Universal Gravitation

Transcription

1 chapte 13 Univesal Gavitation 13.1 Newton s Law of Univesal Gavitation 13.2 Fee-Fall Acceleation and the Gavitational Foce 13.3 Keple s Laws and the Motion of Planets 13.4 The Gavitational Field 13.5 Gavitational Potential Enegy 13.6 Enegy Consideations in Planetay and atellite Motion Befoe 1687, a lage amount of data had been collected on the motions of the Moon and the planets, but a clea undestanding of the foces elated to these motions was not available. In that yea, Isaac Newton povided the key that unlocked the secets of the heavens. He knew, fom his fist law, that a net foce had to be acting on The Whilpool Galaxy (left) was discoveed in 1774 by Chales the Moon because without such a foce the Moon would Messie and is listed in his astonomical catalog as M51. It is move in a staight-line path athe than in its almost known fo its shaply defined spial ams. The smalle companion galaxy (ight) is at the same distance fom Eath as the Whilpool cicula obit. Newton easoned that this foce was the Galaxy. The visual appeaance of the pai indicates a gavitational gavitational attaction exeted by the Eath on the inteaction between them. (NAA, EA,. Beckwith (TcI), and The Hubble Heitage Team (TcI/AURA)) Moon. He ealized that the foces involved in the Eath Moon attaction and in the un planet attaction wee not something special to those systems, but athe wee paticula cases of a geneal and univesal attaction between objects. In othe wods, Newton saw that the same foce of attaction that causes the Moon to follow its path aound the Eath also causes an apple to fall fom a tee. It was the fist time that eathly and heavenly motions wee unified. In this chapte, we study the law of univesal gavitation. We emphasize a desciption of planetay motion because astonomical data povide an impotant test of this law s validity. We then show that the laws of planetay motion developed by Johannes Keple follow fom the law of univesal gavitation and the pinciple of consevation of angula momentum. We conclude by deiving a geneal expession fo the gavitational potential enegy of a system and examining the enegetics of planetay and satellite motion. 374

2 13.1 Newton s Law of Univesal Gavitation Newton s Law of Univesal Gavitation You may have head the legend that, while napping unde a tee, Newton was stuck on the head by a falling apple. This alleged accident supposedly pompted him to imagine that pehaps all objects in the Univese wee attacted to each othe in the same way the apple was attacted to the Eath. Newton analyzed astonomical data on the motion of the Moon aound the Eath. Fom that analysis, he made the bold assetion that the foce law govening the motion of planets was the same as the foce law that attacted a falling apple to the Eath. In 1687, Newton published his wok on the law of gavity in his teatise Mathematical 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 masses and invesely popotional to the squae of the distance between them. The law of univesal gavitation If the paticles have masses m 1 and m 2 and ae sepaated by a distance, the magnitude of this gavitational foce is F g 5 G m 1m 2 2 (13.1) whee G is a constant, called the univesal gavitational constant. Its value in I units is G N? m 2 /kg 2 (13.2) Heny Cavendish ( ) measued the univesal gavitational constant in an impotant 1798 expeiment. Cavendish s appaatus consists of two small sphees, each of mass m, fixed to the ends of a light, hoizontal od suspended by a fine fibe o thin metal wie as illustated in Figue When two lage sphees, each of mass M, ae placed nea the smalle ones, the attactive foce between smalle and lage sphees causes the od to otate and twist the wie suspension to a new equilibium oientation. The angle of otation is measued by the deflection of a light beam eflected fom a mio attached to the vetical suspension. The fom of the foce law given by Equation 13.1 is often efeed to as an invese-squae law because the magnitude of the foce vaies as the invese squae of the sepaation of the paticles. 1 We shall see othe examples of this type of foce law in subsequent chaptes. We can expess this foce in vecto fom by defining a unit vecto ^ 12 (Active Fig. 13.2). Because this unit vecto is diected fom paticle 1 towad paticle 2, the foce exeted by paticle 1 on paticle 2 is 12 F 52G m 1m 2 2 ^ 12 (13.3) whee the negative sign indicates that paticle 2 is attacted to paticle 1; hence, the foce on paticle 2 must be diected towad paticle 1. By Newton s thid law, the foce exeted by paticle 2 on paticle 1, designated F21, is equal in magnitude to F12 and in the opposite diection. That is, these foces fom an action eaction pai, and F21 52F 12. Two featues of Equation 13.3 deseve mention. Fist, the gavitational foce is a field foce that always exists between two paticles, egadless of the medium that sepaates them. Because the foce vaies as the invese squae of the distance between the paticles, it deceases apidly with inceasing sepaation. 1 An invese popotionality between two quantities x and y is one in which y 5 k/x, whee k is a constant. A diect popotion between x and y exists when y 5 kx. The dashed line epesents the oiginal position of the od. M m Mio Light souce Figue 13.1 Cavendish appaatus fo measuing G. m 1 Consistent with Newton s thid law, F 21 F 12. F 21 ˆ12 ACTIVE FIGURE 13.2 F 12 m 2 The gavitational foce between two paticles is attactive. The unit vecto ^ 12 is diected fom paticle 1 towad paticle 2.

3 376 CHAPTER 13 Univesal Gavitation Pitfall Pevention 13.1 Be Clea on g and G The symbol g epesents the magnitude of the fee-fall acceleation nea a planet. At the suface of the Eath, g has an aveage value of 9.80 m/s 2. On the othe hand, G is a univesal constant that has the same value eveywhee in the Univese. Equation 13.3 can also be used to show that the gavitational foce exeted by a finite-size, spheically symmetic mass distibution on a paticle outside the distibution is the same as if the entie mass of the distibution wee concentated at the cente. Fo example, the magnitude of the foce exeted by the Eath on a paticle of mass m nea the Eath s suface is F g 5 G M Em R (13.4) 2 E whee M E is the Eath s mass and its adius. This foce is diected towad the cente of the Eath. Quick Quiz 13.1 A planet has two moons of equal mass. Moon 1 is in a cicula obit of adius. Moon 2 is in a cicula obit of adius 2. What is the magnitude of the gavitational foce exeted by the planet on Moon 2? (a) fou times as lage as that on Moon 1 (b) twice as lage as that on Moon 1 (c) equal to that on Moon 1 (d) half as lage as that on Moon 1 (e) one-fouth as lage as that on Moon 1 Example 13.1 Billiads, Anyone? Thee kg billiad balls ae placed on a table at the cones of a ight tiangle as shown in Figue The sides of the tiangle ae of lengths a m, b m, and c m. Calculate the gavitational foce vecto on the cue ball (designated m 1 ) esulting fom the othe two balls as well as the magnitude and diection of this foce. OLUTION Conceptualize Notice in Figue 13.3 that the cue ball is attacted to both othe balls by the gavitational foce. We can see gaphically that the net foce should point upwad and towad the ight. We locate ou coodinate axes as shown in Figue 13.3, placing ou oigin at the position of the cue ball. Figue 13.3 (Example 13.1) The esultant gavitational foce acting on the cue ball is the vecto sum F21 1 F31. Categoize This poblem involves evaluating the gavitational foces on the cue ball using Equation Once these foces ae evaluated, it becomes a vecto addition poblem to find the net foce. a y m 2 c F F21 F u 31 x m 1 b m 3 Analyze Find the foce exeted by m 2 on the cue ball: F21 5 G m 2m 1 j^ a N? m 2 /kg j^ N Find the foce exeted by m 3 on the cue ball: F31 5 G m 3m 1 i^ b N? m 2 /kg i^ N kg kg m2 2 j^ kg kg m2 2 i^ Find the net gavitational foce on the cue ball by adding these foce vectos: F 5 F 31 1 F i^ j^ N

4 13.2 Fee-Fall Acceleation and the Gavitational Foce cont. Find the magnitude of this foce: Find the tangent of the angle u fo the net foce vecto: F 5 "F F " N N tan u 5 F y F x 5 F 21 F N N Evaluate the angle u: u 5 tan 21 (0.562) Finalize The esult fo F shows that the gavitational foces between eveyday objects have extemely small magnitudes Fee-Fall Acceleation and the Gavitational Foce We have called the magnitude of the gavitational foce on an object nea the Eath s suface the weight of the object, whee the weight is given by Equation 5.6. Equation 13.4 is anothe expession fo this foce. Theefoe, we can set Equations 5.6 and 13.4 equal to each othe to obtain mg 5 G M Em 2 g 5 G M E R (13.5) 2 E Equation 13.5 elates the fee-fall acceleation g to physical paametes of the Eath its mass and adius and explains the oigin of the value of 9.80 m/s 2 that we have used in ealie chaptes. Now conside an object of mass m located a distance h above the Eath s suface o a distance fom the Eath s cente, whee 5 1 h. The magnitude of the gavitational foce acting on this object is TABLE 13.1 Fee-Fall Acceleation g at Vaious Altitudes Above the Eath s uface Altitude h (km) g (m/s 2 ) ` 0 F g 5 G M Em 2 5 G M E m 1 1 h2 2 The magnitude of the gavitational foce acting on the object at this position is also F g 5 mg, whee g is the value of the fee-fall acceleation at the altitude h. ubstituting this expession fo F g into the last equation shows that g is given by g 5 GM E 2 5 GM E 1 1 h2 2 (13.6) Theefoe, it follows that g deceases with inceasing altitude. Values of g at vaious altitudes ae listed in Table Because an object s weight is mg, we see that as `, the weight of the object appoaches zeo. Vaiation of g with altitude Quick Quiz 13.2 upeman stands on top of a vey tall mountain and thows a baseball hoizontally with a speed such that the baseball goes into a cicula obit aound the Eath. While the baseball is in obit, what is the magnitude of the acceleation of the ball? (a) It depends on how fast the baseball is thown. (b) It is zeo because the ball does not fall to the gound. (c) It is slightly less than 9.80 m/s 2. (d) It is equal to 9.80 m/s 2.

