.45). Calculate the foce on a point ass placed at the cente of the seicicle. R c R 1 a b Figue P14.48

33 Pobles 455 ADDITIONAL PROBLEMS 49. Let g M epesent the diffeence in the gavitational fields poduced by the Moon at the points on the Eath s suface neaest to and fathest fo the Moon. Find the faction g M /g, whee g is the Eath s gavitational field. (This diffeence is esponsible fo the occuence of the luna tides on the Eath.) 50. Two sphees having asses M and 2M and adii R and 3R, espectively, ae eleased fo est when the distance between thei centes is 12R. How fast will each sphee be oving when they collide? Assue that the two sphees inteact only with each othe. 51. In Lay Niven s science-fiction novel Ringwold, a igid ing of ateial otates about a sta (Fig. P14.51). The otational speed of the ing is /s, and its adius is (a) Show that the centipetal acceleation of the inhabitants is 10.2 /s 2. (b) The inhabitants of this ing wold expeience a noal contact foce n. Acting alone, this noal foce would poduce an inwad acceleation of 9.90 /s 2. Additionally, the sta at the cente of the ing exets a gavitational foce on the ing and its inhabitants. The diffeence between the total acceleation and the acceleation povided by the noal foce is due to the gavitational attaction of the cental sta. Show that the ass of the sta is appoxiately kg. Sta 52. (a) Show that the ate of change of the fee-fall acceleation with distance above the Eath s suface is dg d 2GM E R 3 E This ate of change ove distance is called a gadient. (b) If h is sall copaed to the adius of the Eath, show that the diffeence in fee-fall acceleation between two points sepaated by vetical distance h is g 2GM Eh R 3 E n Figue P14.51 F g WEB (c) Evaluate this diffeence fo h 6.00, a typical height fo a two-stoy building. 53. A paticle of ass is located inside a unifo solid sphee of adius R and ass M, at a distance fo its cente. (a) Show that the gavitational potential enegy of the syste is U (GM/2R 3 ) 2 3GM/2R. (b) Wite an expession fo the aount of wok done by the gavitational foce in binging the paticle fo the suface of the sphee to its cente. 54. Voyages 1 and 2 suveyed the suface of Jupite s oon Io and photogaphed active volcanoes spewing liquid sulfu to heights of 70 k above the suface of this oon. Find the speed with which the liquid sulfu left the volcano. Io s ass is kg, and its adius is k. 55. As an astonaut, you obseve a sall planet to be spheical. Afte landing on the planet, you set off, walking always staight ahead, and find youself etuning to you spacecaft fo the opposite side afte copleting a lap of 25.0 k. You hold a hae and a falcon feathe at a height of 1.40, elease the, and obseve that they fall togethe to the suface in 29.2 s. Deteine the ass of the planet. 56. A cylindical habitat in space, 6.00 k in diaete and 30 k long, was poposed by G. K. O Neill in Such a habitat would have cities, land, and lakes on the inside suface and ai and clouds in the cente. All of these would be held in place by the otation of the cylinde about its long axis. How fast would the cylinde have to otate to iitate the Eath s gavitational field at the walls of the cylinde? 57. In intoductoy physics laboatoies, a typical Cavendish balance fo easuing the gavitational constant G uses lead sphees with asses of 1.50 kg and 15.0 g whose centes ae sepaated by about 4.50 c. Calculate the gavitational foce between these sphees, teating each as a point ass located at the cente of the sphee. 58. Newton s law of univesal gavitation is valid fo distances coveing an enoous ange, but it is thought to fail fo vey sall distances, whee the stuctue of space itself is uncetain. The cossove distance, fa less than the diaete of an atoic nucleus, is called the Planck length. It is deteined by a cobination of the constants G, c, and h, whee c is the speed of light in vacuu and h is Planck s constant (intoduced biefly in Chapte 11 and discussed in geate detail in Chapte 40) with units of angula oentu. (a) Use diensional analysis to find a cobination of these thee univesal constants that has units of length. (b) Deteine the ode of agnitude of the Planck length. (Hint: You will need to conside nonintege powes of the constants.) 59. Show that the escape speed fo the suface of a planet of unifo density is diectly popotional to the adius of the planet. 60. (a) Suppose that the Eath (o anothe object) has density (), which can vay with adius but is spheically

