6. Gravitation. 6.1 Newton's law of Gravitation

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1 Gvittion / Newton's lw of Gvittion 6. Gvittion Newton's lw of gvittion sttes tht evey body in this univese ttcts evey othe body with foce, which is diectly popotionl to the poduct of thei msses nd invesely popotionl to the sque of the distnce between thei centes. The diection of the foce is long the line joining the pticles. Thus the mgnitude of the gvittionl foce F tht two pticles of msses m1 nd m m1 m septed by distnce exet on ech othe is given by F. m1 m o F G Also cle tht F 1 1. Which is Newton's thid lw of motion. Hee G is constnt of popotionlity which is clled 'Univesl gvittionl constnt'. (i) The vlue of G is N m kg in S.. nd dyne- cm -g in C.G.S. system. (ii) Dimensionl fomul [M 1 L 3 T ]. (iii) The vlue of G does not depend upon the ntue nd size of the bodies. (iv) t lso does not depend upon the ntue of the medium between the two bodies. 6. Popeties of Gvittionl Foce F (1) t is lwys ttctive in ntue. () t is independent of the medium between the pticles. (3) t is found tue fo inteplnety to inte tomic distnces. (4) t is centl foce i.e. cts long the line joining the centes of two intecting bodies. (5) The pinciple of supeposition is vlid. (6) t is the wekest foce in ntue. (7) t is consevtive foce. The lw of gvittion is stted fo two point msses 6.3 Acceletion Due to Gvity. The foce of ttction exeted by the eth on body is clled gvittionl pull o gvity. The cceletion poduced in the motion of body unde the effect of gvity is clled cceletion due to gvity, it is denoted by g. f M mss of the eth nd dius of the eth nd g is the cceletion due to gvity, then 4 g πρ G 3 (i) ts vlue depends upon the mss dius nd density of plnet nd it is independent of mss, shpe nd density of the body plced on the sufce of the plnet.

2 / Physics (ii) Acceletion due to gvity is vecto quntity nd its diection is lwys towds the cente of the plnet. (iii) Dimension [g] [LT ] (iv) t s vege vlue is tken to be 9.8 m/s o 981 cm/sec o 3 feet/sec, on the sufce of the eth t men se level. 6.4 Vition in g Due to Shpe of Eth. Eth is ellipticl in shpe. The equtoil dius is bout 1 km longe thn pol dius. At equto ge At poles gp gpole > g equto e p Theefoe the weight of body inceses s it is tken fom equto to the pole. 6.5 Vition in g with Height. Acceletion due to gvity t height h fom the sufce of the eth g' ( + h) Also g' g + h h (i) f h << g g 1 (ii) f g [As + h] h << Pecentge decese 6.6 Vition in g With Depth g h 100% 100% g Acceletion due to gvity t depth d fom the sufce of the eth 4 g πρg( d) 3 d lso g g 1 (i) The vlue of g deceses on going below the sufce of the eth. (ii) The cceletion due to gvity t the cente of eth becomes zeo. g d (iii) Pecentge decese 100% 100% g

3 Gvittion / 3 (iv) The te of decese of gvity outside the eth (if h << ) is double to tht of inside the eth. 6.7 Vition in g Due to ottion of Eth f the body of mss m lying t point P, whose ltitude is λ, then due to ottion of eth its ppent cceletion cn be given by g g ω cos λ. The ltitude t point on the sufce of the eth is defined s the ngle, which the line joining tht point to the cente of eth mkes with equtoil plne. t is denoted by λ. Fo the poles λ 90 nd fo equto λ 0 (i) gpole g (ii) gequto g ω (iii) f ω is the ngul velocity of ottion of eth fo which body t the equto will become weightless (g 0) ω g π o time peiod of ottion of eth T π ω f eth stts otting 17 times fste then ll objects on equto will become weightless. 6.8 netil nd Gvittionl Msses. (1) netil mss: t is the mss of the mteil body, which mesues its ineti. F mi Hence inetil mss of body my be mesued s the tio of the mgnitude of the extenl foce pplied on it to the mgnitude of cceletion poduced in its motion. (i) Gvity hs no effect on inetil mss of the body. (ii) t is popotionl to the quntity of mtte contined in the body. (iii) When body moves with velocity v, its inetil mss is given by m0 m, whee m0 est mss of body, c velocity of light in vcuum, v 1 c () Gvittionl Mss: Gvittionl mss of body my be mesued s the tio of the mgnitude of the gvittionl foce pplied on it to the mgnitude of cceletion due to gvity. Sping blnce mesue gvittionl mss nd inetil blnce mesue inetil mss. 6.9 Gvittionl Field. The spce suounding mteil body in which gvittionl foce of ttction cn be expeienced is clled its gvittionl field. g