5 378 CHAPTER 13 Univesal Gavitation Example 13.2 Vaiation of g with Altitude h The Intenational pace tation opeates at an altitude of 350 km. Plans fo the final constuction show that mateial of weight N, measued at the Eath s suface, will have been lifted off the suface by vaious spacecaft. What is the weight of the space station when in obit? OLUTION Conceptualize The mass of the space station is fixed; it is independent of its location. Based on the discussion in this section, we ealize that the value of g will be educed at the height of the space station s obit. Theefoe, its weight will be smalle than that at the suface of the Eath. Categoize This example is a elatively simple substitution poblem. Find the mass of the space station fom its weight at the suface of the Eath: Use Equation 13.6 with h km to find g at the obital location: Use this value of g to find the space station s weight in obit: m 5 F g g N 9.80 m/s kg g 5 GM E 1 1 h N? m 2 /kg kg m m m/s 2 mg 5 ( kg)(8.82 m/s 2 ) N Example 13.3 The Density of the Eath Using the known adius of the Eath and that g m/s 2 at the Eath s suface, find the aveage density of the Eath. OLUTION Conceptualize Assume the Eath is a pefect sphee. The density of mateial in the Eath vaies, but let s adopt a simplified model in which we assume the density to be unifom thoughout the Eath. The esulting density is the aveage density of the Eath. Categoize This example is a elatively simple substitution poblem. olve Equation 13.5 fo the mass of the Eath: ubstitute this mass into the definition of density (Eq. 1.1): M E 5 g 2 G E 5 M E V E g 2 /G p g pg 9.80 m/s 2 p N? m 2 /kg m kg/m 3 WHAT IF? What if you wee told that a typical density of ganite at the Eath s suface is kg/m 3. What would you conclude about the density of the mateial in the Eath s inteio? Answe Because this value is about half the density we calculated as an aveage fo the entie Eath, we would conclude that the inne coe of the Eath has a density much highe than the aveage value. It is most amazing that the Cavendish expeiment which detemines G and can be done on a tabletop combined with simple fee-fall measuements of g povides infomation about the coe of the Eath!

6 13.3 Keple s Laws and the Motion of Planets Keple s Laws and the Motion of Planets Humans have obseved the movements of the planets, stas, and othe celestial objects fo thousands of yeas. In ealy histoy, these obsevations led scientists to egad the Eath as the cente of the Univese. This geocentic model was elaboated and fomalized by the Geek astonome Claudius Ptolemy (c. 100 c. 170) in the second centuy and was accepted fo the next 1400 yeas. In 1543, Polish astonome Nicolaus Copenicus ( ) suggested that the Eath and the othe planets evolved in cicula obits aound the un (the heliocentic model). Danish astonome Tycho Bahe ( ) wanted to detemine how the heavens wee constucted and pusued a poject to detemine the positions of both stas and planets. Those obsevations of the planets and 777 stas visible to the naked eye wee caied out with only a lage sextant and a compass. (The telescope had not yet been invented.) Geman astonome Johannes Keple was Bahe s assistant fo a shot while befoe Bahe s death, wheeupon he acquied his mento s astonomical data and spent 16 yeas tying to deduce a mathematical model fo the motion of the planets. uch data ae difficult to sot out because the moving planets ae obseved fom a moving Eath. Afte many laboious calculations, Keple found that Bahe s data on the evolution of Mas aound the un led to a successful model. Keple s complete analysis of planetay motion is summaized in thee statements known as Keple s laws: Johannes Keple Geman astonome ( ) Keple is best known fo developing the laws of planetay motion based on the caeful obsevations of Tycho Bahe. Eich Lessing/At Resouce, NY 1. All planets move in elliptical obits with the un at one focus. 2. The adius vecto dawn fom the un to a planet sweeps out equal aeas in equal time intevals. 3. The squae of the obital peiod of any planet is popotional to the cube of the semimajo axis of the elliptical obit. Keple s laws Keple s Fist Law We ae familia with cicula obits of objects aound gavitational foce centes fom ou discussions in this chapte. Keple s fist law indicates that the cicula obit is a vey special case and elliptical obits ae the geneal situation. This notion was difficult fo scientists of the time to accept because they believed that pefect cicula obits of the planets eflected the pefection of heaven. Active Figue 13.4 shows the geomety of an ellipse, which seves as ou model fo the elliptical obit of a planet. An ellipse is mathematically defined by choosing two points F 1 and F 2, each of which is a called a focus, and then dawing a cuve though points fo which the sum of the distances 1 and 2 fom F 1 and F 2, espectively, is a constant. The longest distance though the cente between points on the ellipse (and passing though each focus) is called the majo axis, and this distance is 2a. In Active Figue 13.4, the majo axis is dawn along the x diection. The distance a is called the semimajo axis. imilaly, the shotest distance though the cente between points on the ellipse is called the mino axis of length 2b, whee the distance b is the semimino axis. Eithe focus of the ellipse is located at a distance c fom the cente of the ellipse, whee a 2 5 b 2 1 c 2. In the elliptical obit of a planet aound the un, the un is at one focus of the ellipse. Thee is nothing at the othe focus. The eccenticity of an ellipse is defined as e 5 c/a, and it descibes the geneal shape of the ellipse. Fo a cicle, c 5 0, and the eccenticity is theefoe zeo. The smalle b is compaed with a, the shote the ellipse is along the y diection compaed with its extent in the x diection in Active Figue As b deceases, c inceases and the eccenticity e inceases. Theefoe, highe values of eccenticity coespond to longe and thinne ellipses. The ange of values of the eccenticity fo an ellipse is 0, e, 1. The semimajo axis has length a, and the semimino axis has length b. 1 2 F 1 ACTIVE FIGURE 13.4 Plot of an ellipse. y c a F 2 Each focus is located at a distance c fom the cente. Pitfall Pevention 13.2 Whee Is the un? The un is located at one focus of the elliptical obit of a planet. It is not located at the cente of the ellipse. b x

7 380 CHAPTER 13 Univesal Gavitation Figue 13.5 (a) The shape of the obit of Mecuy, which has the highest eccenticity (e ) among the eight planets in the sola system. (b) The shape of the obit of Comet Halley. The shape of the obit is coect; the comet and the un ae shown lage than in eality fo claity. The un is located at a focus of the ellipse. Thee is nothing physical located at the cente (the black dot) o the othe focus (the blue dot). un Obit of Comet Halley un Cente Obit of Mecuy Comet Halley Cente a b un M a un F g M p v d dt v Eccenticities fo planetay obits vay widely in the sola system. The eccenticity of the Eath s obit is 0.017, which makes it nealy cicula. On the othe hand, the eccenticity of Mecuy s obit is 0.21, the highest of the eight planets. Figue 13.5a shows an ellipse with an eccenticity equal to that of Mecuy s obit. Notice that even this highest-eccenticity obit is difficult to distinguish fom a cicle, which is one eason Keple s fist law is an admiable accomplishment. The eccenticity of the obit of Comet Halley is 0.97, descibing an obit whose majo axis is much longe than its mino axis, as shown in Figue 13.5b. As a esult, Comet Halley spends much of its 76-yea peiod fa fom the un and invisible fom the Eath. It is only visible to the naked eye duing a small pat of its obit when it is nea the un. Now imagine a planet in an elliptical obit such as that shown in Active Figue 13.4, with the un at focus F 2. When the planet is at the fa left in the diagam, the distance between the planet and the un is a 1 c. At this point, called the aphelion, the planet is at its maximum distance fom the un. (Fo an object in obit aound the Eath, this point is called the apogee.) Convesely, when the planet is at the ight end of the ellipse, the distance between the planet and the un is a 2 c. At this point, called the peihelion (fo an Eath obit, the peigee), the planet is at its minimum distance fom the un. Keple s fist law is a diect esult of the invese-squae natue of the gavitational foce. We have aleady discussed cicula and elliptical obits, the allowed shapes of obits fo objects that ae bound to the gavitational foce cente. These objects include planets, asteoids, and comets that move epeatedly aound the un as well as moons obiting a planet. Thee ae also unbound objects, such as a meteooid fom deep space that might pass by the un once and then neve etun. The gavitational foce between the un and these objects also vaies as the invese squae of the sepaation distance, and the allowed paths fo these objects include paabolas (e 5 1) and hypebolas (e. 1). b da The aea swept out by in a time inteval dt is half the aea of the paallelogam. ACTIVE FIGURE 13.6 (a) The gavitational foce acting on a planet is diected towad the un. (b) Duing a time inteval dt, a paallelogam is fomed by the vectos and d 5 v dt. Keple s econd Law Keple s second law can be shown to be a consequence of angula momentum consevation fo an isolated system as follows. Conside a planet of mass M p moving about the un in an elliptical obit (Active Fig. 13.6a). Let us conside the planet as a system. We model the un to be so much moe massive than the planet that the un does not move. The gavitational foce exeted by the un on the planet is a cental foce, always along the adius vecto, diected towad the un (Active Fig. 13.6a). The toque on the planet due to this cental foce is clealy zeo because Fg is paallel to. Recall that the net extenal toque on a system equals the time ate of change of angula momentum of the system; that is, g text 5 dl /dt (Eq ). Theefoe, because the extenal toque on the planet is zeo, it is modeled as an isolated system fo angula momentum, and the angula momentum L of the planet is a constant of the motion:

