GRAVITATION. Contents. Theory Exercise Exercise Exercise Exercise Answer Key

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1 GAVITATION Contents Topic Page No. Theoy 0-0 Execise Execise Execise Execise Answe Key - 4 Syllabus Law of gavitation; Gavitational potential and field; Acceleation due to gavity; Motion of planets and satellites in cicula obits; Escape velocity. Name : Contact No. AIDE LEANING ONLINE E-LEANING ACADEMY A-479 inda Viha, Kota ajasthan 4005 Contact No

2 GAVITATION KEY CONCEPT. Gavitation: Gavitation is the foce of attaction between any two point paticles in the univese. It is given by: m F = whee G is univesal gavitational constant. The value of G = Nm /kg.. Vaiation of 'g': (i) Due to altitude: Acceleation due to gavity at a height h above the suface of eath is given by: (ii) GM é h ù g h = = g ê - ú (fo h << ) ( + h) ë û Due to depth: Acceleation due to gavity at a depth h below the suface of eath is given by ' g h = é h ù g ê - ú ë û (fo all depths) (iii) At h = (i.e. at the cente of eath): ' g h = 0 Due to otation of eath: Acceleation due to gavity at latitude l is given by: g l = g w cos l whee w = angula velocity of the eath (a) At poles: g p = g w cos (p/) = g = g max (iv) (b) At equato: g eq = g w cos 0 = g w = g min Due to non-spheical shape of eath: Due to the shape of the eath, g is maximum at poles and minimum at equato.. Inetial and gavitational mass : (i) (ii) Inetial mass: It is defined as the atio of the magnitude of extenal foce applied on the body to the magnitude of acceleation poduced in it, i.e., a = (F/m). Gavitational mass: Mass of the mateial of the body, which is detemined by gavitational pull acting on it, is F called as gavitational mass, i.e., m = GM (iii) Inetial and gavitational masses ae found to be equal by obsevation. 4. Gavitational intensity: In case of a solid o hollow sphee of mass M and adius : (a) Fo an extenal point ( > ): I 0 = (GM/ ) (b) Fo an intenal point ( < ): (i) of a spheical shell: I i = 0 (ii) of a solid sphee: I i = (GM/ ) A-479 Inda Viha, Kota ajasthan 4005 Page No. #

3 5. Gavitational potential: In case of a solid o hollow sphee: (a) Fo an extenal point ( > ): V 0 = (GM/) (b) Fo an intenal point ( < ): 6. Escape velocity: (i) of a spheical shell: V i = GM/ = constant GM (ii) of a solid sphee: V i = ( - ) (i) It is minimum speed with which a body must be pojected away fom the suface of the eath so that it may neve etun to the eath. (ii) Escape velocity of a body fom the suface of eath is given by : v es = g = (GM / ) 7. Geostationay satellite: (a) A satellite which appeas to be stationay fo a peson on the suface of the eath is called geostationay satellite. (b) It evolves in the equatoial plane fom west to east with a time peiod of 4 hous. (c) Its height fom the suface of the eath is nealy 5600 km and adius of the cicula obit is nealy 4000 km. (d) The obital velocity of this satellite is nealy.08 km/sec (e) The elative velocity of geostationay satellite with espect to eath is zeo. (f) The obit of a geostationay satellite is called as paking obit. 8. Keple's laws: (i) All planets move aound the sun in elliptical obits, with the sun being at est at one focus of the obit. (ii) The position vecto fom the sun to the planet sweeps out equal aea in equal time, i.e., aeal velocity of a planet aound the sun always emains constant. This gives that the angula momentum o moment of momentum emain constant. (iii) The squae of the time peiod of a planet aound the sun is popotional to the cube of the semi-majo axis of the ellipse o mean distance of the fom the sun, i.e. T µ a whee a is the semi-majo axis of the ellipse. A-479 Inda Viha, Kota ajasthan 4005 Page No. #

4 PAT - I : OBJECTIVE QUESTIONS * Maked Questions ae having moe than one coect option. Section : Univesal law of gavitation A. Fou simila paticles each of mass m ae obiting in a cicle of adius in the same sense and same speed because of thei mutual gavitational attactive foce as shown in the figue. Velocity of a paticle is given by : é æ ê ç + ê 4 ë è öù ú øúû ( ) + (D) zeo A-. Thee paticles P, Q and ae placed as pe given figue. Masses of P, Q and ae m, m and m espectively. The gavitational foce on a fouth paticle S of mass m is equal to : d in ST diection only d d (D) d in SQ diection and d in SQ diection only in SQ diection and d in SU diection in ST diection A- A mass is at the cente of a squae, with fou masses at the cones as shown. (D) ank the choices accoding to the magnitude of the gavitational foce on the cente mass. F A = F B < F C = F D F A = F B > F C = F D F A > F B < F D < F C (D) None Section : Gavitational field and potential B-. Let gavitation field in a space be given as E = (k/). If the efeence point is at distance d i whee potential is V i then elation fo potential is : V = k ln + 0 V = k ln + V V i d i V = ln i d i + kv i (D) V = kln d i + V i A-479 Inda Viha, Kota ajasthan 4005 Page No. #

5 B-. Gavitational field at the cente of a semicicle fomed by a thin wie AB of mass m and length l as shown in the figue. is : along +x axis along +y axis l pl p along + l x axis (D) p along + l y axis B-. A vey lage numbe of paticles of same mass M ae kept at hoizontal distances (in metes) of m, m, 4m, 8m and so on fom (0,0) point. The total gavitational potential at this point is : 8G M G M 4G M (D) G M B-4. Two concentic shells of unifom density of mass M and M ae situated as shown in the figue. The foces expeienced by a paticle of mass m when placed at positions A, B and C espectively ae (given OA = p, OB = q and OC = ). Mm ( M + m)m zeo, G and G q p ( M + M)m ( M + M)m G, G p q and G M m G M m G (M + M, )m Mm q, G q p and zeo G (M + M)m Mm (D), G p q and zeo B-5*. In case of eath : field is zeo, both at cente and infinity potential is zeo, both at cente and infinity potential is same, both at cente and infinity but not zeo (D) potential is minimum at the cente B-6. B-7. B-8. A paticle of mass M is at a distance a fom suface of a thin spheical shell of equal mass and having adius a. Gavitational field and potential both ae zeo at cente of the shell. Gavitational field is zeo not only inside the shell but at a point outside the shell also. Inside the shell, gavitational field alone is zeo. (D) Neithe gavitational field no gavitational potential is zeo inside the shell. A hollow spheical shell is compessed to half its adius. The gavitational potential at the cente inceases deceases emains same (D) duing the compession inceases then etuns at the pevious value. Select the coect choice(s): The gavitational field inside a spheical cavity, within a spheical planet must be non zeo and unifom. When a body is pojected hoizontally at an appeciable lage height above the eath, with a velocity less than fo a cicula obit, it will fall to the eath along a paabolic path. A body of zeo total mechanical enegy placed in a gavitational field if it is tavelling away fom souce of field will escape the field (D) Eath s satellite must be in equatoial plane. A-479 Inda Viha, Kota ajasthan 4005 Page No. # 4

6 Section : Gavitational Potential Enegy and Self Enegy C-. A body stats fom est at a point, distance 0 fom the cente of the eath of mass M, adius. The velocity acquied by the body when it eaches the suface of the eath will be GM æ ç è - 0 ö ø æ GM ç è - 0 ö ø æ ö æ GMç - è 0 ø (D) GM ö ç - è 0 ø C-. Thee equal masses each of mass m ae placed at the thee-cones of an equilateal tiangle of side a. (a) If a fouth paticle of equal mass is placed at the cente of tiangle, then net foce acting on it, is equal to : a 4 a a (D) zeo (b) In above poblem, if fouth paticle is at the mid-point of a side, then net foce acting on it, is equal to: a 4 a a (D) zeo (c) If above given thee paticles system of equilateal tiangle side a is to be changed to side of a, then wok done on the system is equal to : a a 4 a (D) a (d) In the above given thee paticle system, if two paticles ae kept fixed and thid paticle is eleased. Then speed of the paticle when it eaches to the mid-point of the side connecting othe two masses: a a a (D) a C-. A satellite of mass m, initially at est on the eath, is launched into a cicula obit at a height equal to the adius of the eath. The minimum enegy equied is mg mg mg (D) mg Section : (D) Keple s law fo Satellites, Obital Velocity and Escape Velocity D-. Peiodic-time of satellite evolving aound the eath is - ( is density of eath) Popotional to Popotional to Popotional to (D) does not depend on. D-. An atificial satellite of the eath eleases a package. If ai esistance is neglected the point whee the package will hit (with espect to the position at the time of elease) will be ahead exactly below behind (D) it will neve each the eath D-*. An obiting satellite will escape if : its speed is inceased by ( -)00% its speed in the obit is made. 5 times of its initial value its KE is doubled (D) it stops moving in the obit A-479 Inda Viha, Kota ajasthan 4005 Page No. # 5

7 D-4*. D-5. A satellite close to the eath is in obit above the equato with a peiod of evolution of.5 hous. If it is above a point P on the equato at some time, it will be above P again afte time.5 hous.6 hous if it is otating fom west to east 4/7 hous if it is otating fom east to west (D) 4/7 hous if it is otating fom west to east The figue shows the vaiation of enegy with the obit adius of a body in cicula planetay motion. Find the coect statement about the cuves A, B and C A shows the kinetic enegy, B the total enegy and C the potential enegy of the system C shows the total enegy, B the kinetic enegy and A the potential enegy of the system C and A ae kinetic and potential enegies espectively and B is the total enegy of the system (D) A and B ae the kinetic and potential enegies and C is the total enegy of the system. D-6*. In case of an obiting satellite if the adius of obit is deceased : its Kinetic Enegy deceases its Potential Enegy deceases its Mechanical Enegy deceases (D) its speed deceases D-7. A planet of mass m evolves aound the sun of mass M in an elliptical obit. The minimum and maximum distance of the planet fom the sun ae & espectively. If the minimum velocity of the planet is GM ( + ) then it's maximum velocity will be : GM ( + ) GM ( + ) ( + ) (D) GM + D-8. D-9. A spheical unifom planet is otating about its axis. The velocity of a point on its equato is V. Due to the otation of planet about its axis the acceleation due to gavity g at equato is / of g at poles. The escape velocity of a paticle on the pole of planet in tems of V. V e = V V e = V V e = V (D) V e = V Two planets A and B have the same mateial density. If the adius of A is twice that of B, then the atio of the escape velocity v va B is (D) D-0. The escape velocity fo a planet is v e. A tunnel is dug along a diamete of the planet and a small body is dopped into it at the suface. When the body eaches the cente of the planet, its speed will be v e v e v e (D) zeo A-479 Inda Viha, Kota ajasthan 4005 Page No. # 6

8 Section (E) : Eath and Othe Planets Gavity E-. Two blocks of masses m each ae hung fom a balance as shown in the figue. The scale pan A is at height H wheeas scale pan B is at height H. Net toque acting on the od of pan, will be (length of the od is l and H & H ae << ) (H > H ) æ- H ö mg mg H H mg ç l (H - H) l (H + H) l (D) mg è ø H + H l E-. If acceleation due to gavity is 0 ms then let acceleation due to gavitational acceleation at anothe planet of ou sola system be 5 ms. An astonaut weighing 50 kg on eath goes to this planet in a spaceship with a constant velocity. The weight of the astonaut with time of flight is oughly given by (D) E-. E-4. At what altitude will the acceleation due to gavity be 5% of that at the eath s suface (given adius of eath is )? /4 /8 (D) / Let w be the angula velocity of the eath s otation about its axis. Assume that the acceleation due to gavity on the eath s suface has the same value at the equato and the poles. An object weighed at the equato gives the same eading as a eading taken at a depth d below eath s suface at a pole (d<<) The value of d is w g g w w g (D) g g E-5. If the adius of the eath be inceased by a facto of 5, by what facto its density be changed to keep the value of g the same? /5 /5 / 5 (D) 5 E-6. The mass and diamete of a planet ae twice those of eath. What will be the peiod of oscillation of a pendulum on this planet if it is a seconds pendulum on eath? second seconds second (D) second A-479 Inda Viha, Kota ajasthan 4005 Page No. # 7

9 E-7. A (nonotating) sta collapses onto itself fom an initial adius i with its mass emaining unchanged. Which cuve in figue best gives the gavitational acceleation a g on the suface of the sta as a function of the adius of the sta duing the collapse? a b c (D) d E-8. In olde times, people used to think that the Eath was flat. Imagine that the Eath is indeed not a sphee of adius, but an infinite plate of thickness H. What value of H is needed to allow the same gavitational acceleation to be expeienced as on the suface of the actual Eath? (Assume that the Eath s density is unifom and equal in the two models.) 4 8 (D) PAT - II : MISCELLANEOUS QUESTIONS Compehension # : Two unifom spheical stas made of same mateial have adii and. Mass of the smalle planet is m. They stat moving fom est towads each othe fom a lage distance unde mutual foce of gavity. The collision between the stas is inelastic with coefficient of estitution /.. Kinetic enegy of the system just afte the collision is: 8 4 (D) cannot be detemined. The maximum sepaation between thei centes afte thei fist collision is : (D) Compehension # : Figue shows the obit of a planet P ound the sun S. AB and CD ae the mino and majo axes of the ellipse.. If t is the time taken by the planet to tavel along ACB and t the time along BDA, then t = t t > t t < t (D) nothing can be concluded 4. If U is the potential enegy and K kinetic enegy then U > K at Only D Only C both D & C (D) neithe D no C A-479 Inda Viha, Kota ajasthan 4005 Page No. # 8

10 Compehension # Many planets ae evolving aound the fixed sun, in cicula obits of diffeent adius () and diffeent time peiod (T). To estimate the mass of the sun, the obital adius () and time peiod (T) of planets wee noted. Then log 0 T v/s log 0 cuve was plotted. The cuve was found to be appoximately staight line (as shown in figue) having y intecept = 6.0 (Neglect the gavitational inteaction among the planets [Take G = in MKS, p = 0] 5. The slope of the line should be : 9 (D) 4 6. Estimate the mass of the sun : kg kg kg (D) 0 5 kg 7. Two planets A and B, having obital adius and 4 ae initially at the closest position and otating in the same diection. If angula velocity of planet B is w 0, then afte how much time will both the planets be again in the closest position? (Neglect the inteaction between planets). p 7w 0 p 9w 0 p w 0 (D) p 5w 0 Compehension # 4 An atificial satellite is moving in a cicula obit aound the eath with a speed equal to half the magnitude of escape velocity fom the suface of eath. is the adius of eath and g is acceleation due to gavity at the suface of eath. ( = 6400 km) 8. Then the distance of satellite fom the suface of eath is 00 km 6400 km 800 km (D) 4800 km 9. The time peiod of evolution of satellite in the given obit is p g p 4 g p 8 g (D) p 6 g 0. If the satellite is stopped suddenly in its obit and allowed to fall feely onto the eath, the speed with which it hits the suface of the eath. g.5g g (D) g A-479 Inda Viha, Kota ajasthan 4005 Page No. # 9

11 Compehension # 5 A pai of stas otates about a common cente of mass. One of the stas has a mass M and the othe has mass m such that M = m. The distance between the centes of the stas is d (d being lage compaed to the size of eithe sta).. The peiod of otation of the stas about thei common cente of mass (in tems of d, m, G.) is 4p d 8p d p d (D) 4p d. The atio of the angula momentum of the two stas about thei common cente of mass ( L m / L M ) is 4 (D) 9. The atio of kinetic enegies of the two stas ( K m /K M.) is 4 (D) 9 Assetion/eason Type 4. Statement- : Moon evolving aound eath does not come close despite eath s gavitational attaction. Statement- : A adially outwad foce balances eath s foce of attaction duing evolution of moon. Statement- is tue, statement- is tue and statement- is coect explanation fo statement-. Statement- is tue, statement- is tue and statement- is NOT the coect explanation fo statement-. Statement- is tue, statement- is false. (D) Statement- is false, statement- is tue. 5. Statement- : Time peiod of simple pendulum in an obiting geostationay satellite is infinite. Statement- : Eath s gavitational field becomes negligible at lage distance fom it. Statement- is tue, statement- is tue and statement- is coect explanation fo statement-. Statement- is tue, statement- is tue and statement- is NOT the coect explanation fo statement-. Statement- is tue, statement- is false. (D) Statement- is false, statement- is tue. 6. Statement- : Geostationay satellites may be setup in equatoial plane in obits of any adius moe than eath s adius. Statement- : Geostationay satellites have peiod of evolution of 4 hs. Statement- is tue, statement- is tue and statement- is coect explanation fo statement-. Statement- is tue, statement- is tue and statement- is NOT the coect explanation fo statement-. Statement- is tue, statement- is false. (D) Statement- is false, statement- is tue. 7. Statement- : Fo the calculation of gavitational foce between any two unifom spheical shells, they can always be eplaced by paticles of same mass placed at espective centes. Statement- : Gavitational field of a unifom spheical shell out side it is same as that of paticle of same mass placed at its cente of mass. Statement- is tue, statement- is tue and statement- is coect explanation fo statement-. Statement- is tue, statement- is tue and statement- is NOT the coect explanation fo statement-. Statement- is tue, statement- is false. (D) Statement- is false, statement- is tue. 8. Statement- : It takes moe fuel fo a spacecaft to tavel fom the eath to moon than fo the etun tip. Statement- : Potential enegy of spacecaft at moon s suface is geate than that at eath suface. Statement- is tue, statement- is tue and statement- is coect explanation fo statement-. Statement- is tue, statement- is tue and statement- is NOT the coect explanation fo statement-. Statement- is tue, statement- is false. (D) Statement- is false, statement- is tue. A-479 Inda Viha, Kota ajasthan 4005 Page No. # 0

12 Match the column 9. A paticle is taken to a distance (> ) fom cente of the eath. is adius of the eath. It is given velocity V which is pependicula to. With the given values of V in column I you have to match the values of total enegy of paticle in column II and the esultant path of paticle in column III. Hee 'G' is the univesal gavitational constant and 'M' is the mass of the eath. Column I (Velocity) Column II (Total enegy) Column III (Path) V = GM / (p) Negative (t) Elliptical V = GM/ (q) Positive (u) Paabolic V > (D) GM/ () Zeo (v) Hypebolic GM / < V < GM/ (s) Infinite (w) Cicula 0. Let V and E denote the gavitational potential and gavitational field espectively at a point due to cetain unifom mass distibution descibed in fou diffeent situations of column-i. Assume the gavitational potential at infinity to be zeo.the value of E and V ae given in column-ii. Match the statement in column-i with esults in column-ii. Column-I Column-II At cente of thin spheical shell (p) E = 0 At cente of solid sphee (q) E ¹ 0 A solid sphee has a non-concentic spheical cavity. At the cente of the spheical cavity () V ¹ 0 (D) At cente of line joining two point masses of equal magnitude (s) V = 0 PAT - I : MIXED OBJECTIVE Single Choice type. A spheical hollow cavity is made in a lead sphee of adius, such that its suface touches the outside suface of the lead sphee and passes though its cente. The mass of the sphee befoe hollowing was M. With d what gavitational foce will the hollowed-out lead sphee attact a small sphee of mass m, which lies at a distance d fom the cente of the lead sphee on the staight line connecting the centes of the sphees and that of the hollow, if d = : 7GMm 8 7GMm 6 7GMm 9. A staight od of length l extends fom x = a to x = l + a. as shown in the figue. If the mass pe unit length is (a + bx ). The gavitational foce it exets on a point mass m placed at x = 0 is given by 7GMm (D) 7 m æ æ ö ö G m çaç - + bl (a + bx ) è è a a + l ø ø l G æ æ ö ö m çaç - + bl è è a a + l ø ø (D) æ æ ö ö çaç - + bl è è a + l a ø ø A-479 Inda Viha, Kota ajasthan 4005 Page No. #

13 . Figue show a hemispheical shell having unifom mass density. The diection of gavitational field intensity at point P will be along: a b c (D) d 4. Mass M is unifomly distibuted only on cuved suface of a thin hemispheical shell. A, B and C ae thee points on the cicula base of hemisphee, such that A is the cente. Let the gavitational potential at points A, B and C be V A, V B, V C espectively. Then A B C V A > V B >V C V C > V B >V A V B >V A and V B > V C (D) V A = V B =V C 5. A unifom ing of mass M is lying at a distance fom the cente of a unifom sphee of mass m just below the sphee as shown in the figue whee is the adius of the ing as well as that of the sphee. Then gavitational foce exeted by the ing on the sphee is : GMm 8 GMm GMm GMm (D) 8 6. The gavitational potential of two homogeneous spheical shells A and B (sepaated by lage distance) of same suface mass density at thei espective centes ae in the atio : 4. If the two shells coalesce into single one such that suface mass density emains same, then the atio of potential at an intenal point of the new shell to shell A is equal to : : 4 : 5 : (D) : 5 7. If a tunnel is cut at any oientation though eath, then a ball eleased fom one end will each the othe end in time(neglect eath otation) 84.6 minutes 4. minutes 8 minutes (D) depends on oientation 8. A satellite of the eath is evolving in cicula obit with a unifom velocity V. If the gavitational foce suddenly disappeas, the satellite will continue to move with the same velocity in the same obit. move tangentially to the oiginal obit with velocity V. fall down with inceasing velocity. (D) come to a stop somewhee in its oiginal obit. 9. A satellite evolves in the geostationay obit but in a diection east to west. The time inteval between its successive passing about a point on the equato is : 48 hs 4 hs hs (D) neve A-479 Inda Viha, Kota ajasthan 4005 Page No. #

14 0. Two point masses of mass 4m and m espectively sepaated by d distance ae evolving unde mutual foce of attaction. atio of thei kinetic enegies will be : : 4 : 5 : (D) :. A satellite of mass 5M obits the eath in a cicula obit. At one point in its obit, the satellite explodes into two pieces, one of mass M and the othe of mass 4M. Afte the explosion the mass M ends up tavelling in the same cicula obit, but in opposite diection. Afte explosion the mass 4M is : In a cicula obit unbound elliptical obit (D) data is insufficient to detemine the natue of the obit.. A satellite can be in a geostationay obit aound eath at a distance fom the cente. If the angula velocity of eath about its axis doubles, a satellite can now be in a geostationay obit aound eath if its distance fom the cente is / (4) (D) / (). A planet of mass m is in an elliptical obit about the sun (m << M sun ) with an obital peiod T. If A be the aea of obit, then its angula momentum would be: ma T mat ma T (D) mat 4. Satellites A and B ae obiting aound the eath in obits of atio and 4 espectively. The atio of thei aeal velocities is: : : 4 : 8 (D) : 6 æ da ö 5. A planet evolves about the sun in elliptical obit. The aial velocity ç è dt ø of the planet is m /s. The least distance between planet and the sun is 0 m. Then the maximum speed of the planet in km/s is : (D) None of these Moe than one choice type 6. Fo a satellite to appea stationay to an obseve on eath It must be otating about the eath s axis. It must be otating in the equatoial plane. Its angula velocity must be fom west to east. (D) Its time peiod must be 4 hous. 7. Which of the following ae coect? An astonant going fom the eath to the Moon will expeience weightlessness once. When a thin unifom spheical shell gadually shinks maintaining its shape, the gavitational potential at its cente deceases. In the case of a spheical shell, the plot of V vesus is contiunous. (D) In the case of a spheical shell, the plot of gavitational field intensity I vesus is continuous. 8. Which of the following statements ae coect about a planet otating aound the sun in an elliptical obit: its mechanical enegy is constant its angula momentum about the sun is constant its aeal velocity about the sun is constant (D) its time peiod is popotional to A-479 Inda Viha, Kota ajasthan 4005 Page No. #

15 9. A tunnel is dug along a chod of the eath at a pependicula distance / fom the eath s cente. The wall of the tunnel may be assumed to be fictionless. A paticle is eleased fom one end of the tunnel. The pessing foce by the paticle on the wall and the acceleation of the paticle vaies with x (distance of the paticle fom the cente) accoding to : (D) 0. Assuming the eath to be a sphee of unifom density the acceleation due to gavity at a point outside the eath is invesely popotional to the squae of its distance fom the cente at a point outside the eath is invesely popotional to its distance fom the cente at a point inside is zeo (D) at a point inside is popotional to its distance fom the cente.. Two masses m and m (m < m ) ae eleased fom est fom a finite distance. They stat unde thei mutual gavitational attaction acceleation of m is moe than that of m acceleation of m is moe than that of m cente of mass of system will emain at est in all the efeences fame (D) total enegy of system emains constant. In side a hollow isolated spheical shell eveywhee gavitational potential is zeo. eveywhee gavitational field is zeo. eveywhee gavitational potential is same. (D) eveywhee gavitational field is same.. A geostationay satellite is at a height h above the suface of eath. If eath adius is : The minimum colatitude on eath upto which the satellite can be used fo communication is sin ( h) +. The maximum colatitudes on eath upto which the satellite can be used fo communication is sin ( h) +. The aea on eath escaped fom this satellite is given as p ( + sinq) (D) The aea on eath escaped fom this satellite is given as p ( + cosq) 4. When a satellite in a cicula obit aound the eath entes the atmospheic egion, it encountes small ai esistance to its motion. Then its kinetic enegy inceases its kinetic enegy deceases its angula momentum about the eath deceases (D) its peiod of evolution aound the eath inceases A-479 Inda Viha, Kota ajasthan 4005 Page No. # 4

16 5. A communications Eath satellite goes ound the eath fom east to west can be in the equatoial plane only can be vetically above any place on the eath (D) goes ound the eath fom west to east 6. An eath satellite is moved fom one stable cicula obit to anothe lage and stable cicula obit. The following quantities incease fo the satellite as a esult of this change gavitational potential enegy angula vleocity linea obital velocity (D) centipetal acceleation 7. A geostationay satellite S is stationed above a point P on the equato. A paticle is fied fom S diectly towads P. With espect to axis of otation of the eath, P and S have the same angula velocity but diffeent linea velocities. The paticle will hit P. The paticle will hit the equato east of P. (D) The paticle will hit the equato west of P. 8. If a satellite obits as close to the eath's suface as possible, its speed is maximum time peiod of its otation is minimum the total enegy of the 'eath plus satellite' system is minimum (D) the total enegy of the 'eath plus satellite'system is maximum 9. Fo a satellite to obit aound the eath, which of the following must be tue? It must be above the equato at some time It cannot pass ove the poles at any time Its height above the suface cannot exceed 6,000 km (D) Its peiod of otation must be > p / g whee is adius of eath 0. Two satellites s & s of equal masses evolve in the same sense aound a heavy planet in coplana cicula obit of adii & 4 the atio of peiod of evolution s & s is : 8. thei velocities ae in the atio : thei angula momentum about the planet ae in the atio : (D) the atio of angula velocities of s w..t. s when all thee ae in the same line is 9 : 5. PAT - II : SUBJECTIVE QUESTIONS. Two unifom solid sphees of same mateial and same adius ae touching each othe. If the density is then find out gavitational foce between them.. The gavitational potential in a egion is given by V = (0x + 40y) J/kg. Find out the gavitational field (in newton / kg) at a point having co-odinates (, 4). Also find out the magnitude of the gavitational foce on a paticle of 0.50 kg placed at the point (, 4).. The gavitational field in a egion is given by E = (î - 4ĵ ) N/kg. Find out the wok done (in joule) in displacing a paticle of mass kg by m along the line 4y = x + 9. A-479 Inda Viha, Kota ajasthan 4005 Page No. # 5

17 4. Two planets A and B ae fixed at a distance d fom each othe as shown in the figue. If the mass of A is M A and that of B is M B, then find out the minimum velocity of a satellite of mass M S pojected fom the mid point of two planets to infinity. 5. A satellite is established in a cicula obit of adius and anothe in a cicula obit of adius.0. How much pecentage the time peiod of second-satellite will be lage than the fist satellite nealy. 6. Two identical stas of mass M obit aound thei cente of mass. Each obit is cicula and has adius, so that the two stas ae always on opposite sides on a diamete. (a) Find the gavitational foce of one sta on the othe. (b) Find the obital speed of each sta and the peiod of the obit. (c) Find thei common angula speed. (d) Find the minimum enegy that would be equied to sepaate the two stas to infinity. (e) If a meteoite passes though this cente of mass pependicula to the obital plane of the stas. What value must its speed exceed at that point if it escapes to infinity fom the sta system. 7. Two stas of mass M & M ae in cicula obits aound thei cente of mass. The sta of mass M has an obit of adius, the sta of mass M has an obit of adius. (assume that thei cente of mass is not acceleating and distance between stas is fixed) (a) Show that the atio of the obital adii of the two stas equals the ecipocal of the atio of thei masses, that is / = M /M. (b) Explain why the two stas have the same obital peiod and show that the peiod, T = p G / + ( M + M ( ) ). (c) The two stas in a cetain binay sta system move in cicula obits. The fist sta, a moves in an obit of adius km. The othe sta, b moves in an obit of adius km. The obital peiod is 44.5 yea. What ae the masses of each of the two stas? 8. In a solid sphee of adius and density thee is a spheical cavity of adius /4 as shown in figue. A paticle of mass m is eleased fom est fom point B (inside the cavity). Find out - (a) (b) The position whee this paticle stikes the cavity. Velocity of the paticle at this instant. 9. (a) What is the escape speed fo an object in the same obit as that of Eath aound sun (Take obital adius ) but fa fom the eath? (mass of the sun = M s ) (b) If an object aleady has a speed equal to the eath s obital speed, what minimum additional speed must it be given to escape as in (a)? A-479 Inda Viha, Kota ajasthan 4005 Page No. # 6

18 0. A cosmic body A moves towads the Sun with velocity v 0 (when fa fom the Sun) and aiming paamete l, the diection of the vecto v 0 elative to the cente of the Sun as shown in the figue. Find the minimum distance by which this body will get to the Sun. (Mass of Sun = M S ). If a pendulum has a peiod of exactly.00 sec. at the equato, what would be its peiod at the south pole? Assume the eath to be spheical and otational effect of the Eath is to be taken.. A small mass and a thin unifom od each of mass ' m ' ae positioned along the same staight line as shown. Find the foce of gavitational attaction exeted by the od on the small mass.. A point P lies on the axis of a fixed ing of mass M and adius a, at a distance a fom its cente C. A small paticle stats fom P and eaches C unde gavitational attaction only. Its speed at C will be. 4. An object is pojected vetically upwad fom the suface of the eath of mass M with a velocity such that the maximum height eached is eight times the adius of the eath. Calculate: (i) the initial speed of pojection (ii) the speed at half the maximum height. 5. Fou masses (each of m)ae placed at the vetices of a egula pyamid tiangula base of side 'a'. Find the wok done by the system while taking them apat so that they fom the pyamid of side 'a'. 6. A thin spheical shell of total mass M and adius is held fixed. Thee is a small hole in the shell. A mass m is eleased fom est a distance fom the hole along a line that passes though the hole and also though the cente of the shell. This mass subsequently moves unde the gavitational foce of the shell. How long does the mass take to tavel fom the hole to the point diametically opposite. 7. A satellite is moving in a cicula obit aound the eath. The total enegy of the satellite is E = 0 5 J. The amount of enegy to be impated to the satellite to tansfe it to a cicula obit whee its potential enegy is U = 0 5 J is equal to. A-479 Inda Viha, Kota ajasthan 4005 Page No. # 7

19 8. A satellite of mass m is obiting the eath in a cicula obit of adius. It stats losing enegy due to small ai esistance at the ate of C J/ s. Then the time taken fo the satellite to each the eath is. 9. A hypothetical planet of mass M has thee moons each of equal mass m each evolving in the same cicula obit of adius. The masses ae equally spaced and thus fom an equilateal tiangle. Find : (i) the total P.E. of the system (ii) the obital speed of each moon such that they maintain this configuation. 0. A emote sensing satellite is evolving in an obit of adius x ove the equato of eath. Find the aea on eath suface in which satellite can not send message.. A pai of stas otates about a common cente of mass. One of the stas has a mass M which is twice as lage as the mass m of the othe. Thei centes ae a distance d apat, d being lage compaed to the size of eithe sta. (a) Deive an expession fo the peiod of otation of the stas about thei common cente of mass in tems of d,m, G. (b) Compae the angula momentum of the two stas about thei common cente of mass by calculating the atio L m / L M. (c) Compae the kinetic enegies of the two stas by calculating the atio K m /K M.. A small body is pojected with a velocity just sufficient to make it each fom the suface of a planet (of adius and mass M) to the suface of anothe planet (of adius and mass M). The distance between the centes of the two spheical planets is 6. the distance of the body fom the cente of bigge planet is x at any moment. Duing the jouney, find the distance x whee the speed of the body is (a) maximum (b) minimum. Assume motion of body along the line joining centes of planets. PAT - I : IIT-JEE : QUESTIONS * Maked Questions ae having moe than one coect option.. Distance between the centes of two stas is 0a. The masses of these stas ae M and 6 M and thei adii a and a espectively. A body of mass m is fied staight fom the suface of the lage sta towads the smalle sta. What should be its minimum initial speed to each the suface of the smalle sta? obtain the expession in tems of G. M and a. [JEE - 996, 5]. A satellite S is moving in an elliptical obit aound the eath. The mass of the satellite is vey small compaed to the mass of the eath : [JEE (Sc) - 98, ] The acceleation of S is always diected towads the cente of the eath The angula momentum of S about the cente of the eath changes in diection, but its magnitude emains constant The total mechanical enegy of S vaies peiodically with time (D) The linea momentum of S emains constant in magnitude.. A simple pendulum has a time peiod T when on the eath s suface, and T when taken to a height above the eath s suface, whee is the adius of the eath. The value of T /T is : [JEE (Sc) - 00, /5] 4 (D) 4. A geostationay satellite obits aound the eath in a cicula obit of adius 6000 km. Then, the time peiod of a spy satellite obiting a few hunded kilometes above the eath s suface ( Eath = 6400 km) will appoximately be : [JEE(Sc) - 0, /84] / h h h (D) 4 h A-479 Inda Viha, Kota ajasthan 4005 Page No. # 8

20 5. A paticle of mass m is taken though the gavitational field poduced by a souce S, fom A to B, along the thee paths as shown in figue. If the wok done along the paths I, II and III is W I, W II and W III espectively, then : [JEE (Sc.)-00,/84] B II I III A W I = W II = W III W II > W III = W II W III = W II > W I (D) W I > W ii > W III 6. A pojectile is fied vetically up fom the bottom of a cate (big hole) on the moon. The depth of the cate is /00, whee is the adius of the moon. If the initial velocity of the pojectile is the same as the escape velocity fom the moon suface, detemine in tems of, the maximum height attained by the pojectile above the luna (moon) suface. [JEE 00(Main),4/60] 7. A double sta system consists of two stas A and B which have time peiod T A and T B. adius A and B and mass M A and M B. Choose the coect option. [JEE 006, +, /84] If T A > T B then A > B If T A > T B then M A > M B æ T ç è T A B ö ø æ = ç è A B ö ø (D) T A = T B 8. A spheically symmetic gavitational system of paticles has a mass density [JEE 008, +, /8] ì0 fo = í î 0 fo > whee 0 is a constant. A test mass can undego cicula motion unde the influence of the gavitational field of paticles. Its speed V as a function of distance (0 < < ) fom the cente of the system is epesented by V V V V (D) 9. STATEMENT - [JEE 008,+, /8] An astonaut in an obiting space station above the Eath expeiences weightlessness. and STATEMENT - An object moving aound the Eath unde the influence of Eath's gavitational foce is in a state of 'fee-fall. STATEMENT - is Tue, STATEMENT - is Tue; STATEMENT - is a coect explanation fo STATEMENT - STATEMENT - is Tue, STATEMENT - is Tue; STATEMENT - is NOT a coect explanation fo STATEMENT - STATEMENT - is Tue, STATEMENT - is False (D) STATEMENT - is False, STATEMENT - is Tue. 0. A thin unifom annula disc (see figue) of mass M has oute adius 4 and inne adius. The wok equied to take a unit mass fom point P on its axis to infinity is : [JEE 00,, /5] P 4 4 GM GM ( 4-5) ( 4-5) 7-7 GM 4 GM (D) ( -) 5 A-479 Inda Viha, Kota ajasthan 4005 Page No. # 9

21 . A binay sta consists of two stas A (mass. M S ) and B ( mass M S ) whee M s is the mass of the sun. They ae sepaated by distance d and ae otating about thei cente of mass, which is stationay. The atio of the total angula momentum of the binay sta to the angula momentum of sta B about the cente of mass is : [JEE 00, +, /5]. Gavitational acceleation on the suface of a planet is 6 g, whee g is the gavitational acceleation on the suface of the eath. The aveage mass density of the planet is times that of the eath. If the escape speed on the suface of the eath is taken to be kms, the escape speed on the suface of the planet in kms will be : [JEE 00, +, /5] PAT - II : AIEEE QUESTIONS. A satellite of the eath is evolving in a cicula obit with a unifom speed v. If the gavitational foce suddenly disappeas, the statellite will : [AIEEE-00, 4/00] () Continue to move with velocity v along the oiginal obit () Move with a velocity v, tangentially to the oiginal obit () Fall down with inceasing velocity (4) Ultimately come to est somewhee on the oiginal obit. The time peiod of a satellite of eath is 5 hous. If the sepaation between the eath and the satellite is inceased to 4 times the pevious value, the new time peiod becomes [AIEEE-00, 4/00] () 0 hou () 80 hou () 40 hou (4) 0 hou. The escape velocity fo a body pojected vetically upwads fom the suface of eath is km/s. If the body is pojected at an angle of 45º with the vetical, the escape velocity will be : [AIEEE-00,4/00] () km/s () km/s () km/s (4) / m/s 4. A satellite of mass m evolves aound eath of adius at a height x fom its suface. If g is the acceleation due to gavity on the suface of the eath, the obital speed of the satellite is : [AIEEE-004, 4/00] () gx () g - x () g + x æ g ö (4) ç x è + ø 5. The time peiod of an eath satellite in cicula obit is independent of : [AIEEE-004, 4/00] () the mass of the satellite () adius of its obit () both the mass and adius of the obit (4) neithe the mass of the satellite no the adius of its obit 6. If g is the acceleation due to gavity on the eath s suface, the gain in the potential enegy of an object of mass m aised fom the suface of the eath to a height equal to the adius of the eath, is : [AIEEE-004, 4/00] () mg () mg () mg 4 (4) mg 7. The change in the value of g at a height h above the suface of the eath is the same as at a depth d below the suface of eath. When both d and h ae much smalle than the adius of eath, then, which one of the following is coect? [AIEEE-005, 4/00] () d = h h () d = () d = h (4) d = h / A-479 Inda Viha, Kota ajasthan 4005 Page No. # 0

22 8. A paticle of mass 0 kg is kept on the suface of a unifom sphee of mass 00 kg and adius 0 cm. Find the wok to be done against the gavitational foce between them, to take the paticle fa away fom the sphee (you may take G = Nm /kg ); [AIEEE-005, 4/00] () J (). 0 0 J () J (4) J 9. If g E and g m ae the acceleations due to gavity on the sufaces of the eath and the moon espectively and if Millikan's oil dop expiment could be pefomed on the two sufaces, one will find the atio to be [AIEEE-007, /0] () () 0 () g E /g M (4) g M /g E 0. A planet in a distant sola system is 0 times moe massive than the eath and its adius is 0 times smalle. Given that the escape velocity fom the eath is km s, the escape velocity fom the suface of the planet would be [AIEEE-008, /05] () km s () 0 km s () 0. km s (4). km s. The height at which the acceleation due to gavity becomes 9 g (whee g = the acceleation due to gavity on the suface of the eath) in tems of, the adius of the eath, is [AIEEE-009, 4/44] () () () (4). Two bodies of masses m and 4 m ae placed at a distance. The gavitational potential at a point on the line joining them whee the gavitational field is zeo is : [AIEEE 0] () 4 - () 6 - () 9 - (4) zeo. The mass of a spaceship is 000 kg. It is to be launched fom the eath s suface out into fee space. The value of g and (adius of eath) ae 0 m/s and 6400 km espectively. The equied enegy fo this wok will be : [AIEEE 0] () J () J () J (4) J NCET QUESTIONS. (i) In the following two execises, choose the coect answe fom among the given ones : The gavitational intensity at the cente of a hemispheical shell of unifom mass density has the diection indicated by the aow (see figue) (i) a, (ii) b, (iii) c, (iv) none. (ii) Fo the above poblem, the diection of the gavitational intensity at an abitay point P is indicated by the aow (i) d, (ii) e, (iii) f, (iv) g.. A ocket is fied fom the eath towads the sun. At what distance fom the eath s cente is the gavitational foce on the ocket zeo? Mass of the sun = 0 0 kg, mass of the eath = kg. Neglect the effect of othe planets etc. (obital adius =.5 0 m).. A ocket is fied vetically fom the suface of mas with a speed of km s. If 0% of its initial enegy is lost due to matian atmospheic esistance, how fa will the ocket go fom the suface of mas befoe etuning to it? Mass of mas = kg ; adius of mas = 95 kg; G = N m kg. A-479 Inda Viha, Kota ajasthan 4005 Page No. #

23 4. A satun yea is 9.5 times the eath yea. How fa is the satun fom the sun if the eath is km away fom the sun? 5. A body weights 6 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? 6. Assuming the eath to be a sphee of unifom mass density, how much would a body weight half way down to the cente of the eath if it weighted 50 N on the suface? 7. A ocket is fied vetically with a speed of 5 km s fom the eath s suface. How fa fom the eath does the ocket go befoe etuning to the eath? Mass of the eath = kg; mean adius of the eath = m ; G = N m kg. 8. The escape speed of a pojectile on the eath s suface is. km s. 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. 9. A satellite obits the eath at a height of 400 km above the suface. How much enegy must be expanded to ocket the satellite out of the eath s gavitational influence? Mass of the satellite = 00 kg; mass of the eath = kg; adius of the eath = m; G = N m kg. 0. Two stas each of the one sola mass (= 0 0 kg) ae appoaching each othe fo a head on collision. When they ae a distance 0 9 km, thei speeds ae negligible. What is the speed with which they collide? The adius of each sta is 0 4 km. Assume the stas to emain undistoted until they collide. (Use the known value of G).. Two heavy sphees, each of mass 00 kg and adius 0.0 m ae placed.0 m apat on a hoizontal table. What is the gavitational foce and potential at the mid point of the line joining the centes of the sphees? Is an object placed at that point in equilibium? If so, is the equilibium stable o unstable?. As you have leant in the text, a geostationay satellite obits the eath at a height of nealy 6,000 km fom the suface of the eath. What is the potential due to eath s gavity at the site of this satellite? (Take the potential enegy at infinity to be zeo). Mass of the eath = kg, adius = 6400 km.. A sta.5 times the mass of the sun and collapsed to a size of km otates with a speed of. ev. pe second. (Extemely compact stas of this kind ae known as neuton stas. Cetain stella objects called pulsas belong to this categoy). Will an object placed on its equato emain stuck to its suface due to gavity? (mass of the sun = 0 0 kg). 4. A spaceship is stationed on Mas. How much enegy must be expanded on the spaceship to launch it out to the sola system? Mass of the space ship = 000 kg; mass of sun = 0 0 kg ; mass of mas = kg ; adius of mas = 95 kgm ; adius of the obit of mas = km; G = N m kg. A-479 Inda Viha, Kota ajasthan 4005 Page No. #

24 Execise # PAT - I A. A-. A- B-. B-. (D) B-. (D) B-4. (D) B-5*. (AD) B-6. (D) B-7. B-8. C-. C-. (a) (D) (b) (c) (d) C-. (D) D-. D-. (D) D-*. (AC) D-4*. (BC) D-5. (D) D-6*. (BC) D-7. D-8. D-9. D-0. E-. E-. E-. E-4. E-5. E-6. E-7. E-8. PAT - II (D) (D) 7. (D) I II III 0. p, p, q, (D) p, A p w B u C q v D p t Execise # PAT - I (D) 5. (D) (ABCD)7. (ACD) 8. (ABC) 9. (B*C) 0. (AD). (AD). (BCD). (AC) 4. (AC) 5. (BD) (AC) 8. (ABC) 9. (AD) 0. (ABD). 4. PAT - II 4 4 G 9 p. - 0 î - 40 ĵ, F = 5 5 N, F = -5î -0 ĵ G(M d M A + B ) 5..5%. zeo 6. (a) F = GM 4 (b) G M ; T = 4 4 p GM (c) GM 4 (d) GM 4 (e) 4GM 4p [.5 0 ] 7. M a = G[ ] = kg, M b = M a = kg 8. (a) Since foce is always acting towads cente of solid sphee. Hence it will stike at A. (b) v = pg 9. (a) GMs GMs - 0. min = (GM S / v 0 ) [ + ( lv /GM ] (b) ( ) 0 S ) A-479 Inda Viha, Kota ajasthan 4005 Page No. #

25 . T = 4p (86400) = s. L GM æ ö. ç - a è ø 4. (i) 4, (ii) 5 5. a GMm æ 6. / GM J 8. t = ö ç - C è e ø æ m ö G æ m ö 9. (i) ç + M, (ii) ç + M è ø è ø 0. æ x ö ç - - 4p ç x è ø. (a) / pd T=, (b), (c)., [ ]. V min = 5 GM a Execise # PAT - I.. (D) (D) PAT - II. (). (). () 4. (4) 5. () 6. () 7. () 8. (4) 9. () 0. (). (4). (). (4) Execise # 4 NCET QUESTIONS. (i) (c), (ii) (e) m. 495 km m 5. 8 N 6. 5 M m fom the eath s cente 8..7 km/s J m/s. 0, J/kg; an object placed at the mid point is in an unstable equilibium J/kg. Yes 4. 0 J A-479 Inda Viha, Kota ajasthan 4005 Page No. # 4