Ch. 14: Gravitation (Beta Version 7/01) 14 Gravitation


 Prudence Little
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1 Ch. 14: Gavitation (Beta Vesion 7/01) 14 Gavitation The Milky Way galaxy is a diskshaped collection of dust, planets, and billions of stas, including ou Sun and sola system. The foce that binds it o any othe galaxy togethe is the same foce that holds Eath s moon in obit and you on Eath the gavitational foce. That foce is also esponsible fo one of natue s stangest objects, the black hole, a sta that has completely collapsed onto itself. The gavitational foce nea a black hole is so stong that not even light can escape it. If that is the case, how can a black hole be detected? The answe is in this chapte. 141
2 Ch. 14: Gavitation (Beta Vesion 7/01) 141 The Wold and the Gavitational Foce The dawing that opens this chapte shows ou view of the Milky Way galaxy. We ae nea the edge of the disk of the galaxy, about lightyeas ( m) fom its cente, which in the dawing lies in the sta collection known as Sagittaius. Ou galaxy is a membe of the Local Goup of galaxies, which includes the Andomeda galaxy (Fig. 141) at a distance of lightyeas, and seveal close dwaf galaxies, such as the Lage Magellanic Cloud shown in the opening dawing. The Local Goup is pat of the Local Supecluste of galaxies. Measuements taken duing and since the 1980s suggest that the Local Supecluste and the supecluste consisting of the clustes Hyda and Centauus ae all moving towad an exceptionally massive egion called the Geat Attacto. This egion appeas to be about 300 million lightyeas away, on the opposite side of the Milky Way fom us, past the clustes Hyda and Centauus. The foce that binds togethe these pogessively lage stuctues, fom sta to galaxy to supecluste, and may be dawing them all towad the Geat Attacto, is the gavitational foce. That foce not only holds you on Eath but also eaches out acoss integalactic space. 14 Newton s Law of Gavitation Physicists like to study seemingly unelated phenomena to show that a elationship can be found if they ae examined closely enough. This seach fo unification has been going on fo centuies. In 1665, the 3yeaold Isaac Newton made a basic contibution to physics when he showed that the foce that holds the Moon in its obit is the same foce that makes an apple fall. We take this so much fo ganted now that it is not easy fo us to compehend the ancient belief that the motions of eathbound bodies and heavenly bodies wee diffeent in kind and wee govened by diffeent laws. Newton concluded that not only does Eath attact an apple and the Moon but evey body in the univese attacts evey othe body; this tendency of bodies to move towad each othe is called gavitation. Newton s conclusion takes a little getting used to, because the familia attaction of Eath fo eathbound bodies is so geat that it ovewhelms the attaction that eathbound bodies have fo each othe. Fo example, Eath attacts an apple with a foce magnitude of about 08. N. You also attact a neaby apple (and it attacts you), but the foce of attaction has less magnitude than the weight of a speck of dust. Quantitatively, Newton poposed a foce law that we call Newton s law of gavitation: Evey paticle attacts any othe paticle with a gavitational foce whose magnitude is given by F = F F G mm 1 = 1 = 1 (Newton s law of gavitation). (141) Fig. 141: The Andomeda galaxy. Located lightyeas fom us, and faintly visible to the naked eye, it is vey simila to ou home galaxy, the Milky Way. m 1 F F m Fig. 14: Two paticles, of masses m 1 and m and with sepaation, attact each othe accoding to Newton s law of gavitation, Eq The foces of attaction, F and F, ae equal in magnitude and in opposite diections. Hee m 1 and m ae the masses of the paticles, is the distance between them, and G is the gavitational constant, whose value is now known to be G = N m kg (14) 11 3 = m kg s. As Fig. 14 shows, a paticle m attacts a paticle m 1 with a gavitational foce F that is diected towad paticle m, and paticle m 1 attacts paticle m with a gavitational foce F that is diected towad m 1. The foces F and F fom a thidlaw foce pai; they ae opposite in diection but equal in magnitude. They depend on the sepaation of the two paticles, but not on thei location: the paticles could be in a deep cave o in deep space. Also foces F and F ae not alteed by the pesence of othe bodies, even if those bodies lie between the two paticles we ae consideing. 14
