HW 7e Chapte Pae o 5 Halliday/enick/Walke 7e Chapte Gaitation The manitude o the oce o one paticle on the othe i ien by F = Gm m /, whee m and m ae the mae, i thei epaation, and G i the unieal aitational contant We ole o : Gm m F ( 667 0 N m / k )( 5k)( 4k) = = = 0 N We ue ubcipt, e, and m o the Sun, ath and oon, epectiely 9 m Gmmm Fm m m em = = F Gm m m em e m e m em Pluin in the numeical alue (ay, om Appendix C) we ind 99 0 8 0 598 0 50 0 0 8 4 The aitational oce between the two pat i ( ) = 6 Gm m G F = = m m ( ) which we dieentiate with epect to m and et equal to zeo: which lead to the eult m/= / df = 0 = G ( m ) = m dm 4 Uin F = Gm/, we ind that the topmot ma pull upwad on the one at the oiin with 9 0 8 N, and the ihtmot ma pull ihtwad on the one at the oiin with 0 0 8 N Thu, the (x, y) component o the net oce, which can be coneted to pola component (hee we ue manitude-anle notation), ae F net = 04 0,85 0 0 606 (a) The manitude o the oce i 0 8 N 8 8 8 ( ) ( )
HW 7e Chapte Pae o 5 (b) The diection o the oce elatie to the +x axi i 606 5 At the point whee the oce balance G em / = G m /, whee e i the ma o ath, i the ma o the Sun, m i the ma o the pace pobe, i the ditance om the cente o ath to the pobe, and i the ditance om the cente o the Sun to the pobe We ubtitute = d, whee d i the ditance om the cente o ath to the cente o the Sun, to ind = d e ( ) Takin the poitie quae oot o both ide, we ole o A little aleba yield ( ) 9 4 d 50 598 0 k e 8 = = = 60 0 4 + e 99 0 k + 598 0 k Value o e,, and d can be ound in Appendix C 7 We equie the manitude o oce (ien by q -) exeted by paticle C on A be equal to that exeted by B on A Thu, Gm A m C = Gm A m B d We ubtitute in m B = m A and m B = m A, and (ate cancelin m A ) ole o We ind = 5d Thu, paticle C i placed on the x axi, to let o paticle A (o it i at a neatie alue o x), at x = 500d 4 We ollow the method hown in Sample Poblem - Thu, which implie that the chane in weiht i G G a = da = d ( ) W W m da top bottom But ince W bottom = Gm / (whee i ath mean adiu), we hae Gm d 6 mda = d = W bottom = ( 600 N) = 00 N 6 67
HW 7e Chapte Pae o 5 o the weiht chane (the minu in indicatin that it i a deceae in W) We ae not includin any eect due to the ath otation (a teated in q -) 5 The acceleation due to aity i ien by a = G/, whee i the ma o ath and i the ditance om ath cente We ubtitute = + h, whee i the adiu o ath and h i the altitude, to obtain a = G /( + h) We ole o h and obtain h= G / a Accodin to Appendix C, = 67 0 6 m and = 598 0 4 k, o 4 ( 667 / k)( 598 0 k) ( 49m / ) h= = 6 6 67 6 5 (a) The denity o a uniom phee i ien by ρ = /4π, whee i it ma and i it adiu The atio o the denity o a to the denity o ath i ρ ρ 065 0 km = = 0 = 074 45 0 km 4 (b) The alue o a at the uace o a planet i ien by a = G/, o the alue o a i a 4 065 0 km = a = 0 ( 98 m/ ) = 8 m/ 45 0 km (c) I i the ecape peed, then, o a paticle o ma m m G m = G = Fo a ( )( ) 4 (667 / k) 0 598 0 k = = 50 / 6 45 0 (a) Fom q -8, we ee that o = we hae G in thi poblem Uin eney coneation, m o Gm/ = Gm/ which yield = 4 / So the multiple o i 4/ o
HW 7e Chapte Pae 4 o 5 (b) Uin the equation in the textbook immediately pecedin q -8, we ee that in thi poblem we hae K i = Gm/, and the aboe manipulation (uin eney coneation) in thi cae lead to = So the multiple o i 00 (c) Aain eein to the equation in the textbook immediately pecedin q -8, we ee that the mechanical eney = 0 o the ecape condition (a) We ue the pinciple o coneation o eney Initially the paticle i at the uace o the ateoid and ha potential eney U i = Gm/, whee i the ma o the ateoid, i it adiu, and m i the ma o the paticle bein ied upwad The initial kinetic eney i m The paticle jut ecape i it kinetic eney i zeo when it i ininitely a om the ateoid The inal potential and kinetic eneie ae both zeo Coneation o eney yield Gm/ + ½m = 0 We eplace G/ with a, whee a i the acceleation due to aity at the uace Then, the eney equation become a + ½ = 0 We ole o : = = = a (0 m/ )(500 ) 7 / (b) Initially the paticle i at the uace; the potential eney i U i = Gm/ and the kinetic eney i K i = ½m Suppoe the paticle i a ditance h aboe the uace when it momentaily come to et The inal potential eney i U = Gm/( + h) and the inal kinetic eney i K = 0 Coneation o eney yield Gm Gm + h + m = We eplace G with a and cancel m in the eney equation to obtain a ( + h) a + = The olution o h i a (0 m/ ) (500 ) h= = a (0 m/ )(500 ) (000 m/) = 5 5 (500 ) (c) Initially the paticle i a ditance h aboe the uace and i at et It potential eney i U i = Gm/( + h) and it initial kinetic eney i K i = 0 Jut beoe it hit the ateoid it potential eney i U = Gm/ Wite m o the inal kinetic eney Coneation o eney yield Gm Gm = + + h m
HW 7e Chapte Pae 5 o 5 We ubtitute a o G and cancel m, obtainin a = a + + h The olution o i a (0 m/ )(500 ) = a = (0 m/ ) (500 ) + h (500 ) + (000 ) 4 / = 8 (a) We note that heiht = ath whee ath = 67 0 6 m With = 598 0 4 k, 0 = 657 0 6 m and = 77 0 6 m, we hae + = + Gm Gm (70 0 ) =, Ki Ui K U m K 0 which yield K = 8 0 7 J (b) Aain, we ue eney coneation Gm Ki + Ui = K + U m (70 0 ) = 0 0 Gm Theeoe, we ind = 740 0 6 m Thi coepond to a ditance o 049 km 0 0 km aboe the ath uace 84 ney coneation o thi ituation may be expeed a ollow: K + U = K + U Gm Gm m = m whee = 70 0 4 k, = = 6 0 6 m and = (which mean that U = 0) We ae told to aume the meteo tat at et, o = 0 Thu, K + U = 0 and the aboe equation i ewitten a Gm G m 4 = = 4