GAVITATION 6.1 Newton s law of Gavitation Newton s law of gavitation states that evey body in this univese attacts evey othe body with a foce, which is diectly popotional to the poduct of thei masses and invesely popotional to the squae of the distance between thei centes. The diection of the foce is along the line joining the paticles. Thus the magnitude of the gavitational foce F that two paticles of masses m1 m1m and m sepaated by a distance exet on each othe is given by Fa. o m m FG 1. Also clea that F1 -F 1. Which is Newton s thid law of motion. Hee G is constant of popotionality which is called Univesal gavitational constant. (i) The value of G is 6.67 10 11 N m kg in S.I. and 6.67 10 8 dyne cm g in C.G.S. system. (ii) Dimensional fomula [M 1 L 3 T ]. (iii) The value of G does not depend upon the natue and size of the bodies. (iv) It does not depend upon the natue of the medium between the two bodies.
6. Acceleation Due to Gavity. The foce of attaction exeted by the eath on a body is called gavitational pull o gavity. The acceleation poduced in the motion of a body unde the effect of gavity is called acceleation due to gavity, it is denoted by g. If Mmass of the eath and adius of the eath and g is the acceleation due to gavity, then g GM 4 3 p G (i) Its value depends upon the mass adius and density of planet and it is independent of mass, shape and density of the body placed on the suface of the planet. (ii) Acceleation due to gavity is a vecto quantity and its diection is always towads the cente of the planet. (iii) Dimension [g] [LT ] (iv) It s aveage value is taken to be 9.8 m/s o 981 cm/sec, on the suface of the eath at mean sea level. 6.3 Vaiation in g with Height. Acceleation due to gavity at height h fom the suface of the eath g' GM ( + h) Also Ê ˆ g' g Á Ë + h g [As +h] È h (i) If h << g g Í1- Î
(ii) If h << Pecentage decease D g h 100% 100% g 6.4 Vaiation in g With Depth Acceleation due to gavity at depth d fom the suface of the eath g 4 ( ) 3 pg-d also g d ' g È Í1- Î (i) The value of g deceases on going below the suface of the eath. (ii) The acceleation due to gavity at the cente of eath becomes zeo. D g d (iii) Pecentage decease 100% 100% g (iv) The ate of decease of gavity outside the eath (if h << ) is double to that of inside the eath. 6.5 Gavitational Field. The space suounding a mateial body in which gavitational foce of attaction can be expeienced is called its gavitational field. Gavitational Geld intensity : The intensity of the gavitational field of a mateial body at any point in its field is defined as the foce expeienced by a unit mass (test mass) placed at that point. If a test mass m at a point in a gavitational F field expeiences a foce F then l. m 6.16 Gavitational Potential. At a point in a gavitational field potential V is defined as negative of wok done pe unit mass in shifting a test mass fom some efeence point (usually at infinity) to the given point.
Negative sign indicates that the diection of intensity is in the diection whee the potential deceases. Gavitational potential V - GM 6.7 Gavitational Potential Enegy The gavitational potential enegy of a body at a point is defined as the amount of wok done in binging the body fom infinity to that point against the gavitational foce. W GMm - This wok done is stoed inside the body as its gavitational potential enegy U GMm - If then it becomes zeo (maximum) 6.18 Escape Velocity. The minimum velocity with which a body must be pojected up so as to enable it to just ovecome the gavitational pull, is known as escape velocity. If v e is the equied escape velocity, then v e GM v e g (i) Escape velocity is independent of the mass and diection of pojection of the body. (ii) Fo the eath v e 11. km/sec (iii) A planet will have atmosphee if the velocity of molecule in its atmosphee is lesse than escape velocity. This is why eath has atmosphee while moon has no atmosphee
6.9 Keple s Laws of Planetay Motion (1) The law of Obits : Evey planet moves aound the sun in an elliptical obit with sun at one of the foci. () The law of Aea : The line joining the sun to the planet sweeps out equal aeas in equal inteval of time. i.e. aeal velocity is constant. Accoding to this law planet will move slowly when it is fathest fom sun and moe apidly when it is neaest to sun. It is simila to law of consevation of angula momentum. Aeal velocity da dt L m (3) The law of peiods : The squae of peiod of evolution (T) of any planet aound sun is diectly popotional to the cube of the semi-majo axis of the obit. T a 3 o T Ê Á Ë whee a semi-majo axis 1 + ˆ 1 Shotest distance of planet fom sun (peigee). Lagest distance of planet fom sun (apogee). Keple s laws ae valid fo satellites also. 6.10 Obital Velocity of Satellite. 3 E Peigee F Sum A C a 1 D Apogee B v GM [ + h] (i) Obital velocity is independent of the mass of the obiting body. (ii) Obital velocity depends on the mass of planet and adius of obit. (iii) Obital velocity of the satellite when it evolves vey close to the suface of the planet
