EXERCISE 01 CHECK YOUR GRASP [OBJECTIVE QUESTIONS ]

Transcription

1 J-Physics XCIS 01 CHCK YOU GASP [OBJCTIV QUSTIONS ] 1. Thee identical point masses, each of mass 1 kg lie in the x y plane at points (0, 0) (0, 0.m) and (0.m,0). The gavitational foce on the mass at the oigin is : 1.67 x (i ˆ ˆj)N 3.34 x (i ˆ ˆj)N 1.67 x 10 9 (i ˆ ˆj)N 3.34 x (i ˆ ˆj)N. If the gavitational foce wee to vay invesely as m th powe of the distance, then the time peiod of a planet in cicula obit of adius aound the Sun will be popotional to 3m/ 3m/ m + 1/ (m+1)/ 3. If the distance between the centes of ath and Moon is D and mass of ath is 81 times that of Moon. At what distance fom the cente of ath gavitational field will be zeo? D D 4D 9D The adius of ath is about 6400 km and that of mas is 300 km. The mass of the ath is 10 times the mass of mas. An object weight 00 N on the suface of ath. Its weight on the suface of mas will be : 80 N 40 N 0 N 8 N 5. Weight of a body of mass m deceases by 1% when it is aised to height h above the ath's suface. If the body is taken to a depth h in a mine, then its weight will : decease by 0.5% decease by % incease by 0.5% incease by 1% 6. Imagine a new planet having the same density as that of ath but it is 3 times bigge than the ath in size. If the acceleation due to gavity on the suface of ath is g and that on the suface of the new planet is g', then g' = 3g g' = g 9 g' = 9g g' = 7 g 7. An object weighs 10 N at the noth pole of the ath. In a geostationay satellite distance 7 fom the cente of the ath (of adius ), the tue weight and the appaent weight ae 0 N, 0 N 0. N, 0 0. N, 9.8 N 0. N, 0. N 8. A stone dop fom height 'h' eaches to ath suface in1 sec. If the same stone taken to Moon and dop feely then it will eaches fom the suface of the Moon in the time (The 'g' of Moon is 1/6 times of ath): 6 second 9 second 3 second 6 second 9. The otation of the ath having adius about its axis speeds upto a value such that a man at latitude angle 60 0 feels weightless. The duation of the day in such case will be 8 g 8 g g 4 g NOD6 : \Data\014\Kota\J-Advanced\SMP\Phy\Unit No-4\Gavitation\ng\xecise.p A small body of supedense mateial, whose mass is twice the mass of the ath but whose size is vey small compaed to the size of the ath, stats fom est at a height H<< above the ath's suface, and each the ath's suface in time t. Then t is equal to H / g H / g H / 3g 4H / 3g 1 1. A man of mass m stats falling towads a planet of mass M and adius. As he eaches nea to the suface, he ealizes that he will pass though a small hole in the planet. As he entes the hole, he seen that the planet is eally made of two pieces a spheical shell of negligible thickness of mass M 3 cente. Change in the foce of gavity expeienced by the man is : m 3 1 m and a point mass M 3 4 m 3 at the

2 J-Physics 1. A body attains a height equal to the adius of the ath when pojected fom ath' suface. The velocity of the body with which it was pojected is : e e 5 e 4 3 e 1 3. If the gavitational acceleation at suface of ath is g, then incease in potential enegy in lifting an object of mass m to a height equal to the adius of ath will be : mg mg mg mg ind the distance between cente of gavity and cente of mass of a two paticle system attached to the ends of a light od. ach paticle has same mass. Length of the od is, whee is the adius of ath / zeo / The intensity of gavitational field at a point situated at a distance 8000 km fom the cente of ath is 6.0 newton /kg. The gavitational potential at that point in newton mete/kg will be : The gavitational field due to a mass distibution is in the x diection. (K is a constant). Taking the 3 x gavitational potential to be zeo at infinity, its value at the distance x is : K x K x K K x K x 1 7. Two bodies of masses m and M ae placed at distance d apat. The gavitational potential (V) at the position whee the gavitational field due to them is zeo V is : m m G V (m M) d G V d V d G V ( m M ) d 1 8. Gavitation on Moon is 1 th 6 of that on ath. When a balloon filled with hydogen is eleased on Moon then this will ise with an acceleation less then g 6 will ise with acceleation g 6 will fall down with an acceleation less than 5g 6 6 will fall down with acceleation g scape velocity of a body fom the suface of ath is 11. km/sec. fom the ath suface. If the mass of ath becomes double of its pesent mass and adius becomes half of its pesent adius, then escape velocity will become 5.6 km/sec 11. km/sec.4 km/sec 44.8km/sec 0. A body of mass m is situated at distance 4 e above the ath's suface, whee e is the adius of ath how much minimum enegy be given to the body so that it may escape : mg e mg e 1. The atmospheic pessue and height of baomete column is 10 5 P a and 760mm espectively on the ath suface. If the baomete is taken to the Moon then column height will be : mg e 5 mg e 16 zeo 76 mm 16.6 mm 760 mm NOD6 : \Data\014\Kota\J-Advanced\SMP\Phy\Unit No-4\Gavitation\ng\xecise.p65