8 13.3 Keple s Laws and the Motion of Planets 381 L 5 3 p 5 M p 3 v 5 constant We can elate this esult to the following geometic consideation. In a time inteval dt, the adius vecto in Active Figue 13.6b sweeps out the aea da, which equals half the aea 0 3d 0 of the paallelogam fomed by the vectos and d. Because the displacement of the planet in the time inteval dt is given by d 5 v dt, da d v dt 0 5 L dt 2M p da dt 5 L 2M p (13.7) whee L and M p ae both constants. This esult shows that that the adius vecto fom the un to any planet sweeps out equal aeas in equal time intevals as stated in Keple s second law. This conclusion is a esult of the gavitational foce being a cental foce, which in tun implies that angula momentum of the planet is constant. Theefoe, the law applies to any situation that involves a cental foce, whethe invese squae o not. Keple s Thid Law Keple s thid law can be pedicted fom the invese-squae law fo cicula obits. Conside a planet of mass M p that is assumed to be moving about the un (mass M ) in a cicula obit as in Figue Because the gavitational foce povides the centipetal acceleation of the planet as it moves in a cicle, we model the planet as a paticle unde a net foce and as a paticle in unifom cicula motion and incopoate Newton s law of univesal gavitation, F g 5 M p a GM M p 5 M 2 p a v 2 b v M M p The obital speed of the planet is 2p/T, whee T is the peiod; theefoe, the peceding expession becomes GM 5 12p/T22 2 T 2 5 a 4p 2 b 3 5 K 3 GM whee K is a constant given by Figue 13.7 A planet of mass M p moving in a cicula obit aound the un. The obits of all planets except Mecuy ae nealy cicula. K 5 4p s 2 /m 3 GM This equation is also valid fo elliptical obits if we eplace with the length a of the semimajo axis (Active Fig. 13.4): T 2 5 a 4p 2 ba 3 5 K GM a 3 (13.8) Equation 13.8 is Keple s thid law, which was stated in wods at the beginning of this section. Because the semimajo axis of a cicula obit is its adius, this equation is valid fo both cicula and elliptical obits. Notice that the constant of popotionality K is independent of the mass of the planet. 2 Equation 13.8 is theefoe valid fo any planet. If we wee to conside the obit of a satellite such as the Moon about the Eath, the constant would have a diffeent value, with the un s mass eplaced by the Eath s mass; that is, K E 5 4p 2 /GM E. Keple s thid law 2 Equation 13.8 is indeed a popotion because the atio of the two quantities T 2 and a 3 is a constant. The vaiables in a popotion ae not equied to be limited to the fist powe only.

9 382 CHAPTER 13 Univesal Gavitation Useful Planetay Data TABLE 13.2 Mean Peiod of Mean Distance Body Mass (kg) Radius (m) Revolution (s) fom the un (m) T 2 3 1s2 /m 3 2 Mecuy Venus Eath Mas Jupite atun Uanus Neptune Pluto a Moon un a In August 2006, the Intenational Astonomical Union adopted a definition of a planet that sepaates Pluto fom the othe eight planets. Pluto is now defined as a dwaf planet like the asteoid Cees. Table 13.2 is a collection of useful data fo planets and othe objects in the sola system. The fa-ight column veifies that the atio T 2 / 3 is constant fo all objects obiting the un. The small vaiations in the values in this column ae the esult of uncetainties in the data measued fo the peiods and semimajo axes of the objects. Recent astonomical wok has evealed the existence of a lage numbe of sola system objects beyond the obit of Neptune. In geneal, these objects lie in the Kuipe belt, a egion that extends fom about 30 AU (the obital adius of Neptune) to 50 AU. (An AU is an astonomical unit, equal to the adius of the Eath s obit.) Cuent estimates identify at least objects in this egion with diametes lage than 100 km. The fist Kuipe belt object (KBO) is Pluto, discoveed in 1930 and fomely classified as a planet. tating in 1992, many moe have been detected. eveal have diametes in the km ange, such as Vauna (discoveed in 2000), Ixion (2001), Quaoa (2002), edna (2003), Haumea (2004), Ocus (2004), and Makemake (2005). One KBO, Eis, discoveed in 2005, is believed to be significantly lage than Pluto. Othe KBOs do not yet have names, but ae cuently indicated by thei yea of discovey and a code, such as 2006 QH181 and 2007 UK126. A subset of about KBOs ae called Plutinos because, like Pluto, they exhibit a esonance phenomenon, obiting the un two times in the same time inteval as Neptune evolves thee times. The contempoay application of Keple s laws and such exotic poposals as planetay angula momentum exchange and migating planets suggest the excitement of this active aea of cuent eseach. Quick Quiz 13.3 An asteoid is in a highly eccentic elliptical obit aound the un. The peiod of the asteoid s obit is 90 days. Which of the following statements is tue about the possibility of a collision between this asteoid and the Eath? (a) Thee is no possible dange of a collision. (b) Thee is a possibility of a collision. (c) Thee is not enough infomation to detemine whethe thee is dange of a collision. Example 13.4 The Mass of the un Calculate the mass of the un, noting that the peiod of the Eath s obit aound the un is s and its distance fom the un is m. OLUTION Conceptualize Based on Keple s thid law, we ealize that the mass of the un is elated to the obital size and peiod of a planet.

10 13.3 Keple s Laws and the Motion of Planets cont. Categoize This example is a elatively simple substitution poblem. olve Equation 13.8 fo the mass of the un: M 5 4p 2 3 ubstitute the known values: M 5 GT 2 4p m N? m 2 /kg s kg In Example 13.3, an undestanding of gavitational foces enabled us to find out something about the density of the Eath s coe, and now we have used this undestanding to detemine the mass of the un! Example 13.5 A Geosynchonous atellite Conside a satellite of mass m moving in a cicula obit aound the Eath at a constant speed v and at an altitude h above the Eath s suface as illustated in Figue (A) Detemine the speed of satellite in tems of G, h, (the adius of the Eath), and M E (the mass of the Eath). OLUTION Conceptualize Imagine the satellite moving aound the Eath in a cicula obit unde the influence of the gavitational foce. This motion is simila to that of the space shuttle, the Hubble pace Telescope, and othe objects in obit aound the Eath. Categoize The satellite must have a centipetal acceleation. Theefoe, we categoize the satellite as a paticle unde a net foce and a paticle in unifom cicula motion. Analyze The only extenal foce acting on the satellite is the gavitational foce, which acts towad the cente of the Eath and keeps the satellite in its cicula obit. Apply the paticle unde a net foce and paticle in unifom cicula motion models to the satellite: F g 5 ma G M Em 2 5 m a v 2 b F g m h v Figue 13.8 (Example 13.5) A satellite of mass m moving aound the Eath in a cicula obit of adius with constant speed v. The only foce acting on the satellite is the gavitational foce F g. (Not dawn to scale.) olve fo v, noting that the distance fom the cente of the Eath to the satellite is 5 1 h: (1) v 5 Å GM E 5 Å GM E 1 h (B) If the satellite is to be geosynchonous (that is, appeaing to emain ove a fixed position on the Eath), how fast is it moving though space? OLUTION To appea to emain ove a fixed position on the Eath, the peiod of the satellite must be 24 h s and the satellite must be in obit diectly ove the equato. olve Keple s thid law (Equation 13.8, with a 5 and M M E ) fo : 5 a GM ET 2 1/3 4p 2 b ubstitute numeical values: 5 c N? m 2 /kg kg s m 4p 2 1/3 d continued

11 384 CHAPTER 13 Univesal Gavitation 13.5 cont N? m 2 /kg kg2 Use Equation (1) to find the speed of the satellite: v 5 Å m m/s Finalize The value of calculated hee tanslates to a height of the satellite above the suface of the Eath of almost km. Theefoe, geosynchonous satellites have the advantage of allowing an eathbound antenna to be aimed in a fixed diection, but thee is a disadvantage in that the signals between the Eath and the satellite must tavel a long distance. It is difficult to use geosynchonous satellites fo optical obsevation of the Eath s suface because of thei high altitude. WHAT IF? What if the satellite motion in pat (A) wee taking place at height h above the suface of anothe planet moe massive than the Eath but of the same adius? Would the satellite be moving at a highe speed o a lowe speed than it does aound the Eath? Answe If the planet exets a lage gavitational foce on the satellite due to its lage mass, the satellite must move with a highe speed to avoid moving towad the suface. This conclusion is consistent with the pedictions of Equation (1), which shows that because the speed v is popotional to the squae oot of the mass of the planet, the speed inceases as the mass of the planet inceases The Gavitational Field When Newton published his theoy of univesal gavitation, it was consideed a success because it satisfactoily explained the motion of the planets. It epesented stong evidence that the same laws that descibe phenomena on the Eath can be used on lage objects like planets and thoughout the Univese. ince 1687, Newton s theoy has been used to account fo the motions of comets, the deflection of a Cavendish balance, the obits of binay stas, and the otation of galaxies. Nevetheless, both Newton s contempoaies and his successos found it difficult to accept the concept of a foce that acts at a distance. They asked how it was possible fo two objects such as the un and the Eath to inteact when they wee not in contact with each othe. Newton himself could not answe that question. An appoach to descibing inteactions between objects that ae not in contact came well afte Newton s death. This appoach enables us to look at the gavitational inteaction in a diffeent way, using the concept of a gavitational field that exists at evey point in space. When a paticle of mass m is placed at a point whee the gavitational field is g, the paticle expeiences a foce F g 5 mg. In othe wods, we imagine that the field exets a foce on the paticle athe than conside a diect inteaction between two paticles. The gavitational field g is defined as Gavitational field g ; F g m (13.9) 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 mass of the test paticle. We call the object ceating the field the souce paticle. (Although the Eath is not a paticle, it is possible to show that we can model the Eath as a paticle fo the pupose of finding the gavitational field that it ceates.) Notice that the pesence of the test paticle is not necessay fo the field to exist: the souce paticle ceates the gavitational field. We can detect the pesence of the field and measue its stength by placing a test paticle in the field and noting the foce exeted on it. In essence, we ae descibing the effect that any object (in this case, the Eath) has on the empty space aound itself in tems of the foce that would be pesent if a second object wee somewhee in that space. 3 3 We shall etun to this idea of mass affecting the space aound it when we discuss Einstein s theoy of gavitation in Chapte 39.