GM v g 8 km/sec 6.11 Time Peiod of Satellite T P 3 ( + h) g P Ê hˆ 1 g Á + Ë 3/ [As + h] (i) Time peiod is independent of the mass of obiting body (ii) T 3 (Keple s thid law) (iii) Time peiod of neaby satellite, T q g Fo eath T 84.6 minute 1.4 h. 6.1 Height of Satellite h Ê T g ˆ Á Ë 4p 1/ 3-6.13 Geostationay Satellite. The satellite which appeas stationay elative to eath is called geostationay o geosynchonous satellite, communication satellite. A geostationay satellite always stays ove the same place above the eath. The obit of a geostationay satellite is known as the paking obit. (i) It should evolve in an obit concentic and coplana with the equatoial plane. (ii) It sense of otation should be same as that of eath. (iii) Its peiod of evolution aound the eath should be same as that of eath. (iv) Height of geostationay satellite fom the suface of eath h 6 36000 km
(v) Obital velocity v 3.08 km/sec (vi) Angula momentum of satellite depend on both the mass of obiting and planet as well as the adius of obit. 6.14 Enegy of Satellite (1) Potential enegy : U mv -GMm -L m () Kinetic enegy : K 1 m v GMm L m (3) Total enegy : E U + K -GMm GMm -GMm -L + - m (4) Enegy gaph fo a satellite (5) Binding Enegy : The enegy equied to emove the satellite its obit to infinity is called Binding Enegy of the system, i.e., GMm Binding Enegy (B.E.) E O l E K U 6.15 Weightlessness The state of weightlessness (zeo weight) can be obseved in the following situations. (1) When objects fall feely unde gavity () When a satellite evolves in its obit aound the eath (3) When bodies ae at null points in oute space. The zeo gavity egion is called null point.
UNIT VI GAVITATION VEY SHOT ANSWE TYPE QUESTIONS (1 MAK) Q1. The mass of moon is nealy 10% of the mass of the eath. What will be the gavitational foce of the eath on the moon, in compaison to the gavitational foce of the moon on the eath? Q. Why does one feel giddy while moving on a mey go ound? Q3. Name two factos which detemine whethe a planet would have atmosphee o not. Q4. The foce of gavity due to eath on a body is popotional to its mass, then why does a heavy body not fall faste than a lighte body? Q5. The foce of attaction due to a hollow spheical shell of unifom density on a point mass situated inside is zeo, so can a body be shielded fom gavitational influence? Q6. The gavitational foce between two bodies in 1 N if the distance between them is doubled, what will be the foce between them? Q7. A body of mass 5 kg is taken to the cente of the eath. What will be its (i) mass (ii) weight thee. Q8. Why is gavitational potential enegy negative? Q9. A satellite evolves close to the suface of a planet. How is its obital velocity elated with escape velocity of that planet. Q10. Does the escape velocity of a body fom the eath depend on (i) mass of the body (ii) diection of pojection Q11. Identify the position of sun in the following diagam if the linea speed of the planet is geate at C than at D. C A D C
Q1. A satellite does not equie any fuel to obit the eath. Why? Q13. A satellite of small mass buns duing its desent and not duing ascent. Why? Q14. Is it possible to place an atificial satellite in an obit so that it is always visible ove New Delhi? Q15. If the density of a planet is doubled without any change in its adius, how does g change on the planet. Q16. Why is the weight of a body at the poles moe than the weight at the equato? Explain. Q17. Why an astonaut in an obiting space caft is not in zeo gavity although he is in weight lessness? Q18. Wite one impotant use of (i) geostationay satellite (ii) pola satellite. Q19. A binay sta system consists of two stas A and B which have time peiods T A and T B, adius A and B and masses m A and m B which of the thee quantities ae same fo the stas. Justify. Q0. The time peiod of the satellite of the eath is 5 h. If the sepaation between eath and satellite is inceased to 4 times the pevious value, then what will be the new time peiod of satellite. Q1. Why does the eath impat the same acceleation to evey bodies? Q. If suddenly the gavitational foce of attaction between eath and satellite become zeo, what would happen to the satellite? SHOT ANSWE TYPE QUESTIONS ( MAKS) Q3. If the adius of the eath wee to decease by 1%, keeping its mass same, how will the acceleation due to gavity change? Q4. Which of the following symptoms is likely to afflict an astonaut in space (a) swollen feet, (b) swollen face, (c) headache, (d) oientation poblem. Q5. A satellite is moving ound the eath with velocity v 0 what should be the minimum pecentage incease in its velocity so that the satellite escapes.