3 Ch. 14: Gavitation (Beta Vesion 7/01) The stength of the gavitational foce that is, how stongly two paticles with given masses at a given sepaation attact each othe depends on the value of the gavitational constant G. If G by some miacle wee suddenly multiplied by a facto of 10, you would be cushed to the floo by Eath s attaction. If G wee divided by this facto, Eath s attaction would be weak enough that you could jump ove a building. Although Newton s law of gavitation applies stictly to paticles, we can also apply it to eal objects as long as the sizes of the objects ae small compaed to the distance between them. The Moon and Eath ae fa enough apat so that, to a good appoximation, we can teat them both as paticles but what about an apple and Eath? Fom the point of view of the apple, the boad and level Eath, stetching out to the hoizon beneath the apple, cetainly does not look like a paticle. Newton solved the appleeath poblem by poving an impotant theoem called the shell theoem: A unifom spheical shell of matte attacts a paticle that is outside the shell as if all the shell s mass wee concentated at its cente. Eath can be thought of as a nest of such shells, one within anothe, and each attacting a paticle outside Eath s suface as if the mass of that shell wee at the cente of the shell. Thus, fom the apple s point of view, Eath does behave like a paticle, one that is located at the cente of Eath and has a mass equal to that of Eath. Suppose, as in Fig. 143, that Eath pulls down on an apple with a foce of magnitude 0.80 N. The apple must then pull up on Eath with a foce of magnitude 0.80 N, which we take to act at the cente of Eath. Although the foces ae matched in magnitude, they poduce diffeent acceleations when the apple is eleased. Fo the apple, the acceleation is about 98. ms, the familia acceleation of a falling body nea Eath s suface. Fo Eath, the acceleation measued in a efeence fame attached to the cente of mass of the appleeath system is only about ms. F Fig. 143: The apple pulls up on Eath just as had as Eath pulls down on the apple. F EADING EXECISE 141: A paticle is to be placed, in tun, outside fou objects, each of mass m: (1) a lage unifom solid sphee, () a lage unifom spheical shell, (3) a small unifom solid sphee, and (4) a small unifom shell. In each situation, the distance between the paticle and the cente of the object is d. ank the objects accoding to the magnitude of the gavitational foce they exet on the paticle, geatest fist Gavitation and the Pinciple of Supeposition Given a goup of paticles, we find the net (o esultant) gavitational foce on any one of them fom the othes by using the pinciple of supeposition. This is a geneal pinciple that says a net effect is the sum of the individual effects. Hee, the pinciple means that we fist compute the gavitational foce that acts on ou selected paticle due to each of the othe paticles, in tun. We then find the net foce by adding these foces vectoially, as usual. Fo n inteacting paticles, we can wite the pinciple of supeposition fo gavitational foces as F F F F F F 1, net = n 1. (143) Hee F 1,net is the net foce on paticle 1 and, fo example, F 3 1 is the foce on paticle 1 fom paticle 3. We can expess this equation moe compactly as a vecto sum: F n F i (144) 1, net = 1. i= 143
4 Ch. 14: Gavitation (Beta Vesion 7/01) What about the gavitational foce on a paticle fom a eal extended object? The foce is found by dividing the object into pats small enough to teat as paticles and then using Eq to find the vecto sum of the foces on the paticle fom all the pats. In the limiting case, we can divide the extended object into diffeential pats of mass dm, each of which poduces only a diffeential foce df on the paticle. In this limit, the sum of Eq becomes an integal and we have F1 =z df, (145) in which the integal is taken ove the entie extended object and we dop the subscipt net. If the object is a unifom sphee o a spheical shell, we can avoid the integation of Eq by assuming that the object s mass is concentated at the object s cente and using Eq Touchstone Example , at the end of this chapte, illustates how to use what you leaned in this section. TE 144 Gavitation Nea Eath s Suface Let us assume that Eath is a unifom sphee of mass M. The magnitude of the gavitational foce fom Eath on a paticle of mass m, located outside Eath a distance fom Eath s cente, is then given by Eq as F = G Mm (146) If the paticle is eleased, it will fall towad the cente of Eath, as a esult of the gavitational foce F, with a gavitational acceleation a g. Newton s second law tells us that magnitudes F and a g ae elated by F = ma g (147) Now, substituting F fom Eq into Eq and solving fo a g, we find GM ag = (148) Density (10 3 kg/m 3 ) 14 Inne 1 coe 10 8 Oute coe 6 4 Mantle Distance fom cente (10 6 m) Suface Fig. 144: The density of Eath as a function of distance fom the cente. The limits of the solid inne coe, the lagely liquid oute coe, and the solid mantle ae shown, but the cust of Eath is too thin to show clealy on this plot. Table 141 shows values of a g computed fo vaious altitudes above Eath s suface. Since Section 63, we have assumed that Eath is an inetial fame by neglecting its actual otation. This simplification has allowed us to assume that the local gavitational stength g of a paticle is the same as the magnitude of the gavitational acceleation (which we now call a g ). Futhemoe, we assumed that g has the constant value of 98. ms ove Eath s suface. Howeve, the g we would measue diffes fom the a g we would calculate with Eq fo thee easons: (1) Eath is not unifom, () it is not a pefect sphee, and (3) it otates. Moeove, because g diffes fom a g, the measued weight mg of the paticle diffes fom the magnitude of the gavitational foce on the paticle as given by Eq fo the same thee easons. Let us now examine those easons. 1. Eath is not unifom. The density (mass pe unit volume) of Eath vaies adially as shown in Fig. 144, and the density of the cust (o oute section) of Eath vaies fom egion to egion ove Eath s suface. Thus, g vaies fom egion to egion ove the suface. 144
