Hono Claical Phyic I PHY141 Lectue 9 Newton Law of Gavity Pleae et you Clicke Channel to 1 9/15/014 Lectue 9 1
Newton Law of Gavity Gavitational attaction i the foce that act between object that have a gavitational chage, i.e. gavitational ma (one of the myteiou fact of natue i that inetial ma and gavitation chage ae popotional (o identical in SI unit) to anothe! Thi, among othe fact, led intein to the Theoy of Relativity) Foce of Gavity between point-mae m 1 and m at ditance 1 fom one anothe: mm 1 FG,1 on G extemely weak fo 1 nomal mae 1 Magnitude: Gm 1 m / 1, INIFINT RANG with G = 6.673 10 11 Nm /kg Diection: attactive; 1 i pointing outwad Fo SPHRICALLY SYMMTRIC object: a if all ma wa at cente mm F G j e.g. between ma m and ath at ealevel: 9/15/014 mg G R 11 4 GM 6.6710 5.9710 g R 6 6.3810 Lectue 9 j 9.80 m/
Fit example two pheical mae, each 5.0 kg and with 4.95 cm adiu, ae 1.0 mm apat. Calculate the magnitude of the foce of gavity. Cente-to-cente ditance: 10.0 cm = 0.100 m. Foce: F = Gm /d = 6.7 10 11 Nm /kg 5.0 kg / 0.010 m =1.68 10 7 N cf. 50 N fo the weight i.e. extemely weak: no appeciable angle of the ting away fom vetical! 9/15/014 Lectue 9 3
The magnitude of the foce of gavity on a ma m, poitioned a hown between two mae M and M i: M d A. GmM / d to the ight B. GmM / d to the left C. zeo D. GmM / (d ) to the ight. GmM / (d ) to the left d M 9% 38% 5% 4% 5% A. B. C. D.. 9/15/014 Lectue 9 4
The Invee Squae Law The empiical fact that the tength of gavity vaie like the invee quae of the ditance between the ma ouce i VRY IMPORTANT: Allow deciption with gavitational field line adiating fom the ouce of the field like the light ay adiating out of the un and going to infinity Line tat and end ONLY on ouce, and nowhee ele The Foce-pe-unit-tet-ma i diected adially inwad (attactive) The magnitude of the field dop like 1/ ; jut like the denity of the line dawn (in fact, the 1/ dependence i what make thi pictue poible): Taking an imaginay pheical uface with adiu (dotted phee), the numbe of line though the pheical uface emain contant, and the numbe of line pe-unit-uface-aea (i.e. DNSITY of line), deceae like 1/Aea = 1/(4π ), i.e. like 1/! 9/15/014 M Lectue 9 5
Gavity of a Sphee Gavity act between point-like object a 1/ditance-quaed. Thi mean that all extended SPHRICALLY SYMMTRIC object behave a point-like! Take a thin unifom pheical hell (M,R); conide the gavitational foce fom a thin band of the hell with ma dm, on a ma m at ditance : mdm mm m M df G co G daco G Rin A 4 R R d co mm Rco G ind R Note: i R Rco co in d d co d R R the vaiable; and R ae mm ind R contant FG df G co 9/15/014 mm d R G R R mm R G d 1 4R R R mm R G 4R R mm mm G RR G 4R Lectue 9 6
Gavity of a Spheically Symmetic object: A pheically ymmetic ma ditibution equal a eie of thin unifom hell, and give exactly the ame potential outide (>R) a a point ma. Similaly, we can look at the Gavitational Foce INSID a pheical hell (<R); eveything emain the ame except fo the integation limit, now: R to R+: FG df mm R G 4R R R mm R G 4R R R i.e. thee i NO foce of gavity inide the hell! mm G 0! 4R The latte mean that F G = 0 INSID any hollow pheically ymmetic ma ditibution! 9/15/014 Lectue 9 7
The gavitational foce at the cente of the ath i even lage than on the uface: A. Tue B. Fale 94% 6% A. B. 9/15/014 Lectue 9 8
xample: Fat and Cheap Tavel to Autalia Dill a tunnel thu the cente of ath, fom NYC to Peth, Autalia: the tunnel completed, we don ou heat-eitant abeto uit, and jump in What happen? the F G = GM m/, whee i the ditance to ath cente, and M i the ma of ock contained inide the phee of adiu ; i.e. the cloe I get to the cente of the ath, the malle F G! In numbe: mm m M 4 3 FG G 4 G mg V 3 M k 3 i.e. the foce i like a ping foce! on the upwad pat of the jouney to Peth, the foce pull back, and at the time I each the uface in Autalia I come to a momentay top, and gab the buh (of coue, I aume no dag o fiction!) The time fo uch a jouney: d Fk m dt V co k R t R co t T1/ 1h m T 9/15/014 Lectue 9 9
Satellite Satellite motion: motion of maive bodie influenced by gavity If only ath gavity, and the motion i cicula (adiu ): M m FG G FNet ma ad m unifom cicula motion v tan v GM fo a cicula obit v = π/t and: v GM T T 4 GM 3 Keple Peiod Law (3 d Law) 9/15/014 Keple Aea Law ( nd Law): dv F F in m in F, F, 0 ( F act along cente!) dt mv in i contant! in d da mv in m m i contan t dt dt M θ F m da Lectue 9 10