12 13.5 Gavitational Potential Enegy 385 As an example of how the field concept woks, conside an object of mass m nea the Eath s suface. Because the gavitational foce acting on the object has a magnitude GM E m/ 2 (see Eq. 13.4), the field g at a distance fom the cente of the Eath is g 5 F g m 52GM E 2 ^ (13.10) whee the negative sign indicates that the field points towad the cente of the Eath as illustated in Figue 13.9a and ^ is a unit vecto pointing adially outwad fom the Eath. The field vectos at diffeent points suounding the Eath vay in both diection and magnitude. In a small egion nea the Eath s suface, the downwad field g is appoximately constant and unifom as indicated in Figue 13.9b. Equation is valid at all points outside the Eath s suface, assuming the Eath is spheical. At the Eath s suface, whee 5, g has a magnitude of 9.80 N/kg. (The unit N/kg is the same as m/s 2.) 13.5 Gavitational Potential Enegy In Chapte 8, we intoduced the concept of gavitational potential enegy, which is the enegy associated with the configuation of a system of objects inteacting via the gavitational foce. We emphasized that the gavitational potential enegy function U 5 mgy fo a paticle Eath system 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 moe geneal potential enegy function one that is valid without the estiction of having to be nea the Eath s suface will be diffeent fom U 5 mgy. Recall fom Equation 7.26 that the change in the potential enegy of a system associated with a given displacement of a membe of the system is defined as the negative of the intenal wok done by the foce on that membe duing the displacement: The field vectos point in the diection of the acceleation a paticle would expeience if it wee placed in the field. The magnitude of the field vecto at any location is the magnitude of the fee-fall acceleation at that location. a b Figue 13.9 (a) The gavitational field vectos in the vicinity of a unifom spheical mass such as the Eath vay in both diection and magnitude. (b) The gavitational field vectos in a small egion nea the Eath s suface ae unifom in both diection and magnitude. f DU 5 U f 2 U i 52 3 F 12 d (13.11) i We can use this esult to evaluate the geneal gavitational potential enegy function. Conside a paticle of mass m moving between two points and above the Eath s suface (Fig ). The paticle is subject to the gavitational foce given by Equation We can expess this foce as F12 52 GM Em 2 whee the negative sign indicates that the foce is attactive. ubstituting this expession fo F() into Equation 13.11, we can compute the change in the gavitational potential enegy function fo the paticle Eath system: U f 2 U i 5 GM E m 3 f i d 2 5 GM Em c2 1 d f i M E i m F g f F g U f 2 U i 52GM E m a 1 f 2 1 i b (13.12) As always, the choice of a efeence configuation fo the potential enegy is completely abitay. It is customay to choose the efeence configuation fo zeo potential enegy to be the same as that fo which the foce is zeo. Taking U i 5 0 at i 5 `, we obtain the impotant esult U12 52 GM Em Figue As a paticle of mass m moves fom to above the Eath s suface, the gavitational potential enegy of the paticle Eath system changes accoding to Equation (13.13) Gavitational potential enegy of the Eath paticle system

13 386 CHAPTER 13 Univesal Gavitation U M E Eath The potential enegy goes to zeo as appoaches infinity. This expession applies when the paticle is sepaated fom the cente of the Eath by a distance, povided that $. The esult is not valid fo paticles inside the Eath, whee,. Because of ou choice of U i, the function U is always negative (Fig ). Although Equation was deived fo the paticle Eath system, it can be applied to any two paticles. That is, the gavitational potential enegy associated with any pai of paticles of masses m 1 and m 2 sepaated by a distance is O GM E m Figue Gaph of the gavitational potential enegy U vesus fo the system of an object above the Eath s suface Figue Thee inteacting paticles. 3 U 52 Gm 1m 2 (13.14) This expession shows that the gavitational potential enegy fo any pai of paticles vaies as 1/, wheeas the foce between them vaies as 1/ 2. Futhemoe, the potential enegy is negative because the foce is attactive and we have chosen the potential enegy as zeo when the paticle sepaation is infinite. Because the foce between the paticles is attactive, an extenal agent must do positive wok to incease the sepaation between the paticles. The wok done by the extenal agent poduces an incease in the potential enegy as the two paticles ae sepaated. That is, U becomes 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 1Gm 1 m 2 / 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 system. If the extenal agent supplies an enegy geate than the binding enegy, the excess enegy of the system is in the fom of kinetic enegy of the paticles when the paticles ae at an infinite sepaation. We can extend this concept to thee o moe paticles. In this case, the total potential enegy of the system is the sum ove all pais of paticles. Each pai contibutes a tem of the fom given by Equation Fo example, if the system contains thee paticles as in Figue 13.12, U total 5 U 12 1 U 13 1 U 23 52Ga m 1m 2 1 m 1m 3 1 m 2m 3 b (13.15) The absolute value of U total epesents the wok needed to sepaate the paticles by an infinite distance. Example 13.6 The Change in Potential Enegy A paticle of mass m is displaced though a small vetical distance Dy nea the Eath s suface. how that in this situation the geneal expession fo the change in gavitational potential enegy given by Equation educes to the familia elationship DU 5 mg Dy. OLUTION Conceptualize Compae the two diffeent situations fo which we have developed expessions fo gavitational potential enegy: (1) a planet and an object that ae fa apat fo which the enegy expession is Equation and (2) a small object at the suface of a planet fo which the enegy expession is Equation We wish to show that these two expessions ae equivalent. Categoize This example is a substitution poblem. Combine the factions in Equation 13.12: (1) DU 52GM E m a b 5 GM f E m a f 2 i b i i f Evaluate f 2 i and i f if both the initial and final positions of the paticle ae close to the Eath s suface: ubstitute these expessions into Equation (1): whee g 5 GM E / 2 (Eq. 13.5). f 2 i 5Dy i f < 2 DU < GM Em 2 Dy 5 mg Dy

14 13.6 Enegy Consideations in Planetay and atellite Motion cont. WHAT IF? uppose you ae pefoming uppe-atmosphee studies and ae asked by you supeviso to find the height in the Eath s atmosphee at which the suface equation DU 5 mg Dy gives a 1.0% eo in the change in the potential enegy. What is this height? Answe Because the suface equation assumes a constant value fo g, it will give a DU value that is lage than the value given by the geneal equation, Equation et up a atio eflecting a 1.0% eo: ubstitute the expessions fo each of these changes DU: DU suface DU geneal mg Dy GM E m1dy/ i f 2 5 g i f GM E ubstitute fo i, f, and g fom Equation 13.5: 1GM E /R 2 E 2 1 1Dy2 GM E 5 1Dy Dy olve fo Dy: Dy m m km 13.6 Enegy Consideations in Planetay and atellite Motion Conside an object of mass m moving with a speed v in the vicinity of a massive object of mass M, whee M.. m. The system might be a planet moving aound the un, a satellite in obit aound the Eath, o a comet making a one-time flyby of the un. If we assume the object of mass M is at est in an inetial efeence fame, the total mechanical enegy E of the two-object system when the objects ae sepaated by a distance is the sum of the kinetic enegy of the object of mass m and the potential enegy of the system, given by Equation 13.14: E 5 K 1 U E 5 1 2mv 2 2 GMm (13.16) Equation shows that E may be positive, negative, o zeo, depending on the value of v. Fo a bound system such as the Eath un system, howeve, 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 system consisting of an object of mass m moving in a cicula obit about an object of mass M.. m (Fig ). Newton s second law applied to the object of mass m gives F g 5 ma GMm 5 mv 2 2 Multiplying both sides by and dividing by 2 gives 1 2mv 2 5 GMm 2 ubstituting this equation into Equation 13.16, we obtain E 5 GMm 2 2 GMm (13.17) M v Figue An object of mass m moving in a cicula obit about a much lage object of mass M. m E 52 GMm 2 1cicula obits2 (13.18) Total enegy fo cicula obits

15 388 CHAPTER 13 Univesal Gavitation This esult shows that the total mechanical enegy is negative in the case of cicula obits. Notice that the kinetic enegy is positive and equal to half the absolute value of the potential enegy. The absolute value of E is also equal to the binding enegy of the system because this amount of enegy must be povided to the system to move the two objects infinitely fa apat. The total mechanical enegy is also negative in the case of elliptical obits. The expession fo E fo elliptical obits is the same as Equation with eplaced by the semimajo axis length a: Total enegy fo elliptical obits E 52 GMm 2a 1elliptical obits2 (13.19) Futhemoe, the total enegy is constant if we assume the system is isolated. Theefoe, as the object of mass m moves fom to in Figue 13.10, the total enegy emains constant and Equation gives E 5 1 2mv i 2 2 GMm i 5 1 2mv f 2 2 GMm f (13.20) Combining this statement of enegy consevation with ou ealie discussion of consevation of angula momentum, we see that both the total enegy and the total angula momentum of a gavitationally bound, two-object system ae constants of the motion. Quick Quiz 13.4 A comet moves in an elliptical obit aound the un. Which point in its obit (peihelion o aphelion) epesents the highest value of (a) the speed of the comet, (b) the potential enegy of the comet un system, (c) the kinetic enegy of the comet, and (d) the total enegy of the comet un system? Example 13.7 Changing the Obit of a atellite A space tanspotation vehicle eleases a 470-kg communications satellite while in an obit 280 km above the suface of the Eath. A ocket engine on the satellite boosts it into a geosynchonous obit. How much enegy does the engine have to povide? OLUTION Conceptualize Notice that the height of 280 km is much lowe than that fo a geosynchonous satellite, km, as mentioned in Example Theefoe, enegy must be expended to aise the satellite to this much highe position. Categoize This example is a substitution poblem. Find the initial adius of the satellite s obit when it is still in the vehicle s cago bay: Use Equation to find the diffeence in enegies fo the satellite Eath system with the satellite at the initial and final adii: ubstitute numeical values, using f m fom Example 13.5: i km m DE 5 E f 2 E i 52 GM Em 2 f 2 a2 GM Em b 52 GM Em a b 2 i 2 f i DE N? m 2 /kg kg21470 kg a m m b J which is the enegy equivalent of 89 gal of gasoline. NAA enginees must account fo the changing mass of the spacecaft as it ejects buned fuel, something we have not done hee. Would you expect the calculation that includes the effect of this changing mass to yield a geate o a lesse amount of enegy equied fom the engine?

