Objectives. Chapter 6. Learning Outcome. Newton's Laws in Action. Reflection: Reflection: 6.2 Gravitational Field

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1 Chapte 6 Gataton Objectes 6. Newton's Law o nesal Gataton 6. Gatatonal Feld 6. Gatatonal Potental 6. Satellte oton n Ccula Obts 6.5 scape Velocty Leanng Outcoe (a and use the oula / (b explan the eanng o gatatonal eld (c dene gatatonal eld stength as oce o gaty pe unt ass (d use the equaton g = / o a gatatonal eld (e dene the potental at a pont n a gatatonal eld; ( dee and use the oula V = - / electon: g use the oula o potental enegy = - / h show that = g = gh s a specal case o = -/ o stuatons nea to the suace o the ath use the elatonshp g = - dv/d j explan, wth gaphcal llustatons, the aatons o gatatonal eld stength and gatatonal potental wth dstance o the suace o the ath electon: Fo eey acton thee s an equal and opposte eacton. o Whenee on object exets a oce on a second object, the second object exets an equal and opposte oce on the st object. acton = opposte eacton F = -F o a = - a F = - F o a = - a Jet populson exhaust gases n one decton and the ocket n the othe decton Gaty jup o a table and you wll acceleate to eath. In ealty BOH you and the eath ae acceleatng towads each othe You sall ass, huge acceleaton ( a ath huge ass, ey sall acceleaton (- a Souce: 5 Souce: 6 Newton's Laws n Acton Secton. Fcton on the tes podes necessay centpetal acceleaton. Passenges contnue staght ahead n ognal decton and as ca tuns the doo coes towad passenge st Law As ca tuns you push aganst doo and the doo equally pushes aganst you d Souce: Law Secton. B a = - a Secton. Gaty s a undaental oce o natue We do not know what causes t We can only descbe t Law o nesal Gataton ey patcle n the unese attacts eey othe patcle wth a oce that s dectly popotonal to the poduct o the asses and nesely popotonal to the squae o the dstance between the G quaton o: /ps/ch.ppt 7 8 Souce: Secton.5

2 G quaton o: G s the unesal gatatonal constant G = 6.67 x - N. /kg G: s a ey sall quantty thought to be ald thoughout the unese was easued by Caendsh 7 yeas ate G / F F he oces that attact patcles togethe ae equal and opposte F = - F o a = - a Souce: 9 Souce: Secton.5 6. Newton's Law o Gataton G Fo a hoogeneous sphee the gatatonal oce acts as all the ass o the sphee wee at ts cente Souce: Secton.5 Souce: Secton.5 Secton.5 wo objects wth asses o. kg and. kg ae. apat. What s the agntude o the gatatonal oce between the asses?. Neglgble. kg. kg oce Foce o Foce o 9.8 N 9.6 N ath wo objects wth asses o. kg and. kg ae. apat. What s the agntude o the gatatonal oce between the asses? G (6.67 x - N- /kg (. kg(. kg (.. x - N 6. Gatatonal Feld Gatatonal eld: a egon whee gatatonal oce acts on a body. Gatatonal eld stength, g at a pont n a gatatonal eld s the gatatonal pull pe unt ass on a body at that pont, thus g = F/. Whee -/ ; then, g = / Souce: Souce: Secton.5 Secton.5 6. Gatatonal Feld Gatatonal eld stength = Foce o Gaty on ath [oce o gaty on object o ass ] and ae the ass and adus o ath w = g = g = cancels out Souce: weght (w = g g s ndependent o ass Secton.5 5 Vaaton o g wth Alttude Fo a ass, on the suace o the ath, g = / g = / = g I > suace, g = / g = / g = g / ( g = ( / g h = + h, h = alttude 6

3 Vaaton o g wth Alttude g = ( / g g = g /( + h g /( + h I < suace, the gatatonal pull s due to the sphee o adus ; = (/ o g o = / ; g = (/ G ; thus g/g = / ; o g = + h, h = alttude 6. Gatatonal Potental negy As beoe, the gatatonal potental enegy deceases when the sepaaton deceases. We assue that the gatatonal potental enegy p s zeo o =, whee s the sepaaton dstance. he potental enegy s negate o any nte sepaaton and becoes pogessely oe negate as the patcles oe close togethe. We take the gatatonal potental enegy o the two-patcle syste to be p = - / h 7 9 g- Gaph Fo > : g /( + h and o < : g Acceleaton due to gaty, g g g / 6. G. potental negy -Poo Let a baseball, statng o est at a geat (nnte dstance o ath, all towad pont P. he potental enegy o the baseball-ath syste s ntally zeo. When the baseball eaches P, the potental enegy s the negate o the wok W done by the gatatonal oce as the baseball oes to P o ts dstant poston. 8 F dx x P hus p W Fdx F dx A baseball o ass alls towads ath o nnty, along a adal lne (an x axs passng though pont P at a dstance o th ecente o ath. dx x x dx x Acceleaton due to Gaty o a Sphecal no Object g = g = acceleaton due to gaty = ass o any sphecal uno object ath Obt - Centpetal Foce Pope angental Velocty Centpetal Foce F c = a c = / (snce a c = / he pope cobnaton wll keep the oon o an atcal satellte n stable obt Souce: Secton.5 Souce: Secton.5