Q6. The adii of two planets ae and espectively and thei densities ρ and ρ/ espectively. What is the atio of acceleation due to gavity at thei sufaces? Q7. If eath has a mass 9 times and adius 4 times than that of a planet P. Calculate the escape velocity at the planet P if its value on eath is 11. kms 1 Q8. At what height fom the suface of the eath will the value of g be educed by 36% of its value at the suface of eath. Q9. At what depth is the value of g same as at a height of 40 km fom the suface of eath. Q30. The mean obital adius of the eath aound the sun is 1.5 10 8 km. Calculate mass of the sun if G 6.67 10 11 N m /kg? Q31. Daw gaphs showing the vaiation of acceleation due to gavity with (i) height above eath is suface (ii) depth below the eath s suface. Q3. A ocket is fied fom the eath towads the sun. At what point on its path is the gavitational foce on the ocket zeo? Mass of sun 10 30 kg, mass of the eath 6 10 4 kg. Neglect the effect of othe planets etc. Obital adius 1.5 10 11 m. Q33. A Satun yea is 9.5 times the eath yea. How fa is the Satun fom the sun if the eath is 1.50 10 8 km away fom the sun? Q34. A body weighs 63 N on the suface of the eath. What is the gavitational foce on it due to the eath at a height equal to half the adius of the eath? Q35. Why the space ockets ae geneally launched west to East? Q36. Explain why a tennis ball bounces highe on hills than in plane? Q37. The gavitational foce on the eath due to the sun is geate than moon. Howeve tidal effect due to the moon s pull is geate than the tidal effect due to sun. Why? Q38. The mass of moon is M 81 (whee M is mass of eath). Find the distance of the point whee the gavitational field due to eath and moon cancel
each othe. Given distance of moon fom eath is 60, whee is adius of eath. Q39 The figue shows elliptical obit of a planet m about the sun S. The shaded aea of SCD is twice the shaded aea SAB. If t 1 is the time fo the planet to move fom D to C and t, is time to move fom A to B, what is the elation between t 1 and t? B m D D A A 1 S C Q40. Calculate the enegy equied to move a body of mass m fom an obit of adius to 3. Q41. A man can jump 1.5 m high on eath. Calculate the height he may be able to jump on a planet whose density is one quate that of the eath and whose adius is one thid of the eath. SHOT ANSWE TYPE QUESTIONS (3 MAKS) Q4. Define gavitational potential at a point in the gavitational field. Obtain a elation fo it. What is the position at which it is (i) maximum (ii) minimum. Q43. Find the potential enegy of a system of fou paticles, each of mass m, placed at the vetices of a squae of side. Also obtain the potential at the cente of the squae. Q44. Thee mass points each of mass m ae placed at the vetices of an equilateal tiangle of side I. What is the gavitational field and potential at the centoid of the tiangle due to the thee masses. Q45. Biefly explain the pinciple of launching an atificial satellite. Explain the use of multistage ockets in launching a satellite. Q46. In a two stage launch of a satellite, the fist stage bings the satellite to a height of 150 km and the nd stage gives it the necessay citical