16 13.6 Enegy Consideations in Planetay and atellite Motion 389 Escape peed uppose an object of mass m is pojected vetically upwad fom the Eath s suface with an initial speed v i as illustated in Figue We can use enegy consideations to find the minimum value of the initial speed needed to allow the object to move infinitely fa away fom the Eath. Equation gives the total enegy of the system fo any configuation. As the object is pojected upwad fom the suface of the Eath, v 5 v i and 5 i 5. When the object eaches its maximum altitude, v 5 v f 5 0 and 5 f 5 max. Because the total enegy of the isolated object Eath system is constant, substituting these conditions into Equation gives 1 2mv 2 i 2 GM E m 52 GM Em max olving fo v 2 i gives v 2 i 5 2GM E a b (13.21) max Fo a given maximum altitude h 5 max 2, we can use this equation to find the equied initial speed. We ae now in a position to calculate escape speed, which is the minimum speed the object must have at the Eath s suface to appoach an infinite sepaation distance fom the Eath. Taveling at this minimum speed, the object continues to move fathe and fathe away fom the Eath as its speed asymptotically appoaches zeo. Letting max ` in Equation and taking v i 5 v esc gives v esc 5 Å 2GM E (13.22) This expession fo v esc is independent of the mass of the object. In othe wods, a spacecaft has the same escape speed as a molecule. Futhemoe, 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, the total enegy of the system is equal to zeo. Notice that when `, the object s kinetic enegy and the potential enegy of the system ae both zeo. If v i is geate than v esc, the total enegy of the system is geate than zeo and the object has some esidual kinetic enegy as `. v f v i m M E 0 h max Figue An object of mass m pojected upwad fom the Eath s suface with an initial speed v i eaches a maximum altitude h. Pitfall Pevention 13.3 You Can t Really Escape Although Equation povides the escape speed fom the Eath, complete escape fom the Eath s gavitational influence is impossible because the gavitational foce is of infinite ange. No matte how fa away you ae, you will always feel some gavitational foce due to the Eath. Example 13.8 Escape peed of a Rocket Calculate the escape speed fom the Eath fo a kg spacecaft and detemine the kinetic enegy it must have at the Eath s suface to move infinitely fa away fom the Eath. OLUTION Conceptualize Imagine pojecting the spacecaft fom the Eath s suface so that it moves fathe and fathe away, taveling moe and moe slowly, with its speed appoaching zeo. Its speed will neve each zeo, howeve, so the object will neve tun aound and come back. Categoize This example is a substitution poblem. Use Equation to find the escape speed: v esc 5 Å 2GM E m/s 5 Å N? m 2 /kg kg m Evaluate the kinetic enegy of the spacecaft fom Equation 7.16: K 5 1 2mv 2 esc kg m/s J continued

17 390 CHAPTER 13 Univesal Gavitation 13.8 cont. The calculated escape speed coesponds to about mi/h. The kinetic enegy of the spacecaft is equivalent to the enegy eleased by the combustion of about gal of gasoline. WHAT IF? What if you want to launch a kg spacecaft at the escape speed? How much enegy would that equie? Answe In Equation 13.22, the mass of the object moving with the escape speed does not appea. Theefoe, the escape speed fo the kg spacecaft is the same as that fo the kg spacecaft. The only change in the kinetic enegy is due to the mass, so the kg spacecaft equies one-fifth of the enegy of the kg spacecaft: K J J Equations and can be applied to objects pojected fom any planet. That is, in geneal, the escape speed fom the suface of any planet of mass M and adius R is v esc 5 Å 2GM R (13.23) TABLE 13.3 Escape peeds fom the ufaces of the Planets, Moon, and un Planet v esc (km/s) Mecuy 4.3 Venus 10.3 Eath 11.2 Mas 5.0 Jupite 60 atun 36 Uanus 22 Neptune 24 Moon 2.3 un 618 Escape speeds fo the planets, the Moon, and the un ae povided in Table The values vay fom 2.3 km/s fo the Moon to about 618 km/s fo the un. These esults, togethe with some ideas fom the kinetic theoy of gases (see Chapte 21), explain why some planets have atmosphees and othes do not. As we shall see late, at a given tempeatue the aveage kinetic enegy of a gas molecule depends only on the mass of the molecule. Lighte molecules, such as hydogen and helium, have a highe aveage speed than heavie molecules at the same tempeatue. When the aveage speed of the lighte molecules is not much less than the escape speed of a planet, a significant faction of them have a chance to escape. This mechanism also explains why the Eath does not etain hydogen molecules and helium atoms in its atmosphee but does etain heavie molecules, such as oxygen and nitogen. On the othe hand, the vey lage escape speed fo Jupite enables that planet to etain hydogen, the pimay constituent of its atmosphee. Black Holes In Example 11.7, we biefly descibed a ae event called a supenova, the catastophic explosion of a vey massive sta. The mateial that emains in the cental coe of such an object continues to collapse, and the coe s ultimate fate depends on its mass. If the coe has a mass less than 1.4 times the mass of ou un, it gadually cools down and ends its life as a white dwaf sta. If the coe s mass is geate than this value, howeve, it may collapse futhe due to gavitational foces. What emains is a neuton sta, discussed in Example 11.7, in which the mass of a sta is compessed to a adius of about 10 km. (On the Eath, a teaspoon of this mateial would weigh about 5 billion tons!) An even moe unusual sta death may occu when the coe has a mass geate than about thee sola masses. The collapse may continue until the sta becomes a vey small object in space, commonly efeed to as a black hole. In effect, black holes ae emains of stas that have collapsed unde thei own gavitational foce. If an object such as a spacecaft comes close to a black hole, the object expeiences an extemely stong gavitational foce and is tapped foeve. The escape speed fo a black hole is vey high because of the concentation of the sta s mass into a sphee of vey small adius (see Eq ). If the escape speed exceeds the speed of light c, adiation fom the object (such as visible light) cannot escape and the object appeas to be black (hence the oigin of the teminology

18 13.6 Enegy Consideations in Planetay and atellite Motion 391 black hole ). The citical adius R at which the escape speed is c is called the chwazschild adius (Fig ). The imaginay suface of a sphee of this adius suounding the black hole is called the event hoizon, which is the limit of how close you can appoach the black hole and hope to escape. Thee is evidence that supemassive black holes exist at the centes of galaxies, with masses vey much lage than the un. (Thee is stong evidence of a supemassive black hole of mass 2 3 million sola masses at the cente of ou galaxy.) Black hole Event hoizon Dak Matte Equation (1) in Example 13.5 shows that the speed of an object in obit aound the Eath deceases as the object is moved fathe away fom the Eath: R v 5 Å GM E (13.24) Using data in Table 13.2 to find the speeds of planets in thei obits aound the un, we find the same behavio fo the planets. Figue shows this behavio fo the eight planets of ou sola system. The theoetical pediction of the planet speed as a function of distance fom the un is shown by the ed-bown cuve, using Equation with the mass of the Eath eplaced by the mass of the un. Data fo the individual planets lie ight on this cuve. This behavio esults fom the vast majoity of the mass of the sola system being concentated in a small space, i.e., the un. Extending this concept futhe, we might expect the same behavio in a galaxy. Much of the visible galactic mass, including that of a supemassive black hole, is nea the cental coe of a galaxy. The opening photogaph fo this chapte shows the cental coe of the Whilpool galaxy as a vey bight aea suounded by the ams of the galaxy, which contain mateial in obit aound the cental coe. Based on this distibution of matte in the galaxy, the speed of an object in the oute pat of the galaxy would be smalle than that fo objects close to the cente, just like fo the planets of the sola system. That is not what is obseved, howeve. Figue shows the esults of measuements of the speeds of objects in the Andomeda galaxy as a function of distance fom the galaxy s cente. 4 The ed-bown cuve shows the expected speeds fo these objects if they wee taveling in cicula obits aound the mass concentated in the cental coe. The data fo the individual objects in the galaxy shown by the black dots ae all well above the theoetical cuve. These data, as well as an extensive amount of data taken ove the past half centuy, show that fo objects outside the cental coe of the galaxy, the cuve of speed vesus distance fom the cente of the galaxy is appoximately flat athe than deceasing at lage distances. Theefoe, these objects (including ou own ola ystem in the Milky Way) ae otating faste than can be accounted fo by gavity due to the visible galaxy! This supising esult means that thee must be additional mass in a moe extended distibution, causing these objects to obit so fast, and has led scientists to popose the existence of dak matte. This matte is poposed to exist in a lage halo aound each galaxy (with a adius up to 10 times as lage as the visible galaxy s adius). Because it is not luminous (i.e., does not emit electomagnetic adiation) it must be eithe vey cold o electically neutal. Theefoe, we cannot see dak matte, except though its gavitational effects.. The poposed existence of dak matte is also implied by ealie obsevations made on lage gavitationally bound stuctues known as galaxy clustes. 5 These 4 V. C. Rubin and W. K. Fod, Rotation of the Andomeda Nebula fom a pectoscopic uvey of Emission Regions, Astophysical Jounal 159: (1970). 5 F. Zwicky, On the Masses of Nebulae and of Clustes of Nebulae, Astophysical Jounal 86: (1937). Any event occuing within the event hoizon is invisible to an outside obseve. Figue A black hole. The distance R equals the chwazschild adius. v (km/s) Mecuy 40 Venus Eath Mas 20 Jupite atun Uanus Neptune (10 12 m) Figue The obital speed v as a function of distance fom the un fo the eight planets of the sola system. The theoetical cuve is in ed-bown, and the data points fo the planets ae in black. v (km/s) 600 Cental coe (10 19 m) Figue The obital speed v of a galaxy object as a function of distance fom the cente of the cental coe of the Andomeda galaxy. The theoetical cuve is in edbown, and the data points fo the galaxy objects ae in black. No data ae povided on the left because the behavio inside the cental coe of the galaxy is moe complicated.

