F G r. Don't confuse G with g: "Big G" and "little g" are totally different things.


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1 G1 Gavity Newton's Univesal Law of Gavitation (fist stated by Newton): any two masses m 1 and m exet an attactive gavitational foce on each othe accoding to m m G 1 This applies to all masses, not just big ones. G = univesal constant of gavitation = N m / kg (G is vey small, so it is vey difficult to measue!) Don't confuse G with g: "Big G" and "little g" ae totally diffeent things. Newton showed that the foce of gavity must act accoding to this ule in ode to poduce the obseved motions of the planets aound the sun, of the moon aound the eath, and of pojectiles nea the eath. He then had the geat insight to ealize that this same foce acts between all masses. [That gavity acts between all masses, even small ones, was expeimentally veified in 1798 by Cavendish.] Newton couldn't say why gavity acted this way, only how. instein (1915) Geneal Theoy of elativity, explained why gavity acted like this. xample: oce of attaction between two humans. people with masses m 1 m 70 kg, distance = 1 m apat. m 11 1 m ( )( 70) G N This is a vey tiny foce! It is the weight of a mass of gam. A hai weighs 10 3 gams the foce of gavity between two people talking is about 1/60 the weight of a single hai. Computation of g m 1 m Impotant fact about the gavitational foce fom spheical masses: a spheical body exets a gavitational foce on suounding bodies that is the same as if all the sphee's mass wee concentated at its cente. This is difficult to pove (Newton woied about this fo 0 yeas.) 3/1/009 Univesity of Coloado at Boulde
2 G sphee, mass M gav mass m mass m point mass M gav (same as with sphee) We can now compute the acceleation of gavity g! (Befoe, g was expeimentally detemined, and it was a mystey why g was the same fo all masses.) gav = m a = m g ath mass m, dopped nea suface Mm G m g (since = is distance fom m to cente of ath) mass M m's cancel! g GM If you plug in the numbes fo G, M, and, you get g = 9.8 m/s. Newton's Theoy explains why all objects nea the ath's suface fall with the same acceleation (because the m's cancel in gav ma.) Newton's theoy also makes a quantitative pediction fo the value of g, which is coect. xample: g on Planet X. Planet X has the same mass as eath (M X = M ) but has ½ the adius ( X = 0.5 ). What is g x, the acceleation of gavity on planet X? Planet X is dense than eath, so expect g x lage than g. g G M G M 1 G M X x X 1/ g of eath 4 g. Don't need values of G, M, and! Method II, set up a atio: 3/1/009 Univesity of Coloado at Boulde
3 G3 GM X x X X g M g GM M X 1 4, g 4 g X * At height h above the suface of the eath, g is less, since we ae futhe fom the suface, futhe fom the eath's cente. = + h h eath g G M G M ( h) The space shuttle obits eath at an altitude of about 00 mi 1.6 km/mi 30 km. ath's adius is = 6380 km. So the space shuttle is only about 5% futhe fom the eath's cente than we ae. If is 5% lage, then is about 10% lage, and M m ( on mass m in shuttle) G about 10% less than on eath's suface gav ( h) Astonauts on the shuttle expeience almost the same gav as when on eath. So why do we say the astonauts ae weightless?? "Weightless" does not mean "no weight". "Weightless" means "feefall" means the only foce acting is gavity. If you fall down an ailess elevato shaft, you will feel exactly like the astonauts. You will be weightless, you will be in feefall. ath v astonaut gav N gav An astonaut falls towad the eath, as she moves fowad, just as a bullet fied hoizontally fom a gun falls towad eath. Obits Conside a planet like ath, but with no ai. ie pojectiles hoizontally fom a mountain top, with faste and faste initial speeds. 3/1/009 Univesity of Coloado at Boulde
4 G4 Planet would go staight, if no gavity The obit of a satellite aound the eath, o of a planet aound the sun obeys Keple's 3 Laws. Keple, Geman ( ). Befoe Newton. Using obsevational data fom Danish astonome Tycho Bahe ("Bahay"), Keple discoveed that the obits of the planets obey 3 ules. obits! KI : A planet's obit is an ellipse with the Sun at one focus. KII : A line dawn fom planet P to sun S sweeps out equal aeas in equal times. faste S same time intevals, same aeas slowe Sun Planet KIII: o planets aound the sun, the peiod T and the mean distance fom the sun ae elated T TA TB by constant. That is fo any two planets A and B,. This means that A B planets futhe fom the sun (lage ) have longe obital peiods (longe T). Keple's Laws wee empiical ules, based on obsevations of the motions of the planets in the sky. Keple had no theoy to explain these ules. 3/1/009 Univesity of Coloado at Boulde