speed to put it in a cicula obit. Which stage equies moe expenditue of fuel? Given mass of eath 6.0 10 4 kg, adius of eath 6400 km Q47. The escape velocity of a pojectile on eath s suface is 11. kms 1 A body is pojected out with thice this speed. What is the speed of the body fa away fom the eath? Ignoe the pesence of the sun and othe planets. Q48. A satellite obits the eath at a height fom the suface. How much enegy must be expended to ocket the satellite out of eath s gavitational influence? Q49. Define gavitational potential. Give its SI units. Q50. What do you mean by gavitational potential enegy of a body? Obtain an expession fo it fo a body of mass m lying at distance fom the cente of the eath. Q51. State and explain Keple s laws of planetay motion. Name the physical quantities which emain constant duing the planetay motion. LONG ANSWE TYPE QUESTIONS (5 MAKS) Q5. What is acceleation due to gavity? Obtain elations to show how the value of g changes with (i) attitude (ii) depth Q53. Define escape velocity obtain an expession fo escape velocity of a body fom the suface of eath? Does the escape velocity depend on (i) location fom whee it is pojected (ii) the height of the location fom whee the body is launched. Q54. State Keple s thee laws of planetay motion. Pove the second and thid law. Q55. Deive expession fo the obital velocity of a satellite and its time peiod. What is a geostatinay satellite. Obtain the expession fo the height of the geostationay satellite. Q56. State and deive Keple s law of peiods (o hamonic law) fo cicula obits.
Q57. A black hole is a body fom whose suface nothing may eve escape. What is the condition fo a unifom spheical mass M to be a black hole? What should be the adius of such a black hole if its mass is the same as that of the eath? NUMEICALS Q58. The mass of planet Jupite is 1.9 10 7 kg and that of the sun is 1.99 10 30 kg. The mean distance of Jupite fom the Sun is 7.8 10 11 m. Calculate gavitational foce which sun exets on Jupite, and the speed of Jupite. Q59. A mass M is boken into two pats of masses m 1, and m. How ae m 1, and m elated so that foce of gavitational attaction between the two pats is maximum. Q60. If the adius of eath shinks by %, mass emaining constant. How would the value of acceleation due to gavity change? Q61. Find the value of g at a height of 400 km above the suface of the eath. Given adius of the eath, 6400 km and value of g at the suface of the eath 9.8 ms. [Ans. 8.575 ms ] Q6. How fa away fom the suface of eath does the acceleation due to gavity become 4% of its value on the suface of eath? adius of eath 6400 km. [Ans. 5,600 km] Q63. The gavitational field intensity at a point 10,000 km fom the cente of the eath is 4.8 N kg 1. Calculate gavitational potential at that point. Q64. A geostationay satellite obits the eath at a height of nealy 36000 km. What is the potential due to eath s gavity at the site of this satellite (take the potential enegy at to be zeo). Mass of eath is 6 10 4 kg, adius of eath is 6400 km. Q65. Jupite has a mass 318 times that of the eath, and its adius is 11. times the eath s adius. Estimate the escape velocity of a body fom Jupite s suface, given that the escape velocity fom the eath s suface is 11. km s 1.
Q66. The distance of Neptune and Satun fom the sun is nealy 10 13 m and 10 1 m espectively. Assuming that they move in cicula obits, then what will be the atio of thei peiods. Q67. Let the speed of the planet at peihelion P in fig be v p and Sun planet distance SP be p elate ( A v A ) to the coesponding quantities at the aphelion ( A, v A ). Will the planet take equal times to tavese BAC and CPB? B P S A C UNIT VI GAVITATION ANSWE FO VEY SHOT QUESTIONS (1 MAK) 1. Both foces will be equal in magnitude as gavitational foce is a mutual foce between the two bodies.. When moving in a mey go ound, ou weight appeas to decease when we move down and inceases when we move up, this change in weight makes us feel giddy. 3. (i) Value of acceleation due to gavity (ii) suface tempeatue of planet. GMm Gm 4. F F m but g they bodies fall with same g. and does not depend on m hence 5. No, the gavitational foce is independent of intevening medium.