19 392 CHAPTER 13 Univesal Gavitation obsevations show that the obital speeds of galaxies in a cluste ae, on aveage, too lage to be explained by the luminous matte in the cluste alone. The speeds of the individual galaxies ae so high, they suggest that thee is 50 times as much dak matte in galaxy clustes as in the galaxies themselves! Why doesn t dak matte affect the obital speeds of planets like it does those of a galaxy? It seems that a sola system is too small a stuctue to contain enough dak matte to affect the behavio of obital speeds. A galaxy o galaxy cluste, on the othe hand, contains huge amounts of dak matte, esulting in the supising behavio. What, though, is dak matte? At this time, no one knows. One theoy claims that dak matte is based on a paticle called a weakly inteacting massive paticle, o WIMP. If this theoy is coect, calculations show that about 200 WIMPs pass though a human body at any given time. The new Lage Hadon Collide in Euope (see Chapte 46) is the fist paticle acceleato with enough enegy to possibly geneate and detect the existence of WIMPs, which has geneated much cuent inteest in dak matte. Keeping an eye on this eseach in the futue should be exciting. Definitions ummay The gavitational field at a point in space is defined as the gavitational foce expeienced by any test paticle located at that point divided by the mass of the test paticle: g ; F g m (13.9) Concepts and Pinciples Newton s law of univesal gavitation states that the gavitational foce of attaction between any two paticles of masses m 1 and m 2 sepaated by a distance has the magnitude F g 5 G m 1m 2 (13.1) 2 whee G N? m 2 /kg 2 is the univesal gavitational constant. This equation enables us to calculate the foce of attaction between masses unde many cicumstances. An object at a distance h above the Eath s suface expeiences a gavitational foce of magnitude mg, whee g is the fee-fall acceleation at that elevation: g 5 GM E 2 5 GM E 1 1 h2 2 (13.6) In this expession, M E is the mass of the Eath and is its adius. Theefoe, the weight of an object deceases as the object moves away fom the Eath s suface.

20 Objective Questions 393 Keple s laws of planetay motion state: 1. All planets move in elliptical obits with the un at one focus. 2. The adius vecto dawn fom the un to a planet sweeps out equal aeas in equal time intevals. 3. The squae of the obital peiod of any planet is popotional to the cube of the semimajo axis of the elliptical obit. Keple s thid law can be expessed as T 2 5 a 4p 2 GM ba 3 (13.8) whee M is the mass of the un and a is the semimajo axis. Fo a cicula obit, a can be eplaced in Equation 13.8 by the adius. Most planets have nealy cicula obits aound the un. The gavitational potential enegy associated with a system of two paticles sepaated by a distance is U 52 Gm 1m 2 whee U is taken to be zeo as `. (13.14) If an isolated system consists of an object of mass m moving with a speed v in the vicinity of a massive object of mass M, the total enegy E of the system is the sum of the kinetic and potential enegies: E 5 1 2mv 2 2 GMm (13.16) The total enegy of the system is a constant of the motion. If the object moves in an elliptical obit of semimajo axis a aound the massive object and M.. m, the total enegy of the system is E 52 GMm 2a Fo a cicula obit, this same equation applies with a 5. (13.19) The escape speed fo an object pojected fom the suface of a planet of mass M and adius R is v esc 5 Å 2GM R (13.23) Objective Questions 1. Rank the magnitudes of the following gavitational foces fom lagest to smallest. If two foces ae equal, show thei equality in you list. (a) the foce exeted by a 2-kg object on a 3-kg object 1 m away (b) the foce exeted by a 2-kg object on a 9-kg object 1 m away (c) the foce exeted by a 2-kg object on a 9-kg object 2 m away (d) the foce exeted by a 9-kg object on a 2-kg object 2 m away (e) the foce exeted by a 4-kg object on anothe 4-kg object 2 m away 2. The gavitational foce exeted on an astonaut on the Eath s suface is 650 N diected downwad. When she is in the space station in obit aound the Eath, is the gavitational foce on he (a) lage, (b) exactly the same, (c) smalle, (d) nealy but not exactly zeo, o (e) exactly zeo? 3. Imagine that nitogen and othe atmospheic gases wee moe soluble in wate so that the atmosphee of the Eath is entiely absobed by the oceans. Atmospheic pessue would then be zeo, and oute space would stat at the planet s suface. Would the Eath then have a gavitational field? (a) Yes, and at the suface it would be lage in magnitude than 9.8 N/kg. (b) Yes, and it would be essentially the same as the cuent value. (c) Yes, and it would be some- denotes answe available in tudent olutions Manual/tudy Guide what less than 9.8 N/kg. (d) Yes, and it would be much less than 9.8 N/kg. (e) No, it would not. 4. uppose the gavitational acceleation at the suface of a cetain moon A of Jupite is 2 m/s 2. Moon B has twice the mass and twice the adius of moon A. What is the gavitational acceleation at its suface? Neglect the gavitational acceleation due to Jupite. (a) 8 m/s 2 (b) 4 m/s 2 (c) 2 m/s 2 (d) 1 m/s 2 (e) 0.5 m/s 2 5. A satellite moves in a cicula obit at a constant speed aound the Eath. Which of the following statements is tue? (a) No foce acts on the satellite. (b) The satellite moves at constant speed and hence doesn t acceleate. (c) The satellite has an acceleation diected away fom the Eath. (d) The satellite has an acceleation diected towad the Eath. (e) Wok is done on the satellite by the gavitational foce. 6. An object of mass m is located on the suface of a spheical planet of mass M and adius R. The escape speed fom the planet does not depend on which of the following? (a) M (b) m (c) the density of the planet (d) R (e) the acceleation due to gavity on that planet 7. A satellite oiginally moves in a cicula obit of adius R aound the Eath. uppose it is moved into a cicula obit

21 394 CHAPTER 13 Univesal Gavitation of adius 4R. (i) What does the foce exeted on the satellite then become? (a) eight times lage (b) fou times lage (c) one-half as lage (d) one-eighth as lage (e) onesixteenth as lage (ii) What happens to the satellite s speed? Choose fom the same possibilities (a) though (e). (iii) What happens to its peiod? Choose fom the same possibilities (a) though (e). 8. The venal equinox and the autumnal equinox ae associated with two points 180 apat in the Eath s obit. That is, the Eath is on pecisely opposite sides of the un when it passes though these two points. Fom the venal equinox, days elapse befoe the autumnal equinox. Only days elapse fom the autumnal equinox until the next venal equinox. Why is the inteval fom the Mach (venal) to the eptembe (autumnal) equinox (which contains the summe solstice) longe than the inteval fom the eptembe to the Mach equinox athe than being equal to that inteval? Choose one of the following easons. (a) They ae eally the same, but the Eath spins faste duing the summe inteval, so the days ae shote. (b) Ove the summe inteval, the Eath moves slowe because it is fathe fom the un. (c) Ove the Mach-to-eptembe inteval, the Eath moves slowe because it is close to the un. (d) The Eath has less kinetic enegy when it is wame. (e) The Eath has less obital angula momentum when it is wame. 9. A system consists of five paticles. How many tems appea in the expession fo the total gavitational potential enegy of the system? (a) 4 (b) 5 (c) 10 (d) 20 (e) Rank the following quantities of enegy fom lagest to the smallest. tate if any ae equal. (a) the absolute value of the aveage potential enegy of the un Eath system (b) the aveage kinetic enegy of the Eath in its obital motion elative to the un (c) the absolute value of the total enegy of the un Eath system 11. Halley s comet has a peiod of appoximately 76 yeas, and it moves in an elliptical obit in which its distance fom the un at closest appoach is a small faction of its maximum distance. Estimate the comet s maximum distance fom the un in astonomical units (AUs) (the distance fom the Eath to the un). (a) 6 AU (b) 12 AU (c) 20 AU (d) 28 AU (e) 35 AU Conceptual Questions 1. A satellite in low-eath obit is not tuly taveling though a vacuum. Rathe, it moves though vey thin ai. Does the esulting ai fiction cause the satellite to slow down? 2. Explain why it takes moe fuel fo a spacecaft to tavel fom the Eath to the Moon than fo the etun tip. Estimate the diffeence. 3. Why don t we put a geosynchonous weathe satellite in obit aound the 45th paallel? Wouldn t such a satellite be moe useful in the United tates than one in obit aound the equato? 4. (a) Explain why the foce exeted on a paticle by a unifom sphee must be diected towad the cente of the sphee. (b) Would this statement be tue if the mass distibution of the sphee wee not spheically symmetic? Explain. 5. (a) At what position in its elliptical obit is the speed of a planet a maximum? (b) At what position is the speed a minimum? denotes answe available in tudent olutions Manual/tudy Guide 6. You ae given the mass and adius of planet X. How would you calculate the fee-fall acceleation on this planet s suface? 7. (a) If a hole could be dug to the cente of the Eath, would the foce on an object of mass m still obey Equation 13.1 thee? (b) What do you think the foce on m would be at the cente of the Eath? 8. In his 1798 expeiment, Cavendish was said to have weighed the Eath. Explain this statement. 9. Each Voyage spacecaft was acceleated towad escape speed fom the un by the gavitational foce exeted by Jupite on the spacecaft. (a) Is the gavitational foce a consevative o a nonconsevative foce? (b) Does the inteaction of the spacecaft with Jupite meet the definition of an elastic collision? (c) How could the spacecaft be moving faste afte the collision? Poblems The poblems found in this chapte may be assigned online in Enhanced WebAssign 1. denotes staightfowad poblem; 2. denotes intemediate poblem; 3. denotes challenging poblem 1. full solution available in the tudent olutions Manual/tudy Guide 1. denotes poblems most often assigned in Enhanced WebAssign; these povide students with tageted feedback and eithe a Maste It tutoial o a Watch It solution video. denotes asking fo quantitative and conceptual easoning denotes symbolic easoning poblem denotes Maste It tutoial available in Enhanced WebAssign denotes guided poblem shaded denotes paied poblems that develop easoning with symbols and numeical values

