LECTURE 14. m 1 m 2 b) Based on the second law of Newton Figure 1 similarly F21 m2 c) Based on the third law of Newton F 12
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1 CTU 4 ] NWTON W O GVITY -The gavity law i foulated fo two point paticle with ae and at a ditance between the. Hee ae the fou tep that bing to univeal law of gavitation dicoveed by NWTON. a Baed on expeiental infoation one potulate that gavitation poduce an attactive foce. b Baed on the econd law of Newton igue a _ o _ iilaly c Baed on the thid law of Newton It coe out that ( d Baed on expeiental eult Newton gueed that gavitation foce deceae with ditance a /. o, it coe out that oe peciely G ( eaueent how that G 6.67*0 - N /kg i the univeal contant of gavitation - The gavitational foce i a vecto diected veu the ouce that exet thi foce. o, it vecto fo i G (3 G i the unit vecto with tail at a the ouce of. i the unit vecto with tail at a the ouce of - To apy the gavitational law fo two bodie cloe to each othe one ut ue integation technique and the difficulty of calculi depend on the fo of the two bodie. But, if the bodied ae fa enough to each othe, one ay odel the by point paticle and apy the law in it oiginal fo. In paticula, with oe appoxiation, we ae able to odel the inteaction of eath with an object on it uface a if the whole a of the eath i concentated at it cente and the object ae at ditance eath. -any expeient have hown that; when eveal paticle inteact gavitationally between the the pincie of linea upepoition apie. o, inide a yte of paticle,, 3, n, the foce exeted on a i ] TH GVITTION ND TH INTI n i i (4 -When expeing the econd law of Newton we ue the inetial a NT in a (5 When foulating the gavitation law, Newton wa not ue that the a of paticle in thi law i the ae a thei inetial a. et veify thi iue. We tat by uppoing that the a in the gavitational law ay be diffeent fo in. o, we note it g. et conide now a body in fee fall cloe to eath g ath uface. The eath will exet on it the gavitational foce with agnitude g G (3 ath Hee we aue that the body i cloe to the uface o that it ditance fo eath cente i. ath Thei dienion ae <<< than the ditance between the.
2 thi i the net foce exeted on the body we apy the econd law of Newton g ath ath ath NT ina g G o, we get a G g in g ( g becaue G g ath ath ath g Then, a g (6 in big nube of eaueent how that the acceleation of fee bodie i equal to g 9.8/. Thi ean that g / in, and in g. o, the expeient confi that the gavitational a i the ae a the inetial a. - et apy the gavitation law fo the foce exeted by eath ove a a kg cloe to eath. ath ath G g (7 o, the g-vecto i equal to the gavitation foce exeted ( ath + h ath on a a kg. By eauing the foce exeted on the a kg in diffeent location on the eath one get a whole yte of g-vecto (fig.. The totality of thee vecto fo the gavitational field of the eath. igue (7 Note that the g-vecto agnitude deceae with the inceae of ditance h fo the eath but it i alway diected veu the cente of the eath. The gavitational field of the eath ha a pheical yety. In fact, it i not exactly pheical, becaue the odel of eath a a unifo denity phee i not vey pecie. Now, the object weight i equal to gavitational foce exeted by thi field W g (7 o, the weight of the ae object i a vecto that i diffeent in diffeent point of gavitational field of the eath. 3] K W ON NTY OTION - it law: The anet ove on elliptic obit aound the un that i located at one of it focue. igue 3 un b a eihelion phelion The ino axi i long b and the ajo axi i long a. The cloet ditance to un i called peihelion and the bigget ditance to un i called aphelion (fig.3. econd law: The line un- anet weep out equal aea fo equal inteval of tie. igue 4 Thid law: The quae of the peiod of anet otion i popotional to the cube of aveage ditance fo the un. Calculation how that the aveage of ditance un-anet i equal to half of ajo axi a. Then, atellite otion (ee ectue_8.5 tell that T κ a 3 (8 4π whee κ un G un
