Title 02A.GRAVITATION (57 - 75).pdf ✅

NISHITH Multimedia India (Pvt.) Ltd., 5 7 GRAVITATION NISHITH Multimedia India (Pvt.) Ltd., JEE - ADVANCED - VOL-III GRAVITATION LEVEL -V SINGLE ANSWER TYPE QUESTIONS 1. A solid sphere of radius R/ 2 is cut out of a solid sphere of radius R such that the spherical cavity so formed touches the sur- face on one side and the centre of the sphere on the other side, as shown. The initial mass of the solid sphere was M . If a particle of mass m is placed at a distance 2.5R from the centre of the cavity, then what is the gravitational attraction on the mass m ?   2.5R m R O R / 2 O A) 2 GMm R B) 2 2 GMm R C) 2 8 GMm R D) 2 23 100 GMm R 2. Two sphere of masses m and M are situ- ated in air and gravitational force between them is F . The space around the mass is now filled with a liquid of specific gravity 3. The gravitational force now will be A) F /9 B) 3F C) F D) F /3 3. A spherical hollow is made in a lead sphere of radius R such that its surface touches the outside surface of the lead sphere and passes through the centre. The mass of the lead sphere before hollowing was M . The force of attraction that this sphere would exert on a particle of mass m which lies at a distance d from the centre of the lead sphere on the straight line joining the cen- tre of the sphere and the follow is A) 2 GMm d B) 2 2 1 1 8 1 2 GMm d R d                     C) 2 2 1 1 8 1 2 GMm d R d                     D) 2 8 GMm d 4. A mass m is placed in the cavity inside a hollow sphere of mass M as shown in the figure. What is the gravitational force on mass m ?   R r m A) 2 GMm R B) 2 GMm r C)   2 GMm R r  D) zero 5. A solid sphere of uniform density and radius R appiles a gravitational force of attraction equal to F1 on a particle placed at a distance 2R from the centre of the sphere. A spheri- cal cavity of radius R/ 2 is now made in the sphere as shown in the figure. The sphere with the cavity now applies a gravitational force F2 on the same particle. The ratio 1 2 F F/ is:  R A) 1 2 B) 3 4 C) 7 8 D) 9 7
Jr Chemistry E/M GRAVITATION 5 8 NISHITH Multimedia India (Pvt.) Ltd., JEE - ADVANCED - VOL-III NISHITH Multimedia India (Pvt.) Ltd., 6. The following figure shows two shells of masses m1 and m2 . The shells are concen- tric. At which point, a particle of mass m shall experience zero force? m1 m2  D C B A A) A B) B C) C D) D 7. Consider two solid uniform spherical objects of the same density  . One has radius R and the other has radius 2R . They are in outer space where the gravitational fields from other objects are negligible. If they are arranged with their surface touching, what is the contact force between the objects due to their gravitational attraction? A) 2 4 G R  B) 128 2 4 2 81 G R   C) 128 2 81 G D) 128 2 2 87  R G 8. A thin rod of length L is bent to form a circle. Its mass is M . What force will act on the mass m placed at the centre of the circle? A) 2 2 4 GMm L  B) 2 2 4 GMm  L C) 2 2 GMm L  D) zero 9. Three particles each of mass M are placed at the three corners of an equilateral triangle of side l . What is the force due to this sys- tem of particles on another particle of mass m placed at the midpoint of any side? A) 2 3 4 GMm l B) 2 4 3 GMm l C) 2 4 GMm l D) 2 4GMm l 10. A solid sphere of uniform density and mass M has radius 4m . Its centre is at the origin of the coordinate system. Two spheres of radii 1 m are taken out so that their centres at P0, 2,0   and Q0,2,0 respectively. This leaves two spherical cavities. What is the gravitational field at the origin of the co- ordinate axes?    P O Q Y Z X A) 31 1024 GM B) 1024 Gm C) 31GM D) zero 11. Gravitational field at the centre of a semi- circle formed by a thin wire AB of mass m and length l is O y A B x l A) 2 Gm l along x axis B) 2 Gm l along y axis C) 2 2 Gm l  along x axis D) 2 2 Gm l  along y axis
NISHITH Multimedia India (Pvt.) Ltd., 5 9 GRAVITATION NISHITH Multimedia India (Pvt.) Ltd., JEE - ADVANCED - VOL-III 12. Four similar particles of mass m are orbit- ing in a circle of radius r in the same angu- lar direction because of their mutual gravi- tational attractive forces. Velocity of a par- ticle is given by      r m m m m A) 1 2 1 2 2 4 Gm r                B) 3 Gm r C) 1 2 2  Gm r  D) 1 2 1 1 2 2 2 2 Gm r              13. The gravitational potential due to earth at infinite distance from it is zero. Let the gravi- tational potential at a point P be 1 5 J kg   . Suppose, we arbitrarily assume the gravita- tional potential at infinity to be 1 10 J kg   , then the gravitational potential at P will be A) 1 5 J kg   B) 1 5 J kg   C) 1 15 J kg   D) 1 15 J kg   14. A body starts from rest from a point distant 0 r from the centre of the earth. It reaches the surface of the earth whose radius is R . The velocity acquired by the body is A) 0 1 1 2GM R r  B) 0 1 1 2GM R r        C) 0 1 1 GM R r  D) 0 1 1 GM R R        15. A smooth tunnel is dug along the radius of the earth that ends at the centre. A ball is released from the surface of earth along the tunnel if the coefficient of restitution is 0.2 between the surface and ball then the dis- tance travelled by the ball before second collision at the centre. A) 6 5 R B) 7 5 R C) 9 5 R D) 3 2 R 16. Two rings having masses M and 2M , re- spectively, having same radius are placed coaxially as shown in figure. 