4 4 / Physics Gvittionl field intensity: The intensity of the gvittionl field of mteil body t ny point in its field is defined s the foce expeienced by unit mss (test mss) plced t tht point. f test mss m t point in gvittionl field expeiences foce F F then. m (i) t is vecto quntity nd is lwys diected towds the cente of gvity of body whose gvittionl field is consideed. (ii) Units: Newton/kg o m/s (iii) Dimension: [M 0 LT ] (iv) (v) net Gvittionl Field ntensity fo Diffeent Bodies (1) ntensity due to unifom solid sphee Outside the sufce > On the sufce nside the sufce < 3 () ntensity due to spheicl shell / O Outside the sufce > On the sufce nside the sufce < 0 O (3) ntensity due to unifom cicul ing At point on its xis ( + ) 3 / At the cente of the ing 0 P

5 Gvittion / 5 (4) ntensity due to unifom disc At point on its xis At the cente of the disc P θ o (1 cos θ ) 6.11 Gvittionl Potentil. At point in gvittionl field potentil V is defined s negtive of wok done pe unit mss in shifting test mss fom some efeence point (usully t infinity) to the given point i.e., W F. d V m. d m dv d Negtive sign indictes tht the diection of intensity is in the diection whee the potentil deceses. (i) t is scl quntity. (ii) Unit: Joule/kg o m /sec (iii) Dimension: [M 0 L T ] (iv) f the field is poduced by point mss then Gvittionl potentil V (v) Gvittionl potentil diffeence: t is defined s the wok done to move unit mss fom one point to the othe in the gvittionl field. The gvittionl potentil diffeence in binging unit test mss m fom point A to point B unde the gvittionl influence of souce mss M is WA B 1 1 V VB VA m B A (vi) Potentil due to lge numbes of pticle is given by scl ddition of ll the potentils. V V1 + V + V Gvittionl Potentil fo Diffeent Bodies (1) Potentil due to unifom ing At point on its xis V + At the cente V P

6 6 / Physics () Potentil due to spheicl shell Outside the sufce > V On the sufce V (3) Potentil due to unifom solid sphee Outside the sufce V > Vsufce On the sufce nside the sufce < V nside the sufce < V 3 t the cente ( 0) 3 Vcente (mx.) Vcente 3 V sufce / V O 3/ V O 6.13 Gvittionl Potentil Enegy The gvittionl potentil enegy of body t point is defined s the mount of wok done in binging the body fom infinity to tht point ginst the gvittionl foce. m W This wok done is stoed inside the body s its gvittionl potentil enegy m U (i) Potentil enegy is scl quntity. (ii) Unit: Joule (iii) Dimension: [ML T ] (iv) Gvittionl potentil enegy is lwys negtive in the gvittionl field becuse the foce is lwys ttctive in ntue. (v) f then it becomes zeo (mximum) (vi) n cse of discete distibution of msses Gm1m Gmm 3 Gvittionl potentil enegy U ui