6. F 1 F' F 4 7. Mass does not change. 8. Because it aises due to attactive foce of gavitation. 9. v e v o v e GM v o GM when 10. No, v e GM 11. Sun should be at B as speed of planet is geate when it is close to sun. 1. The gavitational foce between satellite and eath povides the necessay centipetal foce fo the satellite to obit the eath. 13. The speed of satellite duing descent is much lage than duing ascent, and so heat poduced is lage. 14. No, A satellite will be always visible only if it evolves in the equatoial plane, but New Delhi does not lie in the egion of equitoial plane. 15. g gets doubled as g ρ (density) 16. As g GM/ and the value of at the poles is less than that the equato, so g at poles is geate that g at the equato. Now, g p > g e, hence mg p > g e i.e., the weight of a body at the poles is moe than the weight at the equato. 17. The astonaut is in the gavitational field of the eath and expeiences gavity. Howeve, the gavity is used in poviding necessay centipetal foce, so is in a state of fee fall towads the eath. 18. Geostationay satellite ae used fo tele communication and pola satellite fo emote ensing. 19. Angula velocity of binay stas ae same is w A w B, π π T T TA T B A B
0. T T 1 1 3 T 64 5 T 40 h 1. The foce of gavitation exeted by the eath on a body of mass m is Mm F G mg Gm Acceleation impated to the body, g Clealy, g does not depend on m. Hence the eath impats same acceleation to all bodies.. The satellite will move tangentially to the oiginal obit with a velocity with which it was evolving. SHOT ANSWES ( MAKS) Gm 3. g If deceases by 1% it becomes 99 100 GM Gm g' 1.0 (.99) (1+ 0.0) Gm Gm. g inceases by 0.0, theefoe inceases by %. 4. (b), (c) and (d) ae affected in space. 5. The maximum obital velocity of a satellite obiting nea its suface is v o g v e Fo the satellite to escape gavitational pull the velocity must become v e But v e v 0 1.414 v 0 (1 + 0.414) v 0 This means that it has to inceases 0.414 in 1 o 41.4%.
The minimum incement is equied, as the velocity of satellite is maximum when it is nea the eath. 6. Hee GM GM 4 3 g. π ρ 3 o g ρ g g 1 ρ 1:1. ρ. 7. v e GM e v p GM p p M e M p, p 9 4 M 4 vp G 9 e GM 3 11..4 3 3 e 7.47 km/sec 8. g 64% of g 64 g 100 g g 64 h 100 ( + ) 8 + h 10 g h 4 1600km 9. g d g h d h g1 g1 d h 40 80 km
30. 1.5 10 8 km 1.5 x 10 11 m T 365 days 365 4 3600 s Centipetal foce gavitational foce Mv GMm m GMm π T M s 11 ( ) 3 4π 4 9.87 1.5 10 GT 6.64 10 365 4 3600 ( ) 11 3 M s.01 10 30 g 1 31. g fo > 0 above suface of eath g g e C O B g ( d) fo < 0 below suface of eath g is max fo 0 on suface. 3. Given M s 10 30 kg, M e 6 10 4 kg, 1.5 10 11 m Let m be the mass of the ocket. Let at distance x fom the eath, the gavitational foce on the ocket be zeo. Then at this distance, Gavitational pull of the eath on the ocket Gavitational pull of the sun on the ocket
i.e., GM m GM m ( -x) M x ( -x) x e e s o Me o -x x M 10 10 577.35 30 3 s Me 4 6 10 3 o x 577.35x o 578.35 x 1.5 10 11 o x 578.35 11 1.5 10.59 10 8 m. 33. Accoding to Keple s law of peiods, T s TE s E 3 But T s T 9.5 and E 1.5 108 km. E (9.5) s 8 1.5 10 3 o s 1.5 10 8 (9.5) /3 14.3 10 8 km. 34. Hee mg 63 N, h / As gh 4. g +h 3 9 + g h 4 g 9 mg h 4 mg 9 4 63 8N. 9 35. Since the eath evolves fom west to east, so when the ocket is launched fom west to east the elative velocity of the ocket inceases which helps it to ise without much consumption of fuel. 36. The value of g on hills is less than at the plane, so the weight of tennis ball on the hills is lesse foce than at planes that is why the eath attact
the ball on hills with lesse foce than at planes. Hence the ball bounces highe. 37. The tidal effect depends invesely on the cube of the distance, while gavitational foce depends on the squae of the distance. 60 38. E (60 - x) C x M Gavitational field at C due to eath Gavitational field at C due to eath moon GM ( 60 - x) GM/81 x 81 x (60 x) 9x 60 x x 6 39. Accoding to Keple s II nd law aea velocity fo the planet is constant A t A A 1 A t 1 1 A t A t 1 1 t 1 t 40. Gavitational P.E of mass m in obit of adius U GMm U i GMm
Uf GMm 3 U U f U i GMm 1-1 3 GMm 6 41. 4 g πgρ 3 4 g' πg ρ' 3 The gain in P.E at the highest point will be same in both cases. Hence mg h mgh 4 m πgρh mgh g' 3 mg 4 m πg ρ' 3 ρ h 3' 4 ρ ' 1.5 ' ρ ' ' ρ' 18 m ANSWES FO 3 MAKS QUESTIONS GM 44. E1 (OA) A m E GM (OB) E 1 E GM (OC) 3 B m E E O 30 D m C
Fom ODB Cos 30 o BD / OB OB OB / BD / 3 OB o Cos 30 3 Gavitational field at O due to m at A, B and C is say E 1,E &E 3 E E + E + E E Cos10 o 3 3 (GM3) 3Gm 3GM 1 + + l l l 3GM along OD l E is equal and opposite to E 1 net gavitational field zeo As gavitational potential is scala V V 1 + V + V 3 GM GM GM OA OB OC 3GM Gm V 3 3 l/ 3 l 46. Wok done on satellite in fist stage W 1 PE at 150 km PE at the suface W GMm GMm 1 + h GMmh ( + h)
Wok done on satellite in nd stage W enegy equied to give obital velocity v o 1 1 GMm + h mν 0 W1 h 150 3 < 1 W 6400 64 W > W 1 so second stage equies moe enegy 47. V e 11. kms 1, velocity of pojection v 3v e Let m be the mass of pojectile and v o the velocity afte it escapes gavitational pull. By law of consevation of enegy 1 1 1 mν mν mν 0 e ν ν ν 9ν ν 8ν 0 e e e e.4 31.68 km s 1 48. The enegy equied to pull the satellite fom eath influence should be equal to the total enegy with which it is evolving aound the eath. The K.E. of satellite The P.E of satellite 1 1 GM GM m ν m ν + h + h GMm + h T.E. 1 mgm GMm 1 GMm ( + h) ( + h) ( + h) Enegy equied will be 1 GMm + ( + h)
ANSWES FO NUMEICALS GMm 58. F 6.67 10 1.99 10 1.9 10 11 (7.8 10 ) 11 30 7 F 4.1 10 3 N mν F GMm m m F ν Gm 6.67 10 1.9 10 ν 11 7.8 10 11 30 ν 1.3 10 4 m s 1 59. Let m 1 m then m M m Foce between them when they ae sepaated by distance Gm(M m) G F (Mm m ) Fo F to be maximum, diffeentiate F w..t m and equate to zeo df G (M m) 0 dm M m; m M m 1 m M 60. inceased by 4% 61. 4 g g 100 h +
10 + h h 4 4 6400 5,600 km. GM 6. Gavitational intensity E Gavitational potential GM V V E o, V E o V 4.8 10,000 10 3 4.8 10 7 J kg 1 GM 63. U Potential at height h + h 6.67 10 6 10 U 6.4 10 + 36 + 10 11 4 6 6 9.44 10 6 J/kp GM 64. Escape velocity fom the eath s suface is ν e 11. kms 1 Escape velocity fom Jupite s suface will be ν e GM But M 318 M, 11. G(318 M) GM 318 ν e 11. 11. 318 318 ν e 11. 59.7 kms 1 11. 11.
65. By keple s III d law s 3 T n n T s 3/ 13 3/ n n 1 5 s T 10 T 10 10 3/ 10 10 10 3.16 31.6 T n : T s 36.6 : 1 66. The magnitude of angula momentum at P is L p m p p v p Similaly magnitude of angula momentum at A is L A m A A v A Fom consevation of angula momentum m p p v p m A v A A ν p A ν A p A > p, ν p > ν A aea bound by SB & SC(SBAC > SBPC) By nd law equal aeas ae swept in equal intevals of time. Time taken to tansvese BAC > time taken to tavese CPB