3R    R P If the mass distribution on both the rings is non-uniform, then gravitational potential at point P is A) 1 2 2 5 GM R         B) 2 1 2 GM R         C) zero D) cannot be determined from given information 17. A point P lies on the axis of a fixed ring of mass M and radius R , at a distance 2R from its centre O. A small particle starts from P and reaches O under gravitational attraction only. Its speed at O will be: A) 2 1 1 5 GM R        B) 2GM R C)   2 5 1 GM R  D) zero 18. Two cylindrical vessels of equal cross-sec- tional area contain water upto heights 1 h and 2 h . The vessels are inter connected so that the levels in them become equal. Calcualte the work done by the force of gravity during this process if  is the density of water: A) 2 1 2 2 h h  A g        B) 2 1 2 2 h h  A g        C) 2 2 1 Ah g h  D) 2 1 2 Ah g h  19. Gravitational potential at the centre of curva ture of a hemispherical bowl of radius R and mass M is V: A) Gravitational potential at the centre of c u r - vature of a thin uniform wire of mass M, be nt into a semicircle of radius R, is also equal to V B) In part (a) if the same wire is bent into a quar- ter circle then also the gravitational potential at the centre of curvature will be V C) In part (a) if the wire mass is non uniformly distributed along its length and it is bent intosemicircle of radius R, gravitational potential at the centre is 2V D) None of these 20. The gravitational field due to a mass
Jr Chemistry E/M GRAVITATION 6 0 NISHITH Multimedia India (Pvt.) Ltd., JEE - ADVANCED - VOL-III NISHITH Multimedia India (Pvt.) Ltd., distribution is given by 3 E K x  / in x  direction. Taking the gravitational potential to be zero at infinity, its value at distance x is: A) 2 2K x B) 2 2 K x C) 2 K x D) 2 3 2 K x 21. An artificial satellite of earth is launched in circular orbit in equatorial plane of the earth and satellite is moving from west to east. With respect to a person on the equator, the satellite is completing one round trip in 24 h. Mass of earth is, 24 M kg  6 10 . For this situation, orbital radius of the satellite is: A) 4 2.66 10  km B) 6400 km C) 36,000 km D) 29,600 km 22. A wreckage of mass m present in space, collides with a satellite of mass 10 m and orbital radius R as shown.  v m R / 2 R As a result of collision, the wreckage sticks to satellite and satellite is transferred to an orbit whose minimum distance from the planet is 2 R . Take mass of earth of mass M. The velocity of the wreckage with which it collides to the satellite is A) GM R B) 58GM R C) 63GM R D) 59GM R 23. A satellite is orbiting around earth in a cir- cular orbit of radius r . A particle of mass m is projected from satellite in forward di- rection with velocity 2 3 v  times orbital velocity (this velocity is given with respect to earth). During subsequent motion of the particle, itsminimum distance from the cen- tre of earth is A) 2 r B) r C) 2 3 r D) 4 5 r 24. The satellite is moving in an elliptical orbit about the earth as shown in figure:   S E P Q  The minimum and maximum distance of sat- ellite from earth are 3 units and 5 units, re- spectively. The distance of satellite from the earth when it is at P is equal to A) 4 units B) 3 units C) 3.75 units D) none of these 25. An exploratory rocket of mass m is in or- bit about the sun at a radius of /10 RES (one tenth of the radius of the earth’s orbit about the sun). To exit this orbit, it fires its engine over a short period of time. This quickly doubles the velocity of the rocket while halv- ing its mass (due to fuel consumption). Im- mediately after the burn, what is the kinetic energy of the rocket? Take mass of sun as MS : A) 2 S ES GM m R B) 10 S ES GM m R C) 20 S ES GM m R D) 5 S ES GM m R 26. A shell is fired vertically from the earth with speed / esc v N , where N is some number greater than one and esc v is escape speed on the earth. Neglecting the rotation of the earth and air resistance, the maximum alti- tude attained by the shell will be (RE is ra- dius of the earth): A) 2 1 RE N  B) 2 RE N C) 2 1 NRE N  D) 2 2 1 N RE N 