3 J-Physics. Two identical satellites ae at the heights and 7 fom the ath's suface. Then which of the following statement is incoect : ( = adius of the ath) atio of total enegy of both is 5 atio of kinetic enegy of both is 4 atio of potential enegy of both 4 atio of total enegy of both is 4 and atio of magnitude of potential to kinetic enegy is 3. A satellite is seen afte each 8 hous ove equato at a place on the ath when its sense of otation is opposite to the ath. The time inteval afte which it can be seen at the same place when the sense of otation of ath & satellite is same will be: 8 hous 1 hous 4 hous 6 hous 4. Potential enegy and kinetic enegy of a two paticle system ae shown by K and P. espectively in figue. This system is bound at : Only point A Only point D e negy A B kinetic enegy potential enegy Points A, B, and C All points A, B, C and D 5. A hollow spheical shell is compessed to half its adius. the gavitational potential at the cente Inceases Deceases emains same Duing the compession inceases then etuns at the pevious value NOD6 : \Data\014\Kota\J-Advanced\SMP\Phy\Unit No-4\Gavitation\ng\xecise.p65 CHCK YOU GASP ANSW KY XCIS -1 Q u e A n s. C D D A A A B A C C A A A B B D D D C C Q u e A n s. A A C C B 7

4 J-Physics XCIS 0 BAIN TASS [MCQs] MCQs with one coect answe 1. One pojectile afte deviating fom its path stats moving ound the ath in a cicula path of adius equal to nine times the adius of ath. Its time peiod will be : g 7 g g Gavitational potential diffeence between suface of a planet and a point situated at a height of 0m above its suface is joule/kg. If gavitational field is unifom, then the wok done in taking a 5kg body of height 4 mete above suface will be : J 0J 40J 10J 3. Two concentic shells of masses M 1 and M ae having adii 1 and. Which of the following is the coect expession fo the gavitational field at a distance : G(M1 M ), fo < 1, fo 1 < < 1, fo 1 < < G(M1 M ), fo < k 4. In a cetain egion of space, the gavitational field is given by, whee is the distance and k is a constant. If the gavitational potential at = 0 be V 0, then what is the expession fo the gavitational potential (V) : k log 0 k log 0 V 0 + k log 0 M V 0 + k log 5. The potential enegy of a body of mass m is U = ax + by the magnitude of acceleation of the body will be : ab m a b m a b m a b m 6. Two metallic balls of mass m ae suspended by two stings of length L. The distance between uppe ends is. The angle at which the sting will be inclined with vetical due to attaction, is (m<<m, whee M is the mass of ath): g M Gm tan g 1 Gm tan gl 1 Gm tan g 7. Thee is a concentic hole of adius in a solid sphee of adius. Mass of the emaining potion is M. What is the gavitational potential at cente? 1 Gm tan gl If thee wee a smalle gavitational effect, which of the following foces do you think would alte in some espect. Viscous foce Achimedes uplift lectostatic foce Magnetic foce 9. Select the coect altenative : 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 ath, with a velocity less than fo a cicula obit, it will fall to the ath along a paabolic path A body of zeo total mechanical enegy placed in a gavitational field will escape the field ath's satellite must be in equatoial plane NOD6 : \Data\014\Kota\J-Advanced\SMP\Phy\Unit No-4\Gavitation\ng\xecise.p65

5 1 0. 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 Neithe gavitational field no gavitational potential is zeo inside the shell a J-Physics 1 1. Thee paticles ae pojected vetically upwad fom a point on the suface of the ath with velocities g / 3, g, 4g / 3 espectively whee is the adius of the ath and g is the acceleation due to gavity on the suface of the ath. The maximum heights attained ae espectively h 1, h, h 3. h 1 : h = : 3 h : h 3 = 3 : 4 h 1 : h 3 = 1 : 4 h = 1. When a satellite in a cicula obit aound the ath 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 ath deceases Its peiod of evolution aound the ath inceases 1 3. o a satellite to be geo stationay, which of the following ae essential conditions? It must always be stationed above the equato It must be otated fom west to east It must be about km above the ath Its obit must be cicula, and not elliptical 1 4. A cavity of adius / is made inside a solid sphee of adius. The cente of the cavity is located at a distance / fom the cente of the sphee. The gavitational foce on a paticle of mass 'm' at a distance / fom the cente of the sphee on the joining both the centes of sphee and cavity is (opposite to the cente of cavity). [Hee g = /, whee M is the mass of the sphee] mg 3mg 8 mg 16 M None of these 1 5. A tunnel is dug along a chod of the ath at a pependicula distance / fom the ath'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 : a M Pessing oce Pessing oce Acceleation Acceleation NOD6 : \Data\014\Kota\J-Advanced\SMP\Phy\Unit No-4\Gavitation\ng\xecise.p65 x=/ x= x x=/ x= x 9 x=/ x= x x=/ x= x 1 6. A double sta is a system of two stas of masses m and m, otating about thei cente of mass only unde thei mutual gavitational attaction. If is the sepaation between these two stas then thei time peiod of otation about thei cente of mass will be popotional to : 3/ m 1/ m 1/ 1 7. A solid sphee of unifom density and adius 4 units is located with its cente at the oigin O of coodinates. Two sphees of equal adii 1 unit, with thei centes at A (, 0, 0) and B (, 0, 0) espectively, ae taken out of the solid leaving behind spheical cavities as shown in figue. Then : The gavitational field due to this object at the oigin is zeo The gavitational field at the point B (, 0, 0) is zeo The gavitational potential is the same at all points of cicle y +z =36 The gavitational potential is the same at all points on the cicle y +z =4 z A y O m B x