22 Poblems 395 ection 13.1 Newton s Law of Univesal Gavitation Poblem 10 in Chapte 1 can also be assigned with this section. 1. Two ocean lines, each with a mass of metic tons, ae moving on paallel couses 100 m apat. What is the magnitude of the acceleation of one of the lines towad the othe due to thei mutual gavitational attaction? Model the ships as paticles. 2. Detemine the ode of magnitude of the gavitational foce that you exet on anothe peson 2 m away. In you solution, state the quantities you measue o estimate and thei values. 3. A 200-kg object and a 500-kg object ae sepaated by 4.00 m. (a) Find the net gavitational foce exeted by these objects on a 50.0-kg object placed midway between them. (b) At what position (othe than an infinitely emote one) can the 50.0-kg object be placed so as to expeience a net foce of zeo fom the othe two objects? 4. Duing a sola eclipse, the Moon, the Eath, and the un all lie on the same line, with the Moon between the Eath and the un. (a) What foce is exeted by the un on the Moon? (b) What foce is exeted by the Eath on the Moon? (c) What foce is exeted by the un on the Eath? (d) Compae the answes to pats (a) and (b). Why doesn t the un captue the Moon away fom the Eath? 5. In intoductoy physics laboatoies, a typical Cavendish balance fo measuing the gavitational constant G uses lead sphees with masses of 1.50 kg and 15.0 g whose centes ae sepaated by about 4.50 cm. Calculate the gavitational foce between these sphees, teating each as a paticle located at the sphee s cente. 6. Thee unifom sphees of masses m kg, m kg, and m kg ae placed at the cones of a ight tiangle as shown in Figue P13.6. Calculate the esultant gavitational foce on the object of mass m 2, assuming the sphees ae isolated fom the est of the Univese. each made of the same element fom the peiodic table. The gavitational foce between the sphees is 1.00 N. 9. Review. A student poposes to study the gavitational foce by suspending two kg spheical objects at the lowe ends of cables fom the ceiling of a tall cathedal and measuing the deflection of the cables fom the vetical. The m-long cables ae attached to the ceiling m apat. The fist object is suspended, and its position is caefully measued. The second object is suspended, and the two objects attact each othe gavitationally. By what distance has the fist object moved hoizontally fom its initial position due to the gavitational attaction to the othe object? uggestion: Keep in mind that this distance will be vey small and make appopiate appoximations. ection 13.2 Fee-Fall Acceleation and the Gavitational Foce 10. When a falling meteooid is at a distance above the Eath s suface of 3.00 times the Eath s adius, what is its acceleation due to the Eath s gavitation? 11. Review. Mianda, a satellite of Uanus, is shown in Figue P13.11a. It can be modeled as a sphee of adius 242 km and mass kg. (a) Find the fee-fall acceleation on its suface. (b) A cliff on Mianda is 5.00 km high. It appeas on the limb at the 11 o clock position in Figue P13.11a and is magnified in Figue P13.11b. If a devotee of exteme spots uns hoizontally off the top of the cliff at 8.50 m/s, fo what time inteval is he in flight? (c) How fa fom the base of the vetical cliff does he stike the icy suface of Mianda? (d) What will be his vecto impact velocity? ( 4.00, 0) m y (0, 3.00) m m 1 F 12 NAA a b NAA/JPL F 32 m O m 2 3 Figue P Two objects attact each othe with a gavitational foce of magnitude N when sepaated by 20.0 cm. If the total mass of the two objects is 5.00 kg, what is the mass of each? 8. Why is the following situation impossible? The centes of two homogeneous sphees ae 1.00 m apat. The sphees ae x Figue P The fee-fall acceleation on the suface of the Moon is about one-sixth that on the suface of the Eath. The adius of the Moon is about ( 5 Eath s adius m). Find the atio of thei aveage densities, Moon / Eath. ection 13.3 Keple s Laws and the Motion of Planets 13. An atificial satellite cicles the Eath in a cicula obit at a location whee the acceleation due to gavity is 9.00 m/s 2. Detemine the obital peiod of the satellite.

23 396 CHAPTER 13 Univesal Gavitation 14. Io, a satellite of Jupite, has an obital peiod of 1.77 days and an obital adius of km. Fom these data, detemine the mass of Jupite. 15. A minimum-enegy tansfe obit to an oute planet consists of putting a spacecaft on an elliptical tajectoy with the depatue planet coesponding to the peihelion of the ellipse, o the closest point to the un, and the aival planet at the aphelion, o the fathest point fom the un. (a) Use Keple s thid law to calculate how long it would take to go fom Eath to Mas on such an obit as shown in Figue P (b) Can such an obit be undetaken at any time? Explain. each sta is 0 v km/s and the obital peiod of each is 14.4 days. Find the mass M of each sta. (Fo compaison, the mass of ou un is kg.) 18. Comet Halley (Fig. P13.18) appoaches the un to within AU, and its obital peiod is 75.6 y. (AU is the symbol fo astonomical unit, whee 1 AU m is the mean Eath un distance.) How fa fom the un will Halley s comet tavel befoe it stats its etun jouney? Aival at Mas Tansfe obit Mas obit un un Launch fom the Eath AU 2a x Eath obit Figue P13.18 (Obit is not dawn to scale.) Figue P A paticle of mass m moves along a staight line with constant velocity v 0 in the x diection, a distance b fom the x axis (Fig. P13.16). (a) Does the paticle possess any angula momentum about the oigin? (b) Explain why the amount of its angula momentum should change o should stay constant. (c) how that Keple s second law is satisfied by showing that the two shaded tiangles in the figue have the same aea when t 2 t 5 t 2 t. m v 0 b O y Figue P Plaskett s binay system consists of two stas that evolve in a cicula obit about a cente of mass midway between them. This statement implies that the masses of the two stas ae equal (Fig. P13.17). Assume the obital speed of M v v CM M Figue P13.17 x 19. Use Keple s thid law to detemine how many days it takes a spacecaft to tavel in an elliptical obit fom a point km fom the Eath s cente to the Moon, km fom the Eath s cente. 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 one moment, they ae aligned as shown in Figue P13.20a, making a staight line with the sta. Duing the next five yeas, the angula displacement of planet X is 90.0 as shown in Figue P13.20b. What is the angula displacement of planet Y at this moment? a Y X b Figue P A synchonous satellite, which always emains above the same point on a planet s equato, is put in obit aound Jupite to study that planet s famous ed spot. Jupite otates once evey 9.84 h. Use the data of Table 13.2 to find the altitude of the satellite above the suface of the planet. 22. Neuton stas ae extemely dense objects fomed fom the emnants of supenova explosions. Many otate vey apidly. uppose the mass of a cetain spheical neuton sta is twice the mass of the un and its adius is 10.0 km. Detemine the geatest possible angula speed it can have so that the matte at the suface of the sta on its equato is just held in obit by the gavitational foce. Y X

24 Poblems uppose the un s gavity wee switched off. The planets would leave thei obits and fly away in staight lines as descibed by Newton s fist law. (a) Would Mecuy eve be fathe fom the un than Pluto? (b) If so, find how long it would take Mecuy to achieve this passage. If not, give a convincing agument that Pluto is always fathe fom the un than is Mecuy. 24. (a) Given that the peiod of the Moon s obit about the Eath is days and the nealy constant distance between the cente of the Eath and the cente of the Moon is m, use Equation 13.8 to calculate the mass of the Eath. (b) Why is the value you calculate a bit too lage? ection 13.4 The Gavitational Field 25. Thee objects of equal mass ae located at thee cones of a squae of edge length, as shown in Figue P Find the magnitude and diection of the gavitational field at the fouth cone due to these objects. mine 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 fom the black hole? (This diffeence in acceleations gows apidly as the ship appoaches the black hole. It puts the body of the ship unde exteme tension and eventually teas it apat.) 100 m Figue P13.27 ection 13.5 Gavitational Potential Enegy 10.0 km In Poblems 28 though 42, assume U 5 0 at 5 `. Black hole y m m O m Figue P13.25 x 28. A satellite in Eath obit has a mass of 100 kg and is at an altitude of m. (a) What is the potential enegy of the satellite Eath system? (b) What is the magnitude of the gavitational foce exeted by the Eath on the satellite? (c) What If? What foce, if any, does the satellite exet on the Eath? 29. How much wok is done by the Moon s gavitational field on a kg meteo as it comes in fom oute space and impacts on the Moon s suface? 26. (a) Compute the vecto gavitational field at a point P on the pependicula bisecto of the line joining two objects of equal mass sepaated by a distance 2a as shown in Figue P (b) Explain physically why the field should appoach zeo as 0. (c) Pove mathematically that the answe to pat (a) behaves in this way. (d) Explain physically why the magnitude of the field should appoach 2GM/ 2 as `. (e) Pove mathematically that the answe to pat (a) behaves coectly in this limit. a a M M Figue P A spacecaft in the shape of a long cylinde has a length of 100 m, and its mass with occupants is kg. It has stayed too close to a black hole having a mass 100 times that of the un (Fig. P13.27). The nose of the spacecaft points towad the black hole, and the distance between the nose and the cente of the black hole is 10.0 km. (a) Dete- P 30. A system consists of thee paticles, each of mass 5.00 g, located at the cones of an equilateal tiangle with sides of 30.0 cm. (a) Calculate the potential enegy of the system. (b) Assume the paticles ae eleased simultaneously. Descibe the subsequent motion of each. Will any collisions take place? Explain. 31. Afte the un exhausts its nuclea fuel, its ultimate fate will be to collapse to a white dwaf state. In this state, it would have appoximately the same mass as it has now, but its adius would be equal to the adius of the Eath. Calculate (a) the aveage density of the white dwaf, (b) the suface fee-fall acceleation, and (c) the gavitational potential enegy associated with a 1.00-kg object at the suface of the white dwaf. 32. An object is eleased fom est at an altitude h above the suface of the Eath. (a) how that its speed at a distance fom the Eath s cente, whee # # 1 h, is v 5 Å 2GM E a h b (b) Assume the elease altitude is 500 km. Pefom the integal f f d Dt 5 3 dt 52 3 v i to find the time of fall as the object moves fom the elease point to the Eath s suface. The negative sign appeas i