5 Ch. 14: Gavitation (Beta Vesion 7/01) TABLE 141: Vaiation of a g with Altitude Altitude (km) a g (m/s ) Altitude Example Mean Eath suface Mt. Eveest Highest manned balloon Space shuttle obit Communications satellite. Eath is not a sphee. Eath is appoximately an ellipsoid, flattened at the poles and bulging at the equato. Its equatoial adius is geate than its pola adius by 1 km. Thus, a point at the poles is close to the dense coe of Eath than is a point on the equato. This is one eason the feefall acceleation g inceases as one poceeds, at sea level, fom the equato towad eithe pole. 3. Eath is otating. The otation axis uns though the noth and south poles of Eath. An object located on Eath s suface anywhee except at those poles must otate in a cicle about the otation axis and thus must have a centipetal acceleation diected towad the cente of the cicle. This centipetal acceleation equies a centipetal net foce that is also diected towad that cente. To see how Eath s otation causes the local gavitational stength at the Eath s suface, g, to diffe fom a g, let us analyze a simple situation in which a cate of mass m is on a scale at the equato. Figue 145a shows this situation as viewed fom a point in space above the noth pole. Figue 145b, a feebody diagam fo the cate, shows the two foces on the cate, both acting along a adial axis that extends fom Eath s cente. The nomal foce N on the cate fom the scale is diected outwad, in the positive diection of axis. The gavitational foce, epesented with its equivalent ma g, is diected inwad. Because the cate tavels in a cicle about the cente of Eath as Eath tuns, it has a centipetal acceleation a diected inwad. Fom Eq. 110, we know the magnitude of this acceleation is equal to ω, whee ω is Eath s angula speed and is the cicle s adius (appoximately Eath s adius). Thus, we can wite Newton s Second Law in component fom fo the axis ( Fnet, = ma) as N + mag = m( ω ). (149) The magnitude N of the nomal foce is equal to the weight mg ead on the scale. With mg substituted fo N and with the fact that N and a g point in opposite diections, Eq gives us mg m a = m( ω ), (1410) o mg = m a m( ω ), which says (measued weight) = (magnitude of gavitational foce) (mass times centipetal acceleation). g g Cate N Cate (a) ma g (b) Scale Noth pole Fig. 145: (a) A cate lies on a scale at Eath s equato, as seen along Eath s otation axis fom above the noth pole. (b) A feebody diagam fo the cate, with a adially outwad axis. The gavitational foce on the cate is epesented with its equivalent ma g. The nomal foce on the cate fom the scale is N. Because of Eath s otation, the cate has a centipetal acceleation a that is diected towad Eath s cente. a Thus, the measued weight is actually less than the magnitude of the gavitational foce on the cate, because of Eath s otation. To find a coesponding expession fo g and a g, we cancel m fom Eq to wite g = a ω, (1411) which says (feefall acceleation) = (local gavitational stength) (centipetal acceleation) g
6 Ch. 14: Gavitation (Beta Vesion 7/01) Thus, the measued feefall acceleation is actually less than the local gavitational stength, because of Eath s otation. The diffeence between the local gavitational stength g and the magnitude of the gavitational acceleation a g is equal to ω and is geatest on the equato (fo one eason, the adius of the cicle taveled by the cate is geatest thee). To find the diffeence, we can use Eq ( ω = θ t ) and Eath s adius = m. Fo one otation of Eath, θ is π ad and the time peiod t is about 4 h. Using these values (and conveting hous to seconds), we find that g is less than a g by only about m s (compaed to 98. ms ). Theefoe, neglecting the diffeence in g and a g is often justified. Similaly, neglecting the diffeence between weight and the magnitude of the gavitational acceleation is also often justified Gavitation Inside Eath Newton s shell theoem can also be applied to a situation in which a paticle is located inside a unifom shell, to show the following: A unifom shell of matte exets no net gavitational foce on a paticle located inside it. Caution: This statement does not mean that the gavitational foces on the paticle fom the vaious elements of the shell