4 ath as the sae ate ycho Bahe/yge Ottesen Bahe de nudstup (56-6 Johannes eple (57-6 Souce: 5 6 Secton.5 he law o obts: All s oe n ellptcal obts, wth the Sun at one ocus he law o aeas: A lne that connects the to the Sun sweeps out equal aeas equal te nteals he law o peods: he squae o the peod o any s popotonal to the cube o the seajo axs o ts obt llptcal obts o s ae descbed by a seajo axs a and an eccentcty e Fo ost s, the eccentctes ae ey sall (ath's e s he law o obts he obt n the gue s descbed by gen ts seajo axs a and ts eccentcty e, the latte dened so that ea s the dstance o the cente o he su o the pehelon (neaest the Sun dstance p and the aphelon (athest o the Sun dstance a s a. he su o the dstance o any poston n the obt to two oc s a. he equaton o any poston (x, y n the obt s he law o obts An eccentcty o zeo coesponds to a ccle, n whch the two oc ege to a sngle cental pont. he eccentctes o the ay obts ae not lage, so the obts look ccula. he eccentcty o ath's obt s only Second law Fo a sta- syste, the total angula oentu s constant (no extenal toques L p da ( ( ( ( ( ( d da dt Fo the eleentay aea swept by ecto d dt const Fo a ccula F a ( ( Fo the denton o a peod da dt L Fo ellptc obts a

5 6. Satellte oton n ccula obts 6. Satellte oton n ccula obts Fo a ccula F a ( netc enegy o a satellte otal echancal enegy o a satellte total 6. Satellte oton n ccula obts Fo an ellptc obt t can be shown a Obts wth deent e but the sae a hae the sae total echancal enegy he Law o Aeas he wll oe ost slowly when t s athest o the Sun and ost apdly when t s neaest to the Sun. he law o aeas s a dect consequence o the dea that all o the oces ae dected exactly towad the sun. 5 6 he Law o Peods Consde a ccula obt wth adus. See gue. Applyng Newton's second law, a, to the obtng yeld ( ( I s eplaced wth /, whee s the peod o the oton, yeld he law holds also o ellptcal obts, poded we xaple A satellte n ccula obt at an alttude h o k aboe ath's suace has a peod o 89 n. What ass o ath ollows o these data? Sol: Fo eple's law o peods we hae he adus o the satellte obt s = + h = 6.7 X 6 + X ( kg G (6.67 (89 6 eplace wth a, the seajo axs o the ellpse. 7 8 xaple = 6.6 X 6, whee = adus o ath. 6 Coet Halley obts about the Sun wth a peod o 76 yeas and, n 986, had a dstance o closest appoach to the Sun, ts pehelon dstance p, o 8.9. (a What s the coet's athest dstance o the Sun, ts aphelon dstance a? (b What s the eccentcty o the obt o coet Halley? Sol:(a Fo eple's law o peod we hae a.7 (6.67 (.99 (. / 9 a = a p = 5. X Snce ea = a p We hae e = (a p / a = p / a = (8.9 X / (.7 X =.97 9

6 6. Satellte oton n a ccula obt he echancal enegy + P o the satellte eans constant. We st assue that the obt o the satellte s ccula. he potental enegy s P whee s the adus o the obt. By Newton's second law, a = / = Whee = s the centpetal acceleaton o the satellte. 6. Satellte oton n a ccula obt Fo a satellte obtng ath, the gatatonal pull o ath upon the satellte, F constant o 6. Satellte oton n a ccula obt he knetc enegy o a satellte s he total echancal enegy s P Fo a satellte n an ellptcal obt o seajo axs a, we hae P a 6.5 scape elocty a c g g Accountng o the shape o ath, pojectle oton has to be oded: 6.5 scape Speed hee s a cetan nu ntal speed that wll cause a pojectle to oe upwad oee, theoetcally cong to est only at nnty. hs ntal speed s called the escape speed. Consde a pojectle o ass, leang the suace o a wth escape speed. When the pojectle eaches nnty, t stops and thus has no knetc enegy. It also has no potental enegy because ths s ou zeopotental enegy conguaton. 6.5 scape elocty I o soe astonocal object Nothng (een lght can escape o the suace o ths object a black hole G object escape 8 / s object c scape elocty scape speed: speed equed o a patcle to escape o the nto nnty (and stop thee escape G G 6.5 scape elocty Fo the pncple o conseaton o enegy, we hae k p Whee s the ass o the and s ts adus. hus 6 7 8

7 xaple An asteod headed dectly towad eath, has a speed o /sec elate to the when t s at a dstance o ath ad o ath's cente. Ignong the eects o the teestal atosphee on the asteod, nd the asteod's speed when t eaches ath's suace. Answe Because the ass o an asteod s uch less than that o ath, we can assgn the gatatonal potental enegy o the asteod- ath syste to the asteod alone, and we can neglect any change n the speed o ath elate to the asteod dung the asteod's all. 9 5 hus, k p Let epesent the ass o the asteod, the ass o the eath (=5.98X kg, and the adus o ath (= 6.7X 6, hus k p ( s (6.67 s ( Suay: Gataton Gatatonal Feld Law o Gataton Gatatonal Potental Satellte scape Velocty V = -/ = (/ / G = / G / = - / = (/ / g = -dv/d 5