7 Gvittion / 7 (vii) f the body of mss m is moved fom point t distnce 1 to (1 > ) then chnge in 1 1 potentil enegy U m 1 (viii) eltion between gvittionl potentil enegy nd potentil U mv (ix) Gvittionl potentil enegy of body t height h fom the eth sufce is given by U h m g m mg + h + h h Wok Done Aginst Gvity. f the body of mess m is moved fom the sufce of eth to point t distnce h bove the sufce of eth, then chnge in potentil enegy o wok done ginst gvity will be mh mgh W h h (i) When the distnce h is not negligible nd is compble to dius of the eth, then we will use bove fomul. 1 (ii) f h then W mg (iv) f h is vey smll s comped to dius of the eth then tem h/ cn be neglected mgh W mgh 1 +h / 6.15 Escpe Velocity. The minimum velocity with which body must be pojected up so s to enble it to just ovecome the gvittionl pull, is known s escpe velocity. f ve is the equied escpe velocity, then ve ve g (i) Escpe velocity is independent of the mss nd diection of pojection of the body. (ii) Fo the eth ve 11. km/sec (iii) A plnet will hve tmosphee if the velocity of molecule in its tmosphee is lesse thn escpe velocity. This is why eth hs tmosphee while moon hs no tmosphee 6.16 Keple s Lws of Plnety Motion (1) The lw of Obits: Evey plnet moves ound the sun in n ellipticl obit with sun t one of the foci. () The lw of Ae: The line joining the sun to the plnet sweeps out equl es in equl intevl of time. i.e. el velocity is constnt. Accoding to this lw plnet will move

8 8 / Physics slowly when it is fthest fom sun nd moe pidly when it is neest to sun. t is simil to lw of consevtion of ngul momentum. da L Ael velocity dt m (3) The lw of peiods: The sque of peiod of evolution (T) of ny plnet ound sun is diectly popotionl to the cube of the semi-mjo xis of the obit. E T o T whee semi-mjo xis 1 Shotest distnce of plnet fom sun (peigee). Lgest distnce of plnet fom sun (pogee). Keple's lws e vlid fo stellites lso Velocity of Plnet in Tems of Eccenticity Speeds of plnet t pogee nd peigee e 3 1 e v 1 + e, 1 + e vp 1 e Angul momentum of plnet o stellite is lwys constnt iespective of shpe of obit Obitl Velocity of Stellite. v [ + h] (i) Obitl velocity is independent of the mss of the obiting body. (ii) Obitl velocity depends on the mss of plnet nd dius of obit. (iii) Obitl velocity of the stellite when it evolves vey close to the sufce of the plnet 6.19 Time Peiod of Stellite T π v g 8 km/sec ( + h) 3 g h π 1 + g 3 / [As + h] (i) Time peiod is independent of the mss of obiting body (ii) T 3 (Keple s thid lw) (iii) Time peiod of neby stellite, T θ Fo eth T 84.6 minute 1.4 h. g Peigee Sun F A C 1 D Apogee B

9 Gvittion / Height of Stellite 1 / 3 T g h 4π 6.1 Geosttiony Stellite. The stellite which ppes sttiony eltive to eth is clled geosttiony o geosynchonous stellite, communiction stellite. A geosttiony stellite lwys stys ove the sme plce bove the eth. The obit of geosttiony stellite is known s the pking obit. (i) t should evolve in n obit concentic nd copln with the equtoil plne. (ii) t sense of ottion should be sme s tht of eth. (iii) ts peiod of evolution ound the eth should be sme s tht of eth. (iv) Height of geosttiony stellite fom the sufce of eth h km (v) Obitl velocity v 3.08 km/sec (vi) Angul momentum of stellite depend on both the mss of obiting nd plnet s well s the dius of obit. 6. Enegy of Stellite m L (1) Potentil enegy: U mv m () Kinetic enegy: K 1 m L m mv m m m L (3) Totl enegy: E U + K + m (4) Enegy gph fo stellite (5) Binding Enegy: The enegy equied to emove the stellite fom its obit to infinity is clled Binding Enegy of m the system, i.e., Binding Enegy (B.E.) E 6.3 Weightlessness The stte of weightlessness (zeo weight) cn be obseved in the following situtions. (1) When objects fll feely unde gvity () When stellite evolves in its obit ound the eth (3) When bodies e t null points in oute spce. The zeo gvity egion is clled null point. Enegy + O E K U

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