magically disappea. athe, it means that the sum of the foce vectos on the paticle fom all the elements is zeo. If the density of Eath wee unifom, the gavitational foce acting on a paticle would be a maximum at Eath s suface and would decease as the paticle moved outwad. If the paticle wee to move inwad, pehaps down a deep mine shaft, the gavitational foce would change fo two easons. (1) It would tend to incease because the paticle would be moving close to the cente of Eath. () It would tend to decease because the thickening shell of mateial lying outside the paticle s adial position would not exet any net foce on the paticle. Fo a unifom Eath, the second influence would pevail and the foce on the paticle would steadily decease to zeo as the paticle appoached the cente of Eath. Howeve, fo the eal (nonunifom) Eath, the foce on the paticle actually inceases as the paticle begins to descend. The foce eaches a maximum at a cetain depth; only then does it begin to decease as the paticle descends fathe. Touchstone Example , at the end of this chapte, illustates how to use what you leaned in this section. TE 146 Gavitational Potential Enegy In Section 103, we discussed the gavitational potential enegy of a paticleeath system. We wee caeful to keep the paticle nea Eath s suface, so that we could egad the gavitational foce as constant. We then chose some efeence configuation of the system as having a gavitational potential enegy of zeo. Often, in this configuation the paticle was on Eath s suface. Fo paticles not on Eath s suface, the gavitational potential enegy deceased when the sepaation between the paticle and Eath deceased. Hee, we boaden ou view and conside the gavitational potential enegy U of two paticles, of masses m and M, sepaated by a distance. We again choose a efeence configuation with U equal to zeo. Howeve, to simplify the equations, the sepaation distance in the efeence configuation is now lage enough to be appoximated as infinite. As befoe, the gavitational potential enegy deceases when the sepaation deceases. Since U = 0 fo =, the potential enegy is negative fo any finite sepaation and becomes pogessively moe negative as the paticles move close togethe. With these facts in mind and as we shall justify next, we take the gavitational potential enegy of the twopaticle system to be 146 m 1 1 m m Fig. 146: Thee paticles fom a system. (The sepaation fo each pai of paticles is labeled with a double subscipt to indicate the paticles.) The gavitational potential enegy of the system is the sum of the gavitational potential enegies of all thee pais of paticles.
7 Ch. 14: Gavitation (Beta Vesion 7/01) U = (gavitational potential enegy). (141) Note that U( ) appoaches zeo as appoaches infinity and that fo any finite value of, the value of U( ) is negative. The potential enegy given by Eq is a popety of the system of two paticles athe than of eithe paticle alone. Thee is no way to divide this enegy and say that so much belongs to one paticle and so much to the othe. Howeve, if M >> m, as is tue fo Eath (mass M) and a baseball (mass m), we often speak of the potential enegy of the baseball. We can get away with this because, when a baseball moves in the vicinity of Eath, changes in the potential enegy of the baseballeath system appea almost entiely as changes in the kinetic enegy of the baseball, since changes in the kinetic enegy of Eath ae too small to be measued. Similaly, in Section 148 we shall speak of the potential enegy of an atificial satellite obiting Eath, because the satellite s mass is so much smalle than Eath s mass. When we speak of the potential enegy of bodies of compaable mass, howeve, we have to be caeful to teat them as a system. If ou system contains moe than two paticles, we conside each pai of paticles in tun, calculate the gavitational potential enegy of that pai with Eq as if the othe paticles wee not thee, and then algebaically sum the esults. Applying Eq to each of the thee pais of Fig. 146, fo example, gives the potential enegy of the system as F HG Gm1m Gm1m3 Gm m U = I KJ. (1413) Poof of Equation 141 Let us shoot a baseball diectly away fom Eath along the path shown in Fig We want to find an expession fo the gavitational potential enegy U of the ball at point P along its path, at adial distance fom Eath s cente. To do so, we fist find the wok W done on the ball by the gavitational foce as the ball tavels fom point P to a geat (infinite) distance fom Eath. Because the gavitational foce F ( ) is a vaiable foce (its magnitude depends on ), we must use the techniques of Section 95 to find the wok. In vecto notation, we can wite W = z F ( ) d. (1414) The integal contains the scala (o dot) poduct of the foce F ( ) and the diffeential displacement vecto d along the ball s path. We can expand that poduct as F () d= F () dcos φ, (1415) whee φ is the angle between the diections of F () and d. When we substitute 180 fo φ and Eq fo F ( ), Eq becomes F () d= d whee M is Eath s mass and m is the mass of the ball. Substituting this into Eq and integating gives us d M F P Fig. 147: A baseball is shot diectly away fom Eath, though point P at adial distance fom Eath s cente. The gavitational foce F and a diffeential displacement vecto d ae shown, both diected along a adial axis. W z 1 d = 0 = = = L N M O Q P. (1416) 147