5 G5 Newton ( ) stated with Keple's Laws and NII ( net = ma) and deduced that MSmP gav G. Newton applied simila easoning to the motion of the athmoon ( Sun planet ) SP M m system (and to an athapple system) and deduced that gav G. ( athmass m ) m Newton then made a mental leap, and ealized that this law applied to any masses, not just to the Sunplanet, the athmoon, and athpojectile systems. Stating with net = ma and gav = G Mm /, Newton was able to deive Keple's Laws (and much moe!). Newton could explain the motion of eveything! Deivation of KIII (fo special case of cicula obits). Conside a small mass m in cicula obit about a lage mass M, with obital adius and peiod T. We aim to show that T / 3 = const. Stat with NII: net = m a M peiod T The only foce acting is gavity, and fo cicula motion a = v / m v M m v M G m G v T [ecall the v = dist / time = / T ] M 4 T 4 G constant, independent of m 3 T G M ( Deiving this esult fo elliptical obits is much hade, but Newton did it. ) An exta esult of this calculation is a fomula fo the speed v of a satellite in cicula obit: GM v. o loweath obit (few hunded miles up), this obital speed is about 7.8 km/s 4.7 miles/second. The Space Shuttle must attain a speed of 4.7 mi/s when it eaches the top of the atmosphee (and it fuel has un out) o else it will fall back to ath. 3/1/009 Univesity of Coloado at Boulde
6 G6 Measuement of Big G The value of G ("big G") was not known until In that yea, Heny Cavendish (nglish) measued the vey tiny gav between lead sphees, using a device called a tosion balance. m1 m gav gav G G = m m 1 ( If gav,, and m's known, can compute G.) GM Befoe Cavendish's expeiment, g and wee known, so using g compute the poduct G M, but G and M could not be detemined sepaately., one could With Cavendish's measuement of G, one could then compute M. Hence, Cavendish "weighed the eath". Gavitational Potential negy Peviously, we showed that P gav = mgh. But to deive P = mgh, we assumed that gav = mg = Mm constant, which is only tue nea the suface of the ath. In geneal, gav G constant (it depends on ). We now show that fo the geneal case, Pgav U( ), [ U( = ) 0 ] This is the gavitational potential fo two masses, M and m, sepaated by a distance. By convention, the zeo of gavitational potential enegy is set at =. [ I will use the common notation U(), instead of P. ] ecall the definition of P: of wok fo the case of 1D motion: x P W () x dx. Hee, we have used the definition i x1 x W d () x dx. ( 1D) x1 3/1/009 Univesity of Coloado at Boulde
7 G7 M gav m 0 x 1 dx x Conside a mass M at the oigin and a mass m at position x 1, as shown in the diagam. We compute the wok done by the foce of gavity as the mass m moves fom x = x 1 to x =. The foce (x) on mass m is in the negative diection, so, indicating diection with a sign, we have x ( ). Hee, the wok done by gavity is negative, since foce and x displacement ae in opposite diections: GMm GMm GMm W ( x) dx dx x x x gav x1 x1 x1 1 GMm om the definition of P, P U U( x= ) U( x1) Wgav. Calling the x initial position (instead of x 1 ), we have U (). 0 1 A slight notation change now: is the adial distance fom the oigin, so is always positive (unlike x which can be positive o negative.) Plotting U() vs., we see a gavitational potential well. 3/1/009 Univesity of Coloado at Boulde
8 G8 U() = 0 U() The "Potential Well" ecall that negative potential enegy simply means less enegy than the zeo of enegy. Question: How is P = mgh a special case of U() = GMm/? U() = eath U(h) U = mgh h = eath scape Speed v escape Thow a ock away fom an (ailess) planet with a speed v. If v < v escape, the ock will ise to a maximum height and then fall back down. If v > v escape, the ock will go to =, and will still have some speed left ove and be moving away fom the planet. If v = v escape, the ock will have just enough initial K to escape the planet: its distance goes to = at the same time its speed appoaches zeo: v 0 as. We can use consevation of enegy to compute the escape speed v esc (often called, incoectly, the "escape velocity" ). 3/1/009 Univesity of Coloado at Boulde
9 G9 Initial configuation: = (suface of planet), v = v esc. inal configuation: =, v = 0. 1 Ki Pi Kf Pf m vesc 0 0 G M vesc Notice that vesc v obit If the ock is thown with speed v > v esc, it will go to =, and will have some K left ove, v final > Ki Pi Kf Pf m vi m vf 0 GM vf vi K P tot constant ( ) ( ) ( ) o ( ) U() tot = K+P P K 3/1/009 Univesity of Coloado at Boulde
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