25 398 CHAPTER 13 Univesal Gavitation because the object is moving opposite to the adial diection, so its speed is v 5 2d/dt. Pefom the integal numeically. 33. At the Eath s suface, a pojectile is launched staight up at a speed of 10.0 km/s. To what height will it ise? Ignoe ai esistance and the otation of the Eath. ection 13.6 Enegy Consideations in Planetay and atellite Motion 34. A space pobe is fied as a pojectile fom the Eath s suface with an initial speed of m/s. What will its speed be when it is vey fa fom the Eath? Ignoe atmospheic fiction and the otation of the Eath. 35. A kg satellite obits the Eath at a constant altitude of 100 km. (a) How much enegy must be added to the system to move the satellite into a cicula obit with altitude 200 km? What ae the changes in the system s (b) kinetic enegy and (c) potential enegy? 36. A teetop satellite moves in a cicula obit just above the suface of a planet, assumed to offe no ai esistance. how that its obital speed v and the escape speed fom the planet ae elated by the expession v esc 5!2v. 37. A comet of mass kg moves in an elliptical obit aound the un. Its distance fom the un anges between AU and 50.0 AU. (a) What is the eccenticity of its obit? (b) What is its peiod? (c) At aphelion, what is the potential enegy of the comet un system? Note: 1 AU 5 one astonomical unit 5 the aveage distance fom the un to the Eath m. 38. (a) What is the minimum speed, elative to the un, necessay fo a spacecaft to escape the sola system if it stats at the Eath s obit? (b) Voyage 1 achieved a maximum speed of km/h on its way to photogaph Jupite. Beyond what distance fom the un is this speed sufficient to escape the sola system? 39. A satellite of mass 200 kg is placed into Eath obit at a height of 200 km above the suface. (a) Assuming a cicula obit, how long does the satellite take to complete one obit? (b) What is the satellite s speed? (c) tating fom the satellite on the Eath s suface, what is the minimum enegy input necessay to place this satellite in obit? Ignoe ai esistance but include the effect of the planet s daily otation. 40. A satellite of mass m, oiginally on the suface of the Eath, is placed into Eath obit at an altitude h. (a) Assuming a cicula obit, how long does the satellite take to complete one obit? (b) What is the satellite s speed? (c) What is the minimum enegy input necessay to place this satellite in obit? Ignoe ai esistance but include the effect of the planet s daily otation. Repesent the mass and adius of the Eath as M E and, espectively. 41. Ganymede is the lagest of Jupite s moons. Conside a ocket on the suface of Ganymede, at the point fathest fom the planet (Fig. P13.41). Model the ocket as a pati- cle. (a) Does the pesence of Ganymede make Jupite exet a lage, smalle, o same size foce on the ocket compaed with the foce it would exet if Ganymede wee not inteposed? (b) Detemine the escape speed fo the ocket fom the planet satellite system. The adius of Ganymede is m, and its mass is kg. The distance between Jupite and Ganymede is m, and the mass of Jupite is kg. Ignoe the motion of Jupite and Ganymede as they evolve about thei cente of mass. Jupite Figue P13.41 v Ganymede 42. A satellite moves aound the Eath in a cicula obit of adius. (a) What is the speed v i of the satellite? (b) uddenly, an explosion beaks the satellite into two pieces, with masses m and 4m. Immediately afte the explosion, the smalle piece of mass m is stationay with espect to the Eath and falls diectly towad the Eath. What is the speed v of the lage piece immediately afte the explosion? (c) Because of the incease in its speed, this lage piece now moves in a new elliptical obit. Find its distance away fom the cente of the Eath when it eaches the othe end of the ellipse. Additional Poblems 43. Review. A cylindical habitat in space 6.00 km in diamete and 30.0 km long has been poposed (by G. K. O Neill, 1974). uch a habitat would have cities, land, and lakes on the inside suface and ai and clouds in the cente. They would all be held in place by otation of the cylinde about its long axis. How fast would the cylinde have to otate to imitate the Eath s gavitational field at the walls of the cylinde? 44. A ocket is fied staight up though the atmosphee fom the outh Pole, buning out at an altitude of 250 km when taveling at 6.00 km/s. (a) What maximum distance fom the Eath s suface does it tavel befoe falling back to the Eath? (b) Would its maximum distance fom the suface be lage if the same ocket wee fied with the same fuel load fom a launch site on the equato? Why o why not? 45. Let Dg M epesent the diffeence in the gavitational fields poduced by the Moon at the points on the Eath s suface neaest to and fathest fom the Moon. Find the faction Dg M /g, whee g is the Eath s gavitational field. (This diffeence is esponsible fo the occuence of the luna tides on the Eath.) 46. Why is the following situation impossible? A spacecaft is launched into a cicula obit aound the Eath and cicles the Eath once an hou.

26 Poblems (a) A space vehicle is launched vetically upwad fom the Eath s suface with an initial speed of 8.76 km/s, which is less than the escape speed of 11.2 km/s. What maximum height does it attain? (b) A meteooid falls towad the Eath. It is essentially at est with espect to the Eath when it is at a height of m above the Eath s suface. With what speed does the meteoite (a meteooid that suvives to impact the Eath s suface) stike the Eath? 48. (a) A space vehicle is launched vetically upwad fom the Eath s suface with an initial speed of v i that is compaable to but less than the escape speed v esc. What maximum height does it attain? (b) A meteooid falls towad the Eath. It is essentially at est with espect to the Eath when it is at a height h above the Eath s suface. With what speed does the meteoite (a meteooid that suvives to impact the Eath s suface) stike the Eath? (c) What If? Assume a baseball is tossed up with an initial speed that is vey small compaed to the escape speed. how that the esult fom pat (a) is consistent with Equation Assume you ae agile enough to un acoss a hoizontal suface at 8.50 m/s, independently of the value of the gavitational field. What would be (a) the adius and (b) the mass of an ailess spheical asteoid of unifom density kg/m 3 on which you could launch youself into obit by unning? (c) What would be you peiod? (d) Would you unning significantly affect the otation of the asteoid? Explain. 50. A sleeping aea fo a long space voyage consists of two cabins each connected by a cable to a cental hub as shown in Figue P The cabins ae set spinning aound the hub axis, which is connected to the est of the spacecaft to geneate atificial gavity in the cabins. A space tavele lies in a bed paallel to the oute wall as shown in Figue P (a) With m, what would the angula speed of the 60.0-kg tavele need to be if he is to expeience half his nomal Eath weight? (b) If the astonaut stands up pependicula to the bed, without holding on to anything with his hands, will his head be moving at a faste, a slowe, o the same tangential speed as his feet? Why? (c) Why is the action in pat (b) dangeous? kg, m, and m. Note: Both the enegy and momentum of the isolated twoplanet system ae constant. 52. Two sphees having masses M and 2M and adii R and 3R, espectively, ae simultaneously eleased fom est when the distance between thei centes is 12R. Assume the two sphees inteact only with each othe and we wish to find the speeds with which they collide. (a) What two isolated system models ae appopiate fo this system? (b) Wite an equation fom one of the models and solve it fo v 1, the velocity of the sphee of mass M at any time afte elease in tems of v 2, the velocity of 2M. (c) Wite an equation fom the othe model and solve it fo speed v 1 in tems of speed v 2 when the sphees collide. (d) Combine the two equations to find the two speeds v 1 and v 2 when the sphees collide. 53. A ing of matte is a familia stuctue in planetay and stella astonomy. Examples include atun s ings and a ing nebula. Conside a unifom ing of mass kg and adius m. An object of mass kg is placed at a point A on the axis of the ing, m fom the cente of the ing (Fig. P13.53). When the object is eleased, the attaction of the ing makes the object move along the axis towad the cente of the ing (point B). (a) Calculate the gavitational potential enegy of the object ing system when the object is at A. (b) Calculate the gavitational potential enegy of the system when the object is at B. (c) Calculate the speed of the object as it passes though B. NAA ω B Figue P Two hypothetical planets of masses m 1 and m 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 fo thei elative speed. (b) Find the kinetic enegy of each planet just befoe they collide, taking m kg, m 2 5 Figue P (a) how that the ate of change of the fee-fall acceleation with vetical position nea the Eath s suface is dg d 522GM E 3 This ate of change with position is called a gadient. (b) Assuming h is small in compaison to the adius of the A