8 Ch. 14: Gavitation (Beta Vesion 7/01) W in Eq is the wok equied to move the ball fom point P (at distance ) to infinity. Equation 104 ( U = W) tells us that we can also wite that wok in tems of potential enegies as U U = W. The potential enegy U at infinity is zeo, and U is the potential enegy at P. Thus, with Eq substituted fo W, the pevious equation becomes U = W = Switching to gives us Eq. 141, which we set out to pove. Path Independence In Fig. 148, we move a baseball fom point A to point G along a path consisting of thee adial lengths and thee cicula acs (centeed on Eath). We ae inteested in the total wok W done by Eath s gavitational foce F on the ball as it moves fom A to G. The wok done along each cicula ac is zeo, because the diection of F is pependicula to the ac at evey point. Thus, the only woks done by F ae along the thee adial lengths, and the total wok W is the sum of those woks. Now, suppose we mentally shink the acs to zeo. We would then be moving the ball diectly fom A to G along a single adial length. Does that change W? No. Because no wok was done along the acs, eliminating them does not change the wok. The path taken fom A to G now is clealy diffeent, but the wok done by F is the same. We discussed such a esult in a geneal way in Section 10. Hee is the point: The gavitational foce is a consevative foce. Thus, the wok done by the gavitational foce on a paticle moving fom an initial point i to a final point f is independent of the actual path taken between the points. Fom Eq. 104, the change U in the gavitational potential enegy fom point i to point f is given by G B E F C D U = U U = W. (1417) f Since the wok W done by a consevative foce is independent of the actual path taken, the change U in gavitational potential enegy is also independent of the actual path taken. Potential Enegy and Foce In the poof of Eq. 141, we deived the potential enegy function U() fom the foce function F ( ). We should be able to go the othe way that is, to stat fom the potential enegy function and deive the foce function. Guided by Eq. 10, we can wite the adial foce component F as du d F = = d d =. i F HG I K J (1418) A Eath Fig. 148: Nea Eath, a baseball is moved fom point A to point G along a path consisting of adial lengths and cicula acs. This is Newton s law of gavitation (Eq. 141). The minus sign indicates that the foce on mass m points adially inwad, towad mass M. Escape Speed If you fie a pojectile upwad, usually it will slow, stop momentaily, and etun to Eath. Thee is, howeve, a cetain minimum initial speed that will cause it to move upwad 148
9 Ch. 14: Gavitation (Beta Vesion 7/01) foeve, theoetically coming to est only at infinity. This initial speed is called the (Eath) escape speed. Conside a pojectile of mass m, leaving the suface of a planet (o some othe astonomical body o system) with escape speed v. It has a kinetic enegy K given by 1 mv and a potential enegy U given by Eq. 141: U = in which M is the mass of the planet, and is its adius. When the pojectile eaches infinity, it stops and thus has no kinetic enegy. It also has no potential enegy because this is ou zeopotentialenegy configuation. Its total enegy at infinity is theefoe zeo. Fom the pinciple of consevation of enegy, its total enegy at the planet s suface must also have been zeo, so This yields F HG I K J = 1 K+ U = mv + v 0 GM =. (1419) The escape speed v does not depend on the diection in which a pojectile is fied fom a planet. Howeve, attaining that speed is easie if the pojectile is fied in the diection the launch site is moving as the planet otates about its axis. Fo example, ockets ae launched eastwad at Cape Canaveal to take advantage of the Cape s eastwad speed of 1500 km/h due to Eath s otation. Equation can be applied to find the escape speed of a pojectile fom any astonomical body, povided we substitute the mass of the body fo M and the adius of the body fo. Table 14 shows escape speeds fom some astonomical bodies. TABLE 14: Some Escape Speeds Body Mass (kg) adius (m) Escape Speed (km/s) Cees a Eath s moon Eath Jupite Sun Siius B b Neuton sta c a The most massive of the asteoids. b A white dwaf (a sta in a final stage of evolution) that is a companion of the bight sta Siius. c The collapsed coe of a sta that emains afte that sta has exploded in a supenova event. EADING EXECISE 14: You move a ball of mass m away fom a sphee of mass M. (a) Does the gavitational potential enegy of the ballsphee system incease o decease? (b) Is positive o negative wok done by the gavitational foce between the ball and the sphee? Touchstone Example , at the end of this chapte, illustates how to use what you leaned in this section. TE 147 Planets and Satellites: Keple s Laws The motions of the planets, as they seemingly wande against the backgound of the stas, have been a puzzle since the dawn of histoy. The looptheloop motion of Mas, 149