34 456 CHAPTER 14 The Law of Gavity WEB syetic. Show that at any paticula adius inside the Eath, the gavitational field stength g() will incease as inceases, if and only if the density thee exceeds 2/3 the aveage density of the potion of the Eath inside the adius. (b) The Eath as a whole has an aveage density of 5.5 g/c 3, while the density at the suface is 1.0 g/c 3 on the oceans and about 3 g/c 3 on land. What can you infe fo this? 61. Two hypothetical planets of asses 1 and 2 and adii 1 and 2, espectively, ae nealy at est when they ae an infinite distance apat. Because of thei gavitational attaction, they head towad each othe on a collision couse. (a) When thei cente-to-cente sepaation is d, find expessions fo the speed of each planet and thei elative velocity. (b) Find the kinetic enegy of each planet just befoe they collide, if kg, kg, , and (Hint: Both enegy and oentu ae conseved.) 62. The axiu distance fo the Eath to the Sun (at ou aphelion) is , and the distance of closest appoach (at peihelion) is If the Eath s obital speed at peihelion is k/s, deteine (a) the Eath s obital speed at aphelion, (b) the kinetic and potential enegies at peihelion, and (c) the kinetic and potential enegies at aphelion. Is the total enegy constant? (Neglect the effect of the Moon and othe planets.) 63. A sphee of ass M and adius R has a nonunifo density that vaies with, the distance fo its cente, accoding to the expession A, fo 0 R. (a) What is the constant A in tes of M and R? (b) Deteine an expession fo the foce exeted on a paticle of ass placed outside the sphee. (c) Deteine an expession fo the foce exeted on the paticle if it is inside the sphee. (Hint: See Section and note that the distibution is spheically syetic.) 64. (a) Deteine the aount of wok (in joules) that ust be done on a 100-kg payload to elevate it to a height of k above the Eath s suface. (b) Deteine the aount of additional wok that is equied to put the payload into cicula obit at this elevation. 65. X-ay pulses fo Cygnus X-1, a celestial x-ay souce, have been ecoded duing high-altitude ocket flights. The signals can be intepeted as oiginating when a blob of ionized atte obits a black hole with a peiod of 5.0 s. If the blob is in a cicula obit about a black hole whose ass is 20M Sun, what is the obital adius? 66. Studies of the elationship of the Sun to its galaxy the Milky Way have evealed that the Sun is located nea the oute edge of the galactic disk, about lightyeas fo the cente. Futheoe, it has been found that the Sun has an obital speed of appoxiately 250 k/s aound the galactic cente. (a) What is the peiod of the Sun s galactic otion? (b) What is the ode of agnitude of the ass of the Milky Way galaxy? Suppose that the galaxy is ade ostly of stas, of which the Sun is typical. What is the ode of agnitude of the nube of stas in the Milky Way? 67. The oldest atificial satellite in obit is Vanguad I, launched Mach 3, Its ass is 1.60 kg. In its initial obit, its iniu distance fo the cente of the Eath was 7.02 M, and its speed at this peigee point was 8.23 k/s. (a) Find its total enegy. (b) Find the agnitude of its angula oentu. (c) Find its speed at apogee and its axiu (apogee) distance fo the cente of the Eath. (d) Find the seiajo axis of its obit. (e) Deteine its peiod. 68. A ocket is given an initial speed vetically upwad of v i 2 Rg at the suface of the Eath, which has adius R and suface fee-fall acceleation g. The ocket otos ae quickly cut off, and theeafte the ocket coasts unde the action of gavitational foces only. (Ignoe atospheic fiction and the Eath s otation.) Deive an expession fo the subsequent speed v as a function of the distance fo the cente of the Eath in tes of g, R, and. 69. Two stas of asses M and, sepaated by a distance d, evolve in cicula obits about thei cente of ass (Fig. P14.69). Show that each sta has a peiod given by T d 3 G(M ) (Hint: Apply Newton s second law to each sta, and note that the cente-of-ass condition equies that M 2 1, whee 1 2 d.) v 1 1 d 2 Figue P14.69 CM v (a) A 5.00-kg ass is eleased fo the cente of the Eath. It oves with what acceleation elative to the Eath? (b) A kg ass is eleased fo the cente of the Eath. It oves with what acceleation elative to the Eath? Assue that the objects behave as pais of paticles, isolated fo the est of the Univese. 71. The acceleation of an object oving in the gavitational field of the Eath is a GM E 3 M

35 Answes to Quick Quizzes 457 whee is the position vecto diected fo the cente of the Eath to the object. Choosing the oigin at the cente of the Eath and assuing that the sall object is oving in the xy plane, we find that the ectangula (catesian) coponents of its acceleation ae Use a copute to set up and cay out a nueical pea x GM Ex GM (x 2 y 2 ) 3/2 a y E y (x 2 y 2 ) 3/2 diction of the otion of the object, accoding to Eule s ethod. Assue that the initial position of the object is x 0 and y 2R E, whee R E is the adius of the Eath. Give the object an initial velocity of /s in the x diection. The tie inceent should be ade as sall as pactical. Ty 5 s. Plot the x and y coodinates of the object as tie goes on. Does the object hit the Eath? Vay the initial velocity until you find a cicula obit. ANSWERS TO QUICK QUIZZES 14.1 Keple s thid law (Eq. 14.7), which applies to all the planets, tells us that the peiod of a planet is popotional to 3/2. Because Satun and Jupite ae fathe fo the Sun than the Eath is, they have longe peiods. The Sun s gavitational field is uch weake at Satun and Jupite than it is at the Eath. Thus, these planets expeience uch less centipetal acceleation than the Eath does, and they have coespondingly longe peiods The ass of the asteoid ight be so sall that you would be able to exceed escape velocity by leg powe alone. You would jup up, but you would neve coe back down! 14.3 Keple s fist law applies not only to planets obiting the Sun but also to any elatively sall object obiting anothe unde the influence of gavity. Any elliptical path that does not touch the Eath befoe eaching point G will continue aound the othe side to point V in a coplete obit (see figue in next colun) The gavitational foce is zeo inside the shell (Eq b). Because the foce on it is zeo, the paticle oves with constant velocity in the diection of its oiginal otion outside the shell until it hits the wall opposite the enty hole. Its path theeafte depends on the natue of the collision and on the paticle s oiginal diection.