3 Note : Kee law ae valid fo elliptical path of any anet aound a cental body; fo exae the 4π oon oving aound the eath but in thi cae κ ath. G ath TH NGY O NT igue 5 - the a of othe anet i uch alle than the a of un we neglect thei action on the otion of the tudied anet. We conide that the yte un-anet i a conevative yte, i.e. the foce oiginated fo outide it ae zeo. In thee cicutance: a The toque of exteio foce i zeo and we can apy the pincie fo conevation of angula oentu. b The wok done by Net exteio foce i zeo and we can apy the pincie of enegy conevation fo the yte un-anet - The pincie of angula oentu conevation tell that (9 o x p B x p (0 0 0 The equality of agnitude bing to condition p in 90 p in 90 anet anet o, we get ( B -The pincie of enegy conevation tell that that ( the echanical enegy i K + U whee K (3 and U We get fte cancelling un By uing the equation ( and the fact that un un and finally un un G (4 un G un ( (5 (6 + a (7, afte oe calculation we find that G (8 and a G (9 a inally, by ubtituting one of thi expeion at the expeion of total enegy (at peihelion o aphelion un we get G (0 a 4] TH BOUND ND UNBOUND TJCTOI -et ee what happen with an object thown up with high initial velocity. The only foce exeted on it i ath the gavitational foce of the eath. G ( extenal net foce i zeo, the yte eath object coneve it enegy. o uch a yte U + K i a contant all tie. Note: the eath i the efeence fae, geneally one talk fo object enegy. 3
4 - By uing the wok done by the gavitation foce (exp. one ay find ath that U ( G ( U( > The gaph of thi function i hown in figue 6. Note that the zeo value of potential enegy of gavitational inteaction i et fo big -value. The zeo-value of potential enegy ean that thee i no inteaction U( between the yte pat, o oe peciely the pat ae not bound to each othe. When the object i on the eath uface, the ditance fo U( C of the eath (at oigin of gaph i equal to the eath adiu and K it potential enegy i U(. When we thow it vetically with initial igue 6 velocity, the echanical enegy of the object (i.e. of object eath yte i ( U( +K( (3 we know, the velocity of the object will deceae with height and will becoe zeo at it axiu height h, i.e. at ditance ( + h. When getting at thi ditance, all it enegy i potential enegy. o, ( U( (4 We ay that the kinetic enegy ake it clib up on a potential well (ee fig.6. The pincie of enegy conevation tell that ( ( (5 By uing thi equation we can find the axiu height a follow; ( +h ( ( +h U( +h and uing (3 we get U( +h ( U( +K( (6 o, ath ath G + and + h G ath ( + h G ath G ath G ath Then, ( _ a _ h << _ ( + h / h + h / G ath g we find that h (7 which i a known eult fo kineatic. g e - When the initial velocity of the body i uch that U( + K( < 0 the object will get to a given ditance fo eath but will eain all tie bound within the yte eath-object; If U( + K( 0 the object will get o fa that the inteaction with eath becoe zeo; it becoe an unbounded object to gavitational field of the eath. The liiting initial velocity neceay to unbind fo gavitational field of eath i known a ecape velocity ec. Thi velocity can be found by condition ec G U ( + K( ec 0 G + 0 ec (8 - uppoe that one ut end a ocket out of eath gavitational field; i.e. ake it unbound object to eath. The fit equieent i to give to ocket an initial velocity ec. ie calculation baed on expeion (8 how that ec 90 /. Thi value of initial velocity can be funihed only by eactive engine. Though a integal calculu. 4
5 What fo ha the tajectoy of an unbound ocket? The atheatical calculation how that if a > ec the ocket tajectoy will not be cloed and it i a hypebola. b ec the ocket tajectoy will not be cloed and it i a paabola. If the initial velocity of ocket < ec the object eain a bound object to the eath gavitation field and it tajectoy will be an ellipe (cloed obit. Note that if the initial velocity i too all << ec, thi obit ay co the eath, i.e. the ocket will tike on eath uface (figue 7. << ec ath < ec ellip -Conide an atificial atellite oving unifoly on a cicula obit at ditance fo C of the eath (fig.8. It otion ha a centipetal acceleation ac and the net foce exeted on it i the eath gavitation. o, G and ac We get G and G (9 What i the equied launch peed fo thi atellite? We apy the pincie of enegy conevation. ( ( (30 > ec ec paabolla G hipebola igue 7 o, G G ( (3 ath G G ( + ( h + h G ( h G ( + (3 o low obit atellite (ay height00-00k fo eath (6378k h/ and fo (3 we can find that igue 8 Cicula obite of atellite. G cicula low obit with adiu i o, the launch velocity fo a tationay G (33 5
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