10 Ch. 14: Gavitation (Beta Vesion 7/01) shown in Fig. 149, was paticulaly baffling. Johannes Keple ( ), afte a lifetime of study, woked out the empiical laws that goven these motions. Tycho Bahe ( ), the last of the geat astonomes to make obsevations without the help of a telescope, compiled the extensive data fom which Keple was able to deive the thee laws of planetay motion that now bea his name. Late, Newton ( ) showed that his law of gavitation leads to Keple s laws. In this section we discuss each of Keple s laws in tun. Although hee we apply the laws to planets obiting the Sun, they hold equally well fo satellites, eithe natual o atificial, obiting Eath o any othe massive cental body. 1. THE LAW OF OBITS: All planets move in elliptical obits, with the Sun at one focus. Figue shows a planet of mass m moving in such an obit aound the Sun, whose mass is M. We assume that M >> m, so that the cente of mass of the planetsun system is appoximately at the cente of the Sun. The obit in Fig is descibed by giving its semimajo axis a and its eccenticity e, the latte defined so that ea is the distance fom the cente of the ellipse to eithe focus f o f. An eccenticity of zeo coesponds to a cicle, in which the two foci mege to a single cental point. The eccenticities of the planetay obits ae not lage, so sketched on pape the obits look cicula. The eccenticity of the ellipse of Fig , which has been exaggeated fo claity, is The eccenticity of Eath s obit is only THE LAW OF AEAS: A line that connects a planet to the Sun sweeps out equal aeas in the plane of the planet s obit in equal times; that is, the ate da/dt at which it sweeps out aea A is constant. Qualitatively, this second law tells us that the planet will move most slowly when it is fathest fom the Sun and most apidly when it is neaest to the Sun. As it tuns out, Keple s second law is totally equivalent to the law of consevation of angula momentum. Let us pove it. The aea of the shaded wedge in Fig a closely appoximates the aea swept out in time t by a line connecting the Sun and the planet, which ae sepaated by a distance. The aea A of the wedge is appoximately the aea of a tiangle with base θ and height. Since the aea of a tiangle is onehalf of the base times the height, A>> 1 θ. This expession fo A becomes moe exact as t (hence θ ) appoaches zeo. The instantaneous ate at which aea is being swept out is then da 1 d θ 1 = = ω, (140) dt dt in which ω is the angula speed of the otating line connecting Sun and planet. Figue 1411b shows the linea momentum p of the planet, along with its adial and pependicula components. Fom Eq. 13 ( L = p ), the magnitude of the angula momentum L of the planet about the Sun is given by the poduct of and p, the component of p pependicula to. Hee, fo a planet of mass m, L = p = ()( mv ) = ()( mω) (141) = m ω, July 6 Octobe 14 June 6 Septembe 4 Fig. 149: The path of the planet Mas as it moved against a backgound of the constellation Capicon duing Its position on fou selected days is maked. Both Mas and Eath ae moving in obits aound the Sun so that we see the position of Mas elative to us; this sometimes esults in an appaent loop in the path of Mas. p M f a θ ea a Fig : A planet of mass m moving in an elliptical obit aound the Sun. The Sun, of mass M, is at one focus F of the ellipse. The othe focus is F, which is located in empty space. Each focus is a distance ea fom the ellipse s cente, with e being the eccenticity of the ellipse. The semimajo axis a of the ellipse, the peihelion (neaest the Sun) distance p, and the aphelion (fathest fom the Sun) distance a ae also shown. ea m f' whee we have eplaced v with its equivalent ω (Eq ). Eliminating ω between Eqs and 141 leads to da dt L = m (14) 1410
11 Ch. 14: Gavitation (Beta Vesion 7/01) If da dt is constant, as Keple said it is, then Eq. 14 means that L must also be constant angula momentum is conseved. Keple s second law is indeed equivalent to the law of consevation of angula momentum. θ p p Sun θ θ A Sun θ p M M m (a) Fig : (a) In time t, the line connecting the planet to the Sun (of mass M) sweeps though an angle θ, sweeping out an aea A (shaded). (b) The linea momentum p of the planet and its components. (b) M θ 3. THE LAW OF PEIODS: The squae of the peiod of any planet is popotional to the cube of the semimajo axis of its obit. Fig. 141: A planet of mass m moving aound the Sun in a cicula obit of adius. To see this, conside the cicula obit of Fig. 141, with adius (the adius of a cicle is equivalent to the semimajo axis of an ellipse). Applying