gravity r2,1 r2 r1 by m 2,1

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1 Gavtaton Many of the foundatons of classcal echancs wee fst dscoveed when phlosophes (ealy scentsts and atheatcans) ted to explan the oton of planets and stas. Newton s ost faous fo unfyng the oton of heavenly bodes wth those on eath (an apple). Newton s law of Unvesal Gavtaton has now been supeseded by Ensten s Geneal Relatvty, but just as eale, Newton s appoxaton s vey good n any stuatons and thus vey wothy of study. G The classcal law of gavtaton s often wtten as F ˆ gavty, whee G s a unvesal constant equal to 6.67 x 0 - N kg -, and s the sepaaton between the asses. Howeve ths s not pecse enough fo us. Consde two sphecal asses,, and the foce of gavty between the, ˆ, F on by G We want to know the foced placed on by o F,. Ths s equal to F, ˆ, whee ˆ s the unt vecto that ponts fo to. (The way to eebe the ode of the subscpts s fnal nus ntal.) The negatve sgn ndcates that the foce s attactve and puts the foce n the opposte decton fo the unt vecto. So knowng the foce placed on by, what about the convese, the foce placed on by, o F? Clealy fo the syety of the expesson these foces ae equal n agntude, but opposte n decton. (Ths s an exaple of Newton s thd laws that states that f an object places a foce on anothe, then t expeences an equal and opposte placed by the othe on t. Newton s thd law s by fa the ost coplcated of hs thee to undestand and apply coectly. An untold nube of

2 ntoductoy physcs students have gotten theselves ted up n knots tyng to fgue out whee to put the eacton foce. You have been foewaned! One hnt, the eacton foce s neve put on the object expeencng the ognal foce.) Takng a close look at the fgue above ases an potant queston. was defned to be the dstance between the centes of the sphecal objects. Why was ths selected; why not the suface to-suface dstance? O what happens when the object s not sphecal? The Unvesal Law of Gavtaton s well defned fo pont asses. Ths s a good appoxaton when the objects ae vey fa apat, such that the ad ae uch salle than the sepaaton. But n the fgue above that s not the case. The sepaaton s only ~0 tes the ad of the asses. Ae we akng a poo appoxaton? To exane ths poble, consde the gavtatonal attacton between a pont ass,, and an abtay extended object, M. Thee s no estcton placed on the sepaaton between the two. M How can we use the Unvesal Law of Gavtaton to detene the foce F M, that body M puts on? You ay be thnkng that you need to fnd the Cente of Mass, o CM of object M, but ths s not coect. (Although t s a thoughtful guess and the appoach to take s sla to that taken to fnd the CM.) Consde a sall poton of ass M, fo conceteness a vey sall cube soewhee wthn the body. The Unvesal Law of Gavtaton allows us to fnd the foce between the sall cube and ass. Specfcally, t s F ˆ, G, whee s the ass of the cube and s the dstance between the cube and. s now well defned because the postons of and ae pecse. Of couse, ths can be done fo a second pont

3 F, F, Now the net foce on the pont ass fo both and s the just the sple vecto addton of the ndvdual foces, F F F net,, Fnet fo and You can know see whee ths s gong. Consde dcng the whole of the extended object M nto lttle peces. (The shape of the lttle peces s abtay, howeve soetes the syety of the object suggests a shape.) Then the net foce on would be gven by, F G. If we know the densty () of the extended object as a functon of ˆ, M, poston then the aount of ass n each lttle pece s ( ) V and the next foce ( ) V becoes: F ˆ, M G,. Now ths s an expesson that can be used on a copute to fnd the net foce placed by M on. The copute can be pogaed to su ove a vast nube of lttle boxes usng the appopate densty, dstance and decton. Howeve, atheatcally t s vey useful to take the lt as the nube of boxes goes to nfnty. Takng ogn of the coodnate syste to be at the pont ass, the expesson becoes ( ) V ( ) F ˆ ˆ, M G l, G dv M. Ths s an ntegal ove the volue of the extended object, wth beng a vecto that ognates at the locaton of the pont ass and

4 tenates wthn ass M. You can agne the end of ths vecto beng oved all ove the ( ) extent of M as t s beng noalzed, ultpled by and sued up. If ths looks coplcated to you, you ae ght, t can be. You wll have a chance to do ths type of ntegal n Calc III. Fo the te beng t s enough to know n pncple how t s done and be able to do t nuecally. It s nteestng to note that because the total foce depends lnealy on the pont ass, t s often convenent to talk about the foce that would be placed on a test patcle of ass as FM, () FM, g o convesely defne gm (). Ths s the concept of a feld, n ths case the gavtatonal feld. The feld fo ass M exsts wthout the pesence of the second patcle to expeence t. One way to pctue the gavtatonal feld s a collecton of aows pontng n the decton that a test ass would acceleate f t wee placed at a locaton and havng a agntude gven by ts acceleaton. Hee s a athe poo endton

5 ( ) ' Fo each pont n space you would need to fnd gm () G dv M ( ) ', whee the ntegal s ove the ped vaables and dv. Ths need not be easy, but t can be done, and s a vey useful concept. What s potant now s that you begn to thnk about such a gavtatonal feld as aows exstng n space aound all assve objects and that these aows tell you the acceleaton that would occu f a test ass was placed thee. We have coe a long way n ou dscusson of gavty, but have not copletely answeed ou queston about how to handle the gavtatonal attacton between extended objects. We know that n geneal the foce nvolves sung up contbutons fo all ove the objects. Consdeng the ognal stuaton, ths looks qute dffcult. We even have to su ove two objects! Thee ust be a way out, ˆ, F on by and thee s, but only n a vey specal case.

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