Chapter 13 Gravitation

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1 Chapte 13 Gavitation In this chapte we will exploe the following topics: -Newton s law of gavitation, which descibes the attactive foce between two point masses and its application to extended objects -The acceleation of gavity on the suface of the Eath, above it as well as below it. -Gavitational potential enegy -Keple s thee laws of planetay motion -Satellites (obits, enegy, escape velocity) (13-1)

2 m m 1 mm F = G 1 Newton's Law of Gavitation Newton ealized that the foce that holds the moon in its obit is of the same natue as the foce that makes an apple dop nea the suface of the Eath. Newton concluded that the Eath attacts apples as well as the moon, and also that evey object in the univese attacts evey othe object. The tendency of objects to move towad each othe is known as gavitation. Newton fomulated a foce law known as Newton's law of gavitation. Evey paticle attacts any othe paticle with a gavitational foce that has the following chaacteistics: 1. The foce acts along the line that connects the two paticles.. Its magnitude is given by the equation mm F = G 1. Hee m and m ae the masses of the two paticles, is thei 1 sepaation, and G is the gavitational constant. Its value is: G = N m / kg 11 (13-)

3 mm F = G F 1 1 F 1 m The gavitational foce F exeted on mby m 1 1 is equal in magnitude to the foce F exeted on mby m1 but opposite in diection. The two foces obey Newton's thid law: F + F = m 1 m m 1 Newton poved that a unifom shell attacts a paticle that is outside the shell as if the shell's mass wee concentated at the shell cente: mm F = G 1 1. F 1 Note: If the paticle is inside the shell, the net foce is zeo. m m 1 Conside the foce F the Eath (adius R, mass M) exets on an apple of mass m. The Eath can be thought of as consisting of concentic shells. Thus fom the apple's point of view the Eath behaves like a point mass at the Eath's cente. The magnitude of the foce is given by the equation: mm F = G R (13-3)

4 Gavitation and the Pinciple of Supeposition The net gavitational foce exeted by a goup of paticles is equal to the vecto sum of the contibution fom each paticle. Fo example, the net foce F1 exeted on m1 by m and m3 is equal to: F = F + F Hee F and F ae the foces exeted on m by m and m, espectively In geneal, the foce exeted on m1 by n paticles is given by the equation: n F1 = F1 + F13+ F F1 n = F1 i. i= The gavitation foce exeted by a continuous extended df m 1 object on a paticle of mass m can be calculated using the 1 pinciple of supeposition. The object is divided into elements of mass dm. The net foce on m1 is the vecto sum of the foces dm exeted by each element. The sum takes the fom of an integal: F = df. Hee df is the foce exeted on m by dm: 1 1 df = Gm1dm. (13-4)

Gavitation Nea the Eath's Suface If we assume that the Eath is a sphee of mass M, the magnitude F of the foce exeted by the Eath on an object of mass m GMm placed at a distance fom the cente of the

5 Gavitation Nea the Eath's Suface If we assume that the Eath is a sphee of mass M, the magnitude F of the foce exeted by the Eath on an object of mass m GMm placed at a distance fom the cente of the Eath is F =. The gavitational foce esults in an acceleation a known as gavitational GM acceleation. Using Newton's second law we have that ag =. Up to this point we have assumed that the fee-fall acceleation g nea the suface of the Eath is constant. Howeve, if we measue g at vaious points on the suface of the Eath we find that its value is not constant. This is attibuted to thee easons: 1. The Eath's mass is not unifomly distibuted. The density of the Eath vaies adially as shown in the figue. The density of the oute section vaies fom egion to egion ove the Eath' s suface. Thus g vaies fom point to point. g (13-5)

. The Eath is not a sphee. Eath is appoximately an ellipsoid flattened at the poles and bulging at the equato. Its equatoial adius is lage than the pola adius by 1 km.

6 . The Eath is not a sphee. Eath is appoximately an ellipsoid flattened at the poles and bulging at the equato. Its equatoial adius is lage than the pola adius by 1 km. Thus the value of g at sea level inceases as one goes fom the equato to the poles. y-axis 3. The Eath is otating. Conside the cate shown in the figue. The cate is esting on a scale at a point on the equato. The net foce along the y-axis F = F F = ma F g y,net g N g N GM Hee ag =, and F i N = mg s the nomal foce exeted R on the cate by the scale. The cate has an acceleation a= ω R due to the otation of the Eath about its axis evey 4 hous. If we apply Newton's second law we get: ma mg = mω R mg = ma mω R g = a ω R. g g Fee-fall acceleation = gavitational acceleation - centipetal acceleation The tem ω R = m/s, which is much smalle than 9. 8 m/s. (13-6)

m m 1 m m 1 F 1 Gavitation Inside the Eath Newton poved that the net gavitational foce on a paticle by a shell depends on the position of the paticle with espect to the shell.

