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Jr Chemistry E/M GRAVITATION 7 6 NISHITH Multimedia India (Pvt.) Ltd., JEE - ADVANCED - VOL-III NISHITH Multimedia India (Pvt.) Ltd., LEVEL-VI SINGLE ANSWER QUESTIONS 1. The figure shows a uniform rod of length l whose mass per unit length is  . What is the gravitational force of the rod on a par- ticle of mass m located a distance d from one end of the rod, as shown in the figure ? A)   Gm l d l d   B)   2 2 2 Gm l d l d   C)   2 Gm l d l d   D)   2 2 2 Gm l d l d   2. A uniorm rod of mass M and length L lies along the x axis with centre at the origin. Consider an element of length dx at dis- tance x from the origin. Show that this ele- ment produces a gravitational field at a point on the acis 0 x ( 0 x grater then 1 2 L ) given by     2 0 GMdx L x x   . Integrate this result over the rod to find the total gravitational field at the point 0 x due to the rod. A) 0 2 2 0 4 x GM E L x          B) 0 2 2 2 4 x GM E L x          C) 0 2 2 0 4 x GM E L x          D) 0 2 0 4 x GM E L x          3. A satellite of mass m is moving in a circu- lar orbit of radius r . Due to atmospheric drag if it loses energy at a constant rate W . Find the time in which satellite will fall to the surface of earth. the mass of the earth is M and radius R. A) 2 1 1 2 GMm W R r        B) 2 2 1 1 2 GMm W R r        C) 2 2 1 1 2 GMm W R r        D) 1 1 2 GMm W R r        4. Find the potential energy of the gravitational interaction of a point mass m and lengh l . if they are acted along a straight line at a distance a from each other. a m A) 2 2 ln 2 Gm a l U l a l           B) 2 2 ln 2 Gm a l U l a l           C) 2 2 2 ln 2 Gm a l U l a l           D) 2 2 2 2 ln 2 Gm a l U l a l           5. Find the internal potential energy (Self - energy) of matter forming (a) a thin uniform sphrical shell of mass M and radius R. (b) a uniform solid sphere of mass M and radius R. A)     2 2 3 2 5 Gm Gm a U b R R    B)     2 3 0 5 Gm a U b R   C)     2 2 3 2 2 3 Gm Gm a U b R R    D)     2 1 0 4 Gm a b U R  
NISHITH Multimedia India (Pvt.) Ltd., 7 7 GRAVITATION NISHITH Multimedia India (Pvt.) Ltd., JEE - ADVANCED - VOL-III 6. Mass M is distributed uniformly along a line of length 2L. A particle of mas m is at a point that is a distance a above the centre of the line on its perpendiculr bisector (Point P in figure). The gravitational force that the line exerts on the particle is P  L L M a A) 2 2 GMm L a  B)   2 2 GMm a L a  C) 2 2 GMm a L a  D)   2 2 2 GMm a L a  7. A planet of mass m moves along an ellipse around the sun so that its maximum and minimum distances from the sun are equal to 1 r and 2 r respectively. Find the aqnglular momentum of this planet relative to the cen- tre of the sum. Mass of the sun is M. A)   1 2 2 1 2 2GMr r m r r  B)   1 2 1 2 2GMr r m r r  C)   2 2 1 2 2 2GMr r m r r  D)   2 1 2 2 2GMr r m r r  8. Inside a uniform shere of density  there is a spherical cavity whose centre is at a dis- tance l from the centre of the sphere. Find the strengh of the gravitational field inside the cavity. A) 2 3 E G l     B) 4 3 E G l     C) 4 2 3 E G l     D) 4 2 2 3 E G l     9. The density inside a solid sphere of radius R is given by 0R r    where 0 is the den- sity at the surface and r is the distance from the centre. find the gravitational field due to this sphere at a distance 2R from its cen- tre. A) 0 4   G R B) 0 3   G R C) 2 0 2   G R D) 0 2   G R 10. Inside a fixed sphere of radius R and uni- form density  , there is spherical cavity of radius 2 R such that surface of the cavity passes through the centre of the sphere as hown in figure. A particle of mass m is re- leased from rest at centre B of the cavity. Calculate velocity with which particle strikes the centre A of the sphere. Neglect earth’s gravity. Initially sphere and particle are at rest.  B A R R / 2 A) 2 2 PR 3 G B) 2 2 2 3 GP R C) 2 2 5 GPR D) 2 2 2 2 2 3  G P R 11. A ring of radius R m  4 is made of a highly dense meterial. Mass of the ring is 9 1 m kg   5.4 10 distributed uniformly over uts circumference. a highly dense partice of mass 8 2 m kg  6 10 is placed on the axis of he ring at a distance 0 x  3 m from the centre. Neglecting all other forces, except mutual gravitational interaction of the two, calculate (i) displacement of the ring when particle is at the centre of ring and (ii) speed of the paricle at that instant. A)    1 i m ii cms 0.4 16  B) i m ii cm  0.3 18 / s   C)i m ii cm  0.2 12 / s   D)i m ii cm  0.6 24 / s   12. A cosmic body A moves to the sun with ve- locity 0 v (when far from the sun) and aiming parameter l the arm of the vector 0 v rela- tive to the centre of the sun . Find the mini- mum distance by which this body will get to the sun. Mass of the sum is M. A) 2 2 0 2 0 1 1 GM lv v GM                 B) 2 0 1 GM v  C) 2 2 0 2 0 1 1 GM lv v GM               D) 2 0 GMlv 1
Jr Chemistry E/M GRAVITATION 7 8 NISHITH Multimedia India (Pvt.) Ltd., JEE - ADVANCED - VOL-III NISHITH Multimedia India (Pvt.) Ltd., 13. Two satellites 1 S and 2 S revolve around a planet in coplanar circular orbits in the opposit sense. The periods of revolutions are T and T rspectively. Find the angular speed of 2 S as observed by an astronaut in 1 S , when they are closest to each other. A) 1 3 1 3 2 1 1 n W T n                  B) 1 3 2 2 3 2 1 1 n W T n                  C) 1 3 2 3 2 1 1 n W T n                  D) 2 3 1 3 2 1 1 n W T n                  14. A particle of mass m is placed on centre of curvature of fixed, uniform semi - circular ring of radius R and mass M as shown in figure. Calculate : (a) interaction force between the ring and the particle and (b) work required to displace the particle from centre of curvature to infinity.  m R M A)     2 2GM GM a F b  R R  B)     2 2GMm GMm a F b  R R  C)     2 2 2 2GMm GMm a F b  R R  D)     2 2GMm GMm a F b  R R  15. Given a thin homogeneous disc of radius a and mass m1 . A particle of mass m2 is placed at a distance l from the disk on its axis of symmetry. Initially both are motionless in free space but they ultimately collide be- cause of graviational attraction. Find the relative velocity at the time of collision. As- sume a<<1. A)   1 2 1 2 2 