Newton s Second Law, ( F = ma), to the obiting planet in Fig yields m = ( )( ω ). (143) Hee we have substituted fom Eq.141 fo the foce magnitude F and used Eq to substitute ω fo the centipetal acceleation. If we use Eq to eplace ω with π T, whee T is the peiod of the motion, we obtain Keple s thid law: T 4π GM = F H G I K J 3 (law of peiods). (144) The quantity in paentheses is a constant that depends only on the mass M of the cental body about which the planet obits. Equation 144 holds also fo elliptical obits, povided we eplace with a, the 3 semimajo axis of the ellipse. This law pedicts that the atio T a has essentially the same value fo evey planetay obit aound a given massive body. Table 143 shows how well it holds fo the obits of the planets of the sola system. TABLE 143: Keple s Law of Peiods fo the Sola System Planet Semimajo Axis a (10 10 m) Peiod T (y) Mecuy Venus Eath Mas Jupite Satun Uanus Neptune Pluto T /a 3 (1034 y /m 3 ) 1411
12 Ch. 14: Gavitation (Beta Vesion 7/01) EADING EXECISE 143: Satellite 1 is in a cetain cicula obit about a planet, while satellite is in a lage cicula obit. Which satellite has (a) the longe peiod and (b) the geate speed? Touchstone Examples and 147, at the end of this chapte, illustate how to use what you leaned in this section. TE 148 Satellites: Obits and Enegy As a satellite obits Eath on its elliptical path, both its speed, which fixes its kinetic enegy K, and its distance fom the cente of Eath, which fixes its gavitational potential enegy U, fluctuate with fixed peiods. Howeve, the mechanical enegy E of the satellite emains constant. (Since the satellite s mass is so much smalle than Eath s mass, we assign U and E fo the Eathsatellite system to the satellite alone.) The potential enegy of the system is given by Eq and is U = (with U = 0 fo infinite sepaation). Hee is the adius of the obit, assumed fo the time being to be cicula, and M and m ae the masses of Eath and the satellite, espectively. To find the kinetic enegy of a satellite in a cicula obit, we wite Newton s second law, in tems of vecto magnitudes as ( F = ma), as m v = (145) whee v / is the centipetal acceleation of the satellite. Then, fom Eq. 145, the kinetic enegy is On Febuay 7, 1984, at a height of 10 km above Hawaii and with a speed of about 9,000 km/h, Buce McCandless stepped (untetheed) into space fom a space shuttle and became the fist human satellite. 1 K = mv = (146) which shows us that fo a satellite in a cicula obit, The total mechanical enegy of the obiting satellite is K E = K+ U = U = (cicula obit). (147) o E = (cicula obit). (148) This tells us that fo a satellite in a cicula obit, the total enegy E is the negative of the kinetic enegy K: E = K (cicula obit). (149) Fo a satellite in an elliptical obit of semimajo axis a, we can substitute a fo in Eq to find the mechanical enegy as E = (elliptical obit). (1430) a 141
13 Ch. 14: Gavitation (Beta Vesion 7/01) Equation tells us that the total enegy of an obiting satellite depends only on the semimajo axis of its obit and not on its eccenticity e. Fo example, fou obits with the same semimajo axis ae shown in Fig ; the same satellite would have the same total mechanical enegy E in all fou obits. Figue shows the vaiation of K, U, and E with fo a satellite moving in a cicula obit about a massive cental body. Enegy M e = K() E() U() Fig : Fou obits about an object of mass M. All fou obits have the same semimajo axis a and thus coespond to the same total mechanical enegy E. Thei eccenticities e ae maked. E = K + U Fig : The vaiation of kinetic enegy K, potential enegy U, and total enegy E with adius fo a satellite in a cicula obit. Fo any value of, the values of U and E ae negative, the value of K is positive, and E = K. As appoaches infinity, all thee enegy cuves appoach a value of zeo. EADING EXECISE 144: In the figue, a space shuttle is initially in a cicula obit of adius about Eath. At point P, the pilot biefly fies a fowadpointing thuste to decease the shuttle s kinetic enegy K and mechanical enegy E. (a) Which of the dashed elliptical obits shown in the figue will the shuttle then take? (b) Is the obital peiod T of the shuttle (the time to etun to P) then geate than, less than, o the same as in the cicula obit? 1 P 149 Einstein and Gavitation Pinciple of Equivalence Albet Einstein once said: I was in the patent office at Ben when all of a sudden a thought occued to me: If a peson falls feely, he will not feel his own weight. I was statled. This simple thought made a deep impession on me. It impelled me towad a theoy of gavitation. Thus Einstein tells us how he began to fom his geneal theoy of elativity. The fundamental postulate of this theoy about gavitation (the gavitating of objects towad each othe) is called the pinciple of equivalence, which says that gavitation and acceleation ae equivalent. If a physicist wee locked up in a small box as in Fig , he would not be able to tell whethe the box was at est on Eath (and subject only to Eath s gavitational foce), as in Fig a, o acceleating though intestella space at 1413
14 Ch. 14: Gavitation (Beta Vesion 7/01) 98. m s (and subject only to the foce poducing that acceleation), as in Fig b. In both situations he would feel the same and would ead the same value fo his weight on a scale. Moeove, if he watched an object fall past him, the object would have the same acceleation elative to him in both situations. Cuvatue of Space We have thus fa explained gavitation as due to a foce between masses. Einstein showed that, instead, gavitation is due to a cuvatue (o shape) of space that is caused by the masses. (As is discussed late in this book, space and time ae entangled so the cuvatue of which Einstein spoke is eally a cuvatue of spacetime, the combined fou dimensions of ou univese.) Pictuing how space (such as vacuum) can have cuvatue is difficult. An analogy might help: Suppose that fom obit we watch a ace in which two boats begin on the equato with a sepaation of 0 km and head due south (Fig a). To the sailos, the boats tavel along flat, paallel paths. Howeve, with time the boats daw togethe until, neae the south pole, they touch. The sailos in the boats can intepet this dawing togethe in tems of a foce acting on the boats. Howeve, we can see that the boats daw togethe simply because of the cuvatue of Eath s suface. We can see this because we ae viewing the ace fom outside that suface. Figue 1416b shows a simila ace: Two hoizontally sepaated apples ae dopped fom the same height above Eath. Although the apples may appea to tavel along paallel paths, they actually move towad each othe because they both fall towad Eath s cente. We can intepet the motion of the apples in tems of the gavitational foce on the apples fom Eath. We can also intepet the motion in tems of a cuvatue of the space nea Eath, due to the pesence of Eath s mass. This time we cannot see the cuvatue because we cannot get outside the cuved space, as we got outside the cuved Eath in the boat example. Howeve, we can depict the cuvatue with a dawing like Fig c; thee the apples would move along a suface that cuves towad Eath because of Eath s mass. Equato N S (a) S C (b) Flat space fa fom Eath Conveging paths Paallel paths (c) Eath Cuved space nea Eath a a (a) (b) Fig : (a) A physicist in a box esting on Eath sees a cantaloupe falling with acceleation a = 9.8 ms. (b) If he and the box acceleate in deep space at 9.8 ms, the cantaloupe has the same acceleation elative to him. It is not possible, by doing expeiments within the box, fo the physicist to tell which situation he is in. Fo example, the platfom scale on which he stands eads the same weight in both situations. Fig : (a) Two objects moving along lines of longitude towad the south pole convege because of the cuvatue of Eath s suface. (b) Two objects falling feely nea Eath move along lines that convege towad the cente of Eath because of the cuvatue of space nea Eath. (c) Fa fom Eath (and othe masses), space is flat and paallel paths emain paallel. Close to Eath, the paallel paths begin to convege because space is cuved by Eath s mass. When light passes nea Eath, its path bends slightly because of the cuvatue of space thee, an effect called gavitational lensing. When it passes a moe massive stuctue, like a galaxy o a black hole having lage mass, its path can be bent moe. If such a massive stuctue is between us and a quasa (an extemely bight, extemely distant souce of light), the light fom the quasa can bend aound the massive stuctue and towad us (Fig a). Then, because the light seems to be coming to us fom a numbe of slightly diffeent diections in the sky, we see the same quasa in all those diffeent diections. In some situations, the quasas we see blend togethe to fom a giant luminous ac, which is called an Einstein ing (Fig b). Should we attibute gavitation to the cuvatue of spacetime due to the pesence of masses o to a foce between masses? O should we attibute it to the actions of a type of 1414
15 Ch. 14: Gavitation (Beta Vesion 7/01) fundamental paticle called a gaviton, as conjectued in some moden physics theoies? We do not know. Paths of light fom quasa Appaent quasa diections Galaxy o lage black hole Final paths Eath detecto (a) Fig : (a) Light fom a distant quasa follows cuved paths aound a galaxy o a lage black hole because the mass of the galaxy o black hole has cuved the adjacent space. If the light is detected, it appeas to have oiginated along the backwad extensions of the final paths (dashed lines). (b) The Einstein ing known as MG on the compute sceen of a telescope. The souce of the light (actually, adio waves, which ae a fom of invisible light) is fa behind the lage, unseen galaxy that poduces the ing; a potion of the souce appeas as the two bight spots seen along the ing
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