7 m m 1 m m 1 F 1 Gavitation Inside the Eath Newton poved that the net gavitational foce on a paticle by a shell depends on the position of the paticle with espect to the shell. If the paticle is inside the shell, the net foce is zeo. mm 1 If the paticle is outside the shell, the foce is given by: F1 = G. Conside a mass m inside the Eath at a distance fom the cente of the Eath. If we divide the Eath into a seies of concentic shells, only the shells with GmM Hee M is the mass of the pat of the Eath inside a sphee of adius : ins adius less than exet a foce on m. The net foce on m is: F =. ins 3 4π 4πGmρ Mins = ρvins = ρ F = F is linea with. 3 3 (13-7)

8 U m GmM = Gavitational Potential Enegy In Chapte 8 we deived the potential enegy U of a mass m nea the suface of the Eath. We will emove this estiction and assume that the mass m can move away fom the suface of the Eath, at a distance fom the cente of the Eath as shown in the figue. In this case the gavitational potential enegy is GmM U =. The negative sign of U expesses the fact that the coesponding gavitational foce is attactive. Note: The gavitational potential enegy is not only associated with the mass m but with M as well, i.e., with both objects. If we have thee masses m, m, and m positioned as shown in the figue, 1 3 the potential enegy U due to the gavitational foces among the objects is U Gm m Gm m Gm m = + +. We take into account each pai once. (13-8)

9 (13-9) Gavitational Potential Enegy U of the Eath and a Mass m Placed at a Distance R fom the Cente of the Eath. We assume that we move the mass m upwad so that it eaches a geat distance (pactically infinite) fom the Eath. The potential enegy U is equal to the wok W that the gavitational foce does on m. GMm 1 GMm U = W = F() d = d = GMm = R R R R We choose an integation path that points adially outwad. We ae allowed to do this because the gavitational foce is consevative. This can also be seen in the figue to the left. In this case we follow a diffeent path of abitay shape. We then beak the path into adial sections ( AB, CD, and EF) and tangential sections ( BC, DE, and FG). The wok in the latte is equal to zeo because the foce is at ight angles to GMm the displacement. Thus we ae left with the sum. d R

10 Escape Speed If a pojectile of mass m is fied upwad at point A as shown in the figue, the pojectile will stop momentaily and etun to the Eath. v = GM R v = 0 B m Thee is, howeve, a minimum initial speed fo which the pojectile will escape fom the gavitational pull of the Eath and will stop at infinity (point B in the figue). This minimum speed is known as escape velocity. We can detemine the escape velocity using enegy consevation between point A and point B. mv GMm EA = K + U = EB = K + U = 0 R mv GMm GM EA = EB = 0 v = R R The escape speed fom the Eath is 11. km/s. Note: The escape speed does not depend on m. (13-10) v A = m

11 Planets and Satellites : Keple's Laws Stas follow egula paths in the evening sky. They otate once evey 4 hous about an axis that passes though the sta Polais. Polais is the only sta that does not move in the sky. The stas have fixed spatial elationships among them. Humans have classified them in goups known as "constellations." Rotation Axis of the Celestial Sphee Polais Sta N S Eath Celestial sphee (13-11)

12 In contast, planets follow complicated paths in the sky. An example of such motion is given in the figue. Tycho Bahe made vey caeful measuements of planetay motions, but he died befoe he had the chance to analyze his data. This task was caied out by his assistant Johannes Keple, who summaized the esults into thee empiical laws known by his name. Late, Newton used his second law of motion with his gavitational law and the newly developed methods of calculus and deived Keple's laws. (13-1)

13 da dt = Keple's Fist Law. with the Sun at one focus. All planets move on elliptical obits The obits ae descibed by two paametes: the semimajo axis a and the eccenticity e. The obit in the figue of the Eath's obit is only constant da has e = The actual eccenticity Keple's Second Law. The line that connects a planet to the Sun sweeps out equal aeas in the plane of the obit in equal da time intevals Δ t : = constant. dt 1 The aea ΔA swept out by the planet (fig. a) is given by the equation ΔA Δθ. da 1 dθ 1 = = ω whee ω is the planet's angula speed. The planet's angula dt dt da L momentum L= p = mv = mω = m ω =. Keple's second law dt m is equivalent to the law of consevation of angula momentum. ΔA (13-13)

14 Keple's Thid Law. The squae of the peiod of any planet is popotional to the cube of the semimajo axis of its obit. Fo the sake of simplicity we will conside the cicula obit shown in the figue. A planet of mass m moves on a cicula obit of adius aound a sta of mass M. We apply Newton's second law to the motion: GMm GM Fg = = ma= ( m)( ω ) = mω = ω (eq. 1) The peiod T can be 3 π 4π expessed in tems of the angula speed ω : T = T = (eq. ). ω ω T 4π If we substitute ω fom eq. 1 into eq. we get: =. 3 MG T Note 1: The atio does not depend on the mass m of the planet but only on the 3 mass M of the cental sta. Note : Fo elliptical obits the atio T emains consta nt. 3 a T 3 4π = MG (13-14)

15 GMm E = Satellites : Obits and Enegy Conside a satellite that follows a cicula obit of adius aound a planet of mass M. We apply Newton's second GMm v GM = ma = m v = law and have:. mv GMm The kinetic enegy: K = = (eq. 1) GMm The potential enegy: U = (eq. ) U If we compae eq. 1 with eq. we have: K = (eq. 3). GMm GMm GMm The total enegy E = K + U = = = K. The enegies EK,, and U ae plotted as functions of in the figue to the left. GMm Note: Fo elliptical obits E =. a (13-15)

Chapter 13: Gravitation

Chapter 13: Gravitation v m m F G Chapte 13: Gavitation The foce that makes an apple fall is the same foce that holds moon in obit. Newton s law of gavitation: Evey paticle attacts any othe paticle with a gavitation foce given