6 J-Physics 1 8. The magnitudes of the gavitational field at distance 1 and fom the cente of a unifom, sphee of adius and mass M ae 1 and espectively. then : if 1 < and < 1 1 if 1 < and < if 1 > and > 1 1 if 1 < and < Mak the coect statement/s : Gavitational potential at cuvatue cente of a thin hemispheical shell of adius and mass M is equal to Gavitational field stength at point lying on the axis of a thin, unifom cicula ing of adius and mass M x is equal to 3 / x whee x is distance of that point fom cente of the ing Newton's law of gavitation fo gavitational foce between two bodies is applicable only when bodies have spheically symmetic distibution of mass None of these 0. Gavitational potential at the cente of cuvatue of a hemispheical bowl of adius and mass M is V : Gavitational potential at the cente of cuvatue of a thin unifom wie of mass M, bent into a semicicle of adius, is also equal to V In pat if the same wie is bent into a quate of a cicle then also the gavitational potential at the cente of cuvatue will be V In pat if the wie mass is non unifomly distibuted along its length audit is bent into a semicicle of adius, gavitational potential at the cente is V None of these 1. Suppose a smooth tunnel is dug along a staight line joining two points on the suface of the ath and a paticle is dopped fom est at its one end. Assume that mass of ath is unifomly distibuted ove its volume. Then The paticle will emege fom the othe end with velocity e e whee M e and e ae ath's mass and adius espectively The paticle will come to est at cente of the tunnel because at this position, paticle is closest to ath cente. Potential enegy of the paticle will be equal to zeo at cente of tunnel if it is along tunnel's diamete Acceleation of the paticle will be popotional to its distance fom midpoint of the tunnel. A small ball of mass 'm' is eleased at a height '' above the ath suface, as shown in the figue. If the maximum depth of the ball to which it goes is / inside the ath though a naow gove befoe coming to est momentaily. M The gove, contain an ideal sping of sping constant K and natual length, m the value of K is ( is adius of ath and M mass of ath) K 3m 3 6m m 3 7 G M m 3 3. A paticle of mass m is tansfeed fom the cente of the base of a unifom solid hemisphee of mass M and adius to infinity. The wok pefomed in the pocess by the gavitational foce exeted on the paticle by the hemisphee is m 1 m 3 m 3 m 4 NOD6 : \Data\014\Kota\J-Advanced\SMP\Phy\Unit No-4\Gavitation\ng\xecise.p65

7 J-Physics 4. Masses and adii of ath and Moon ae M 1, M and 1, espectively. The distance between thei cente is 'd'. The minimum velocity given to mass 'M' fom the mid point of line joining thei cente so that it will escape : 4G M M 1 d 4G M M 1 d M M 1 G M1 M d M M 1 G M d M 1 5. A planet is evolving aound the Sun in an elliptical obit. Its closest distance fom the Sun is min. The fathest distance fom the Sun is max. If the obital angula velocity of the planet when it is neaest to the Sun is, then the obital angula velocity at the point when it is at the fathest distance fom the Sun is HG min max I KJ HG max min I KJ max min min max 6. A satellite is in a cicula obit vey close to the suface of a planet. At some point it is given an impulse along its diection of motion, causing its velocity to incease n times. It now goes into an elliptical obit. The maximum possible value of n fo this to occu is NOD6 : \Data\014\Kota\J-Advanced\SMP\Phy\Unit No-4\Gavitation\ng\xecise.p65 BAIN TASS ANSW KY XCIS - Que A ns. B A D D C A D B C D C,D A,C ABCD B B,C Que A ns. A,D A,C,D A,B B,C A,C D D C A D B 31

8 J-Physics XCIS 03 T ue/alse 1. Gavitational foce is a consevative and cental foce.. o bounded system, potential enegy must be negative. 3. Two ai bubbles in wate attact each othe. ill in the blanks MISCLLANOUS TYP QUSTIONS 1. The escape velocity fo a planet is v e. A paticle stats fom est at a lage distance fom the planet, eaches the planet only unde gavitational attaction, and passes though a smooth tunnel though its cente. Its speed at the cente of the planet will be.... An exploatoy ocket of mass m is in obit about the Sun of mass M s at a adius 10 ( = distance of Sun fom s ath). To exit fom this obit, it fies its engine ove a shot peiod of time. This quickly doubles the velocity of ocket while its mass educes to half due to fuel consumption. Immediately afte fiing the engine, the kinetic enegy of the ocket is Two point masses each of mass m sepaated by a lage distance ae attacted towads each othe due to gavitational inteaction. If they stat fom est then speed of each mass when they ae sepaated by distance is The numeical value of the angula velocity of otation of the ath should be... ad/s in ode to make the effective acceleation due to gavity at equato equal to zeo. 5. Accoding to Keple's second law, the adius vecto to a planet fom the Sun sweeps out equal aeas in equal intevals of time. This law is consequence of the consevation of The atio of ath's obital angula momentum (about the Sun) to its mass is m /s. The aea enclosed by ath's obit appoximately... m. 7. A paticle is pojected vetically upwads fom the suface of ath (adius ) with a kinetic enegy equal to half of the minimum value needed fo it to escape. The height to which it ises above the suface of ath is... Matc h t h e C o l u m n 1. A satellite is in a cicula equatoial obit of adius 7000 km aound the ath. If it is tansfeed to a cicula obit of double the adius Column I Column II Angula momentum (p) Inceases Aea of ath coveed by satellite signal (q) Deceases Potential enegy () Becomes double Kinetic enegy (s) Becomes half s. In elliptical obit of a planet, as the planet moves fom apogee position to peigee position, Column I Column II Speed of planet (p) emains same Distance of planet fom cente of Sun (q) Deceases Potential enegy () Inceases Angula momentum about cente of Sun (s) Can not say 3. Two concentic spheical shells ae as shown in figue. : Column I Column II Potential at A (p) geate than B Gavitational field at A (q) less than B As one moves fom C to D () potential emains constant As one moves fom D to A (s) gavitational field deceases (t) None 3 B D A C NOD6 : \Data\014\Kota\J-Advanced\SMP\Phy\Unit No-4\Gavitation\ng\xecise.p65

9 J-Physics 4. Column I Column II Kinetic enegy of a paticle in (p) wok done by gavitational foce gavitational field is inceasing should be positive Potential enegy of a paticle in (q) wok done by extenal foce gavitational field is inceasing should be non zeo Mechanical enegy of a paticle () wok done by gavitational foce in gavitational field is inceasing should be negative (s) can not say anything ASSTION & ASON TYP QUSTIONS These questions contains, Statement 1 (assetion) and Statement (eason). 1. Statement 1 : Two satellites A & B ae in the same obit aound the ath, B being behind A. B cannot ovetake A by inceasing its speed. Statement : It will then go into a diffeent obit. Statement 1 is Tue, Statement is Tue ; Statement is a coect explanation fo Statement 1 Statement 1 is Tue, Statement is Tue ; Statement is not a coect explanation fo Statement 1 Statement 1 is Tue, Statement is alse. Statement 1 is alse, Statement is Tue.. Statement 1 : The acceleation of a paticle nea the ath suface diffes slightly fom the gavitational acceleation a g = /. Statement : The ath is not a unifom sphee and because the ath otates. Statement 1 is Tue, Statement is Tue ; Statement is a coect explanation fo Statement 1 Statement 1 is Tue, Statement is Tue ; Statement is not a coect explanation fo Statement 1 Statement 1 is Tue, Statement is alse. Statement 1 is alse, Statement is Tue. 3. Statement 1 : Keple's law of aeas is equivalent to the law of consevation of angula momentum. Statement : Aeal velocity da L = constant dt m NOD6 : \Data\014\Kota\J-Advanced\SMP\Phy\Unit No-4\Gavitation\ng\xecise.p65 Statement 1 is Tue, Statement is Tue ; Statement is a coect explanation fo Statement 1 Statement 1 is Tue, Statement is Tue ; Statement is not a coect explanation fo Statement 1 Statement 1 is Tue, Statement is alse. Statement 1 is alse, Statement is Tue. 4. Statement 1 : A satellite is moving in a cicula obit of the ath. If the gavitational pull suddenly disappeas, then it moves with the same speed tangential to the oiginal obit. Statement : The obital speed of a satellite inceases with the incease in adius of the obit. Statement 1 is Tue, Statement is Tue ; Statement is a coect explanation fo Statement 1 Statement 1 is Tue, Statement is Tue ; Statement is not a coect explanation fo Statement 1 Statement 1 is Tue, Statement is alse. Statement 1 is alse, Statement is Tue. 33

10 J-Physics 5. Statement 1 : A peson feels weightlessness in an atificial satellite of the ath. Howeve a peson on the Moon (natual satellite) feels his weight. Statement : Atificial satellite is a feely falling body and on the Moon suface, the weight is mainly due to Moon's gavitational attaction. Statement 1 is Tue, Statement is Tue ; Statement is a coect explanation fo Statement 1 Statement 1 is Tue, Statement is Tue ; Statement is not a coect explanation fo Statement 1 Statement 1 is Tue, Statement is alse. Statement 1 is alse, Statement is Tue. 6. Statement 1 : Moon cannot be used as a satellite fo communication. Statement : Moon doesn't move in the equatoial plane of the ath. Statement 1 is Tue, Statement is Tue ; Statement is a coect explanation fo Statement 1 Statement 1 is Tue, Statement is Tue ; Statement is not a coect explanation fo Statement 1 Statement 1 is Tue, Statement is alse. Statement 1 is alse, Statement is Tue. 7. Statement 1 : Moon has no atmosphee. Statement : Due to less gavity Moon is unable to etain its atmosphee. Statement 1 is Tue, Statement is Tue ; Statement is a coect explanation fo Statement 1 Statement 1 is Tue, Statement is Tue ; Statement is not a coect explanation fo Statement 1 Statement 1 is Tue, Statement is alse. Statement 1 is alse, Statement is Tue. 8. Statement 1 : Pendulum clock stops woking on the spaceship. Statement : Pendulum of the pendulum clock falls down on the spaceship. Statement 1 is Tue, Statement is Tue ; Statement is a coect explanation fo Statement 1 Statement 1 is Tue, Statement is Tue ; Statement is not a coect explanation fo Statement 1 Statement 1 is Tue, Statement is alse. Statement 1 is alse, Statement is Tue. 9. Statement 1 : scape velocity is independent of the angle of pojection. Statement : scape velocity fo vetical pojection fom the suface of ath is Statement 1 is Tue, Statement is Tue ; Statement is a coect explanation fo Statement 1 Statement 1 is Tue, Statement is Tue ; Statement is not a coect explanation fo Statement 1 Statement 1 is Tue, Statement is alse. Statement 1 is alse, Statement is Tue Statement 1 : The plane of the obit of an atificial satellite must contain the cente of the ath. Statement : o an atificial satellite, the necessay centipetal foce is povided by gavity. Statement 1 is Tue, Statement is Tue ; Statement is a coect explanation fo Statement 1 Statement 1 is Tue, Statement is Tue ; Statement is not a coect explanation fo Statement 1 Statement 1 is Tue, Statement is alse. Statement 1 is alse, Statement is Tue. 34. NOD6 : \Data\014\Kota\J-Advanced\SMP\Phy\Unit No-4\Gavitation\ng\xecise.p65

11 J-Physics 1 1. Statement 1 : When a planet appoaches the point which is fathest fom the Sun, its obital speed deceases. Statement : Wok done on the planet by the gavitational foce exeted by the Sun is negative. Statement 1 is Tue, Statement is Tue ; Statement is a coect explanation fo Statement 1 Statement 1 is Tue, Statement is Tue ; Statement is not a coect explanation fo Statement 1 Statement 1 is Tue, Statement is alse. Statement 1 is alse, Statement is Tue. COMPHNSION TYP QUSTIONS Compehension # 1 When a paticle is pojected fom the suface of ath, its mechanical enegy and angula momentum about cente of ath at all time ae constant. 1. A paticle of mass m is pojected fom the suface of ath with velocity v 0 at angle with hoizontal. Suppose h be the maximum height of paticle fom suface of ath and v its speed at that point then v is : v 0 cos > v 0 cos < v 0 cos Zeo. Maximum height h of the paticle is : = v 0 sin g > v 0 sin g < v 0 sin g Can be geate than o less than v 0 sin g Compehension # A solid sphee of mass M and adius is suounded by a spheical shell of same mass M and adius as shown. A small paticle of mass m is eleased fom est fom a height h (<<) above the shell. Thee is a hole in the shell. 1. In what time will it ente the hole at A : m A B h NOD6 : \Data\014\Kota\J-Advanced\SMP\Phy\Unit No-4\Gavitation\ng\xecise.p65 h h. What time will it take to move fom A to B? h h 3. With what appoximate speed will it collide at B? 35 h h 3 None of these None of these

12 J-Physics Compehension # 3 Two stas bound togethe by gavity obit each othe because of thei mutual attaction. Such a pai of stas is efeed to as a binay sta system. One type of binay system is that of a black hole and a companion sta. The black hole is a sta that has collapsed on itself and is so missive that not even light ays can escape its gavitational pull. Theefoe when descibing the elative motion of a black hole and a companion sta, the motion of the black hole can be assumed negligible compaed to that of the companion. The obit of the companion sta is eithe elliptical with the black hole at one of the foci o cicula with the black hole at the cente. The gavitational potential enegy is given by U = GmM/, whee G is the univesal gavitational constant, m is the mass of the companion sta, M is the mass of the block hole, and is the distance between the cente of the companion sta and the cente of the black hole. Since the gavitational foce is consevative, the companion sta's total mechanical enegy is a constant. Because of the peiodic natue of the obit, thee is a simple elation between the aveage kinetic enegy <K> of the companion sta and its aveage potential enegy <U>. In paticula, <K> = < U/>. Two special points along the obit ae single out by astonomes. Paigee is the point at which the companion sta is closest to the black hole, and apogee is the point at which is the fathest fom the black hole. 1. o cicula obits the potential enegy of the companion sta is constant thoughout the obit. If the adius of the obit doubles, what is the new value of the velocity of the companion sta? It is 1/ of the old value It is 1/ of the old value It is the same as old value It is double the old value. Which of the following pevents the companion sta fom leaving its obit and falling into the black hole? The centipetal foce The gavitational foce MISCLLANOUS TYP QUSTION ANSW KY XCIS -3 T ue / alse 1. T. 3. T ill in the Blanks v. e 10 sm S angula momentum h= Match the Column 1. pp p,s q,s. q q p 3. q t s 4. s q Assetion - eason A A A C A A A C Compehension based B A C Compehension #1 : 1. C. B Compehension # : 1. A. C 3. D Compehension #3: 1. B. D 36 NOD6 : \Data\014\Kota\J-Advanced\SMP\Phy\Unit No-4\Gavitation\ng\xecise.p65

13 J-Physics XCIS 04 [A] CONCPTUAL SUBJCTIV XCIS 1. In the given figue, L = 1 mete, if total gavitational foce on 4 kg mass is 1 and on kg mass is, then find out the atio of 1.. Two point like objects, each with mass m, ae connected by a massless ope of length. The objects ae suspended vetically nea the suface of ath, so that one object is hanging below the othe. Then the m objects ae eleased. Show that the tension in the ope is T whee M is the mass of the ath and 3 is its adius. [ <<] 3. A solid sphee of unifom density and adius applies a gavitational foce of attaction equal to 1 on a paticle placed at P, distance fom the cente of the sphee. A spheical cavity of adius / is now made in the sphee as shown O P in figue. The sphee with cavity now applies a gavitational foce on the same paticle placed at P. ind the atio of / The gavitational field in a egion is given by (3i ˆ 4ˆj) N/kg. ind out the wok done (in joule) in displacing a paticle by 1 m, along the line 4y = 3x + 9 : 5. The ath may be egaded as a spheically shaped unifom coe of density and adius suounded by a unifom shell of thickness and density. ind the atio of 1 if the value of acceleation due to gavity is the same at suface as at depth fom the suface. 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 paticle of mass 1 kg is placed at a distance of 4 m fom the cente and on the axis of a unifom ing of mass 5 kg NOD6 : \Data\014\Kota\J-Advanced\SMP\Phy\Unit No-4\Gavitation\ng\xecise.p65 and adius 3 m. Calculate the wok done to incease the distance of the paticle fom 4 m to 3 3 m. 8. ind the potential enegy of a system of fou paticles placed at the vetices of a squae of side. Also obtain the potential at the cente of the squae. 9. ind the potential enegy of a system of eight paticles placed at the vetices of a cube of side L. Neglect the self enegy of the paticles ind the potential enegy of the gavitational inteaction of a point mass of mass m and a thin unifom od of mass M and length, if they ae located along a staight line such that the point mass is at a distance 'a' fom one end. Also find the foce of thei inteaction. 37

14 J-Physics 1 1. A body of mass m be pojected vetically upwad fom the suface of the ath so as to each a height n above the ath s suface. Calculate : (i) The incease in its potential enegy (ii) The velocity with which the body must be pojected 1. Two small dense stas otate about thei common cente of mass as a binay system with the peiod 1 yea fo each. One sta is of double the mass of the othe and the mass of the lighte one is 1 3 of the mass of the Sun. ind the distance between the stas if distance between the ath & the Sun is A comet obits the Sun in a highly elliptical obit. Does the comet have a constant (i) linea speed, (ii) angula speed, (iii) angula momentum, (iv) kinetic enegy, (v) potential enegy, (vi) total enegy thoughout its obit? Neglect any mass loss of the comet when it comes vey close to the Sun An atificial satellite is moving in a cicula obit aound the ath with a speed equal to half the magnitude of escape velocity fom the ath. (i) (ii) Detemine the height of the satellite above the ath's suface. If the satellite is stopped suddenly in its obit and allowed to fall feely on the ath, find the speed with which it hits and suface of ath. Given M = mass of ath & = adius of ath CONCPTUAL SUBJCTIV XCIS ANSW KY X C IS - 4 ( A ) Gm n 11. (i) mg n 1 (ii), 4 Gm n g n Joule G M L All quantities vay ove an obit except angula momentum and total enegy 14. (i) 6400 km (ii) 7.9 km/s 10. m log e 1 a NOD6 : \Data\014\Kota\J-Advanced\SMP\Phy\Unit No-4\Gavitation\ng\xecise.p65

15 J-Physics XCIS 04 [B] BAIN STOMING SUBJCTIV XCIS 1. A small satellite evolves aound a heavy planet in a cicula obit. At cetain point in its obit a shap impulse acts on it and instantaneously inceases its kinetic enegy to 'k' (<) times without change in its diection of motion. Show that in its subsequent motion the atio of its maximum and minimum distances fom the planet is k, assuming the mass of the satellite is negligibly small as compaed to that of the planet. k. A satellite of mass m is in an elliptical obit aound the ath of mass M (M>>m). The speed of the satellite at 6 its neaest point to the ath (peigee) is whee = its closest distance to the ath. It is desied to 5 tansfe this satellite into a cicula obit aound the ath of adius equal its lagest distance fom the ath. ind the incease in its speed to be impated at the apogee (fathest point on the elliptical obit). 3. A body is launched fom the ath's suface a angle 0 30 to the hoizontal at a speed v 0 = 1.5 Neglecting ai esistance and ath's otation, find (i) the height to which the body will ise. (ii) The adius of cuvatue of tajectoy at its top point.. 4. Distance between the centes of two stats is 10a. The masses of these stas ae M and 16M and thei adii a and a espectively. A body of mass m is fied staight fom the suface of the lage sta towads the suface of 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. 5. The minimum and maximum distances of a satellite fom the cente of the ath ae and 4 espectively, whee is the adius of ath and M is the mass of the ath. ind adius of cuvatue at the point of minimum distance. 6. A paticle takes n seconds less and acquies a velocity u m/sec. highe at one place than at anothe place in falling though the same distance. Calculate the poduct of the acceleation due to gavity at these two places. 7. A satellite of mass M s is obiting the ath in a cicula obit of adius s. It stats losing enegy slowly at a constant ate C due to fiction. If M e and e denote the mass and adius of the ath espectively, show that the satellite falls on the ath in a limit time t given by SM t C e 1 1 e S NOD6 : \Data\014\Kota\J-Advanced\SMP\Phy\Unit No-4\Gavitation\ng\xecise.p65 8. A ocket stats vetically upwads with speed v 0. Show that its speed v at a height h is given by v0 v gh / 1 39 Whee is the adius of the ath. Hence deduce the maximum height eached by a ocket fied with speed equal to 90% of escape velocity. 9. A cod of length 64 m is used to connect a 100 kg astonaut to spaceship whose mass is much lage than that of the astonaut. stimate the value of the tension in the cod. Assume that the spaceship is obiting nea ath suface. Assume that the spaceship and the astonaut fall on a staight line fom the ath cente. The adius of the ath is 6400 km. h

16 J-Physics 1 0. A emote sensing satellite is evolving in an obit of adius x on the equato of ath. ind the aea on ath suface in which satellite can not send message Calculate the distance fom the suface of the ath at which above and below the suface, acceleation due to gavity is the same. 1. A body moving adially away fom a planet of mass M, when at distance fom planet, explodes in such a way that two of its many fagments move in mutually pependicula cicula obits aound the planet. What will be (i) Then velocity in cicula obits? (ii) Maximum distance between the two fagments befoe collision and (iii) Magnitude of thei elative velocity just befoe they collide? 1 3. A satellite P is evolving aound the ath at a height h= adius of ath () above equato. Anothe satellite Q is at a height h evolving in opposite diection. At an instant the two ae at same vetical line passing though cente of sphee. ind the least time of afte which again they ae in this situation. ath M P Q 1 4. The fastest possible ate of otation of a planet is that fo which the gavitational foce on mateial at the equato baely povides the centipetal foce needed fo the otation. (Why?) (i) Show then that the coesponding shotest peiod of otation is given by T = (ii) of the planet, assumed to be homogeneous G whee is the density valuate the otation peiod assuming a density of 3.0 gm/cm 3, typical of many planets, satellites, and asteoids. No such object is found to be spinning with a peiod shote than found by this analysis In a paticula double sta system, two stas of mass kg each evolve about thei common cente of mass, m away. (i) (ii) Calculate thei common peiod of evolution, in yeas. Suppose that a meteooid (small solid paticle in space) passes though this cente of mass moving at ight angles to the obital plane of the stas. What must its speed be if it is to escape fom the gavitational field of the double sta? 1 6. Assume that a tunnel is dug acoss the ath (adius =) passing though its cente. ind the time a paticle takes to each cente of ath if it is pojected into the tunnel fom suface of ath with speed needed fo it to escape the gavitational field of ath Ajay can thow a ball at a speed on ath which can coss a ive of width 10 m. Ajay eaches on an imaginay planet whose mean density is twice of the ath. ind out the maximum possible adius of planet so that if Ajay thows the ball at same speed it may escape fom planet. Given adius of ath = m: 1 8. A paticle takes a time t 1 to move down a staight tunnel fom the suface of ath to its cente. If gavity wee to emain constant this time would be t. Calculate the atio t t 1 NOD6 : \Data\014\Kota\J-Advanced\SMP\Phy\Unit No-4\Gavitation\ng\xecise.p65

17 J-Physics 1 9. A satellite close to the ath is in obit above the equato with a peiod of otation of 1.5 hous. If it is above a point P on the equato at some time, it will be above P again afte time Conside two satellites A and B of equal mass m, moving in the same cicula obit of adius aound the ath but in opposite sense of otation and theefoe on a collision couse (see figue). A B M e (i) (ii) (iii) In tems of G, M e, m and find the total mechanical enegy A + B of the two satellite plus ath system befoe collision. If the collision is completely inelastic so that weckage emains as one piece of tangled mateial (mass =m), find the total mechanical enegy immediately afte collision. Descibe the subsequent motion of the weckage. BAIN STOMING SUBJCTIV XCIS ANSW KY X C I S - 4 ( B ) (i) h= 7 1, (ii) a u n NOD6 : \Data\014\Kota\J-Advanced\SMP\Phy\Unit No-4\Gavitation\ng\xecise.p N (i) 15. (i) T (i) GmM e (ii) (iii) 3 Gm (ii) v= 4Gm 1 x x 41 3 / , = d T sin hous if it is otating fom west to east, 4 17, (ii) GmM e (iii) fee fall e g (ii) 1.9 h Km hous if it is otating east to west.

18 J-Physics XCIS If suddenly the gavitational foce of attaction between eath and a satellite evolving aound it becomes zeo, then the satellite will- [AI - 00] (1) continue to move in its obit with same velocity () move tangentially to the oiginal obit with same velocity (3) become stationay in its obit (4) move towads the eath. The kinetic enegy needed to poject a body of mass m fom the eath's suface (adius ) to infinity is- [AI - 00] (1) mg () mg (3) mg (4) mg 4 3. negy equied to move a body of mass m fom an obit of adius to 3 is- [AI - 00] (1) m/1 () m/3 (3) m/8 (4) m/6 4. The time peiod of a satellite of eath is 5 hous. If the sepaation between the cente of eath and the satellite is inceased to 4 times the pevious value, the new time peiod will become- [AI - 003] (1) 10 h () 80 h (3) 40 h (4) 0 h 5. Two spheical bodies of mass M and 5M and adii and espectively ae eleased in fee space with initial sepaation between thei centes equal to 1. If they attact each othe due to gavitational foce only, then the distance coveed by the smalle body just befoe collision is- [AI - 003] (1).5 () 4.5 (3) 7.5 (4) A satellite of mass m evolves aound the 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- [AI - 004] (1) gx () g x (3) g x PVIOUS YA QUSTIONS (4) 1 / g x 7. The time peiod of an eath satellite in cicula obit is independent of- [AI - 004] (1) the mass of the satellite () adius of its obit (3) both the mass and adius of the obit (4) neithe the mass of the satellite no the adius of its obit 8. 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- [AI - 004] (1) mg () 1 mg (3) 1 mg (4) mg 4 9. Suppose the gavitational foce vaies invesely as the n th powe of distance. Then the time peiod of a planet in cicula obit of adius aound the sun will be popotional to- [AI - 004] (1) n 1 () n 1 (3) n (4) 4 n 1 0. Aveage density of the eath- [AI - 005] (1) does not depend on g () is a complex function of g (3) is diectly popotional to g (4) is invesely popotional to g 1 1. 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? [AI ] (1) d = h () d = 3h (3) d = h (4) d = h NOD6 : \Data\014\Kota\J-Advanced\SMP\Phy\Unit No-4\Gavitation\ng\xecise.p65

19 J-Physics 1. A paticle of mass 10 g is kept on the suface of a unifom sphee of mass 100 kg and adius 10 cm. ind 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 ) [AI - 005] (1) J () J (3) J (4) J 1 3. If g 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 expeiment could be pefomed on the two sufaces, one will find the atio electonic ch ag e on the moon to be- [AI - 007] electonic ch ag e on the eath (1) 1 () zeo (3) g /g M (4) g M /g 1 4. A planet in a distance sola system is 10 times moe massive than the eath and its adius is 10 times smalle. Given that the escape velocity fom the eath is 11 km s -1, the escape velocity fom the suface of the planet would be [AI - 008] (1) 1.1 km s -1 () 11 km s -1 (3) 110 km s -1 (4) 0.11 km s This quqestion contains statement-1 and statement - of the fou choices given afte the statements, choose the one that best descibes the two statements. [AI - 008] Statement 1: o a mass M kept at the cente of a cube of side a, the flux of gaviational field passing though its sides is 4. Statement : If the diection of a field due to a point souce is adial and its dependence on the distance I fom the souce is given as, its flux though a closed suface depends only on the stength of the souce enclosed by the suface and not on the size o shape of the suface. (1) Statement 1 is false, Statement is tue () Statement 1 is tue, Statement is tue; Statement is a coect explanation fo Statement 1 (3) Statement 1 is tue, Statement is tue; Statement is not a coect explanation fo Statement 1 (4) Statement 1 is tue, Statement is false 1 6. The height at which the acceleation due to gavity becomes g 9 (whee g = the acceleation due to gavity on the suface of the eath) in tems of, the adius of the eath, is :- [AI - 009] (1) () (3) (4) 1 7. Two bodies of masses m and 4m ae placed at a distance. The gavitational potential at a point on the line joining them whee the gavitational field is zeo is :- [ AI - 011] (1) 6Gm () 9Gm (3) zeo (4) 4Gm 1 8. Two paticles of equal mass 'm' go aound a cicle of adius unde the action of thei mutual gavitational attaction. The speed of each paticle with espect to thei cente of mass is:- [ A I ] (1) Gm () Gm 4 (3) Gm 3 (4) Gm NOD6 : \Data\014\Kota\J-Advanced\SMP\Phy\Unit No-4\Gavitation\ng\xecise.p The mass of a spaceship is 1000 kg. It is to be launched fom the eath's suface out into fee space. The value of 'g' and '' (adius of eath) ae 10 m/s and 6400 km espectively. The equied enegy fo this wok will be :- [ A I ] (1) Joules () Joules (3) Joules (4) Joules ANSW-KY Que Ans

20 J-Physics XCIS 05 PVIOUS YA QUSTIONS MCQs wi th one coect answe 1. A satellite S is moving in an elliptical obit aound the ath. The mass of the satellite is vey small compaed to the mass of the ath : [ IIT-J 1998] The acceleation of S always diected towads the cente of the ath The angula momentum of S about the cente of the ath changes in diection, but its magnitude emain constant The total mechanical enegy of S vaies peiodically with time The linea momentum of S emains constant in magnitude. A geostationay satellite obits aound the ath in a cicula obit of adius 36,000 km. Then, the time peiod of a spy satellite obiting a few hunded km (600 km) above the ath's suface ( e = 6400 km) will appoximately be : [ IIT-J 00] 1 h 1h h 4h 3. 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 : [IIT-J 006] If T A > T B then A > B If T A > T B then M A > M B B 3 T A A T B T A = T B 4. Two bodies, each of mass M, ae kept fixed with a sepaation L. A paticle of mass m is pojected fom the midpoint of the line joining thei centes, pependicula to the line. The gavitational constant is G. The coect statement(s) is (ae) :- [ IIT-J 013] The minimum initial velocity of the mass m to escape the gavitational field of the two bodies is 4 L The minimum initial velocity of the mass m to escape the gavitational field of the two bodies is L The minimum initial velocity of the mass m to escape the gavitational field of the two bodies is L The enegy of the mass m emains constant 44 NOD6 : \Data\014\Kota\J-Advanced\SMP\Phy\Unit No-4\Gavitation\ng\xecise.p65

21 J-Physics Assetion-eason 1. Statement 1 : An astonaut in an obiting space station above the ath expeiences weightlessness. Statement : An object moving aound the ath unde the influence of ath s gavitational foce is in a state of fee-fall. Statement-1 is Tue, Statement- is Tue; Statement- is a coect explanation fo Statement-1 Statement-1 is Tue, Statement- is Tue; Statement- is Not a coect explanation fo Statement-1 Statement-1 is Tue, Statement- is alse [ IIT-J 008] Statement-1 is alse, Statement- is Tue Subjective Questions 1. Thee is a cate of depth 100 on the suface of the Moon (adius ). A pojectile is fied vetically upwad fom the cate with velocity, which is equal to the escape velocity v fom the suface of the Moon. ind the maximum height attained by the pojectile. NOD6 : \Data\014\Kota\J-Advanced\SMP\Phy\Unit No-4\Gavitation\ng\xecise.p65 PVIOUS YAS QUSTIONS XCIS -5 MCQ's One coect answes 1. A. C 3. D 4. B,D Asseti on - eason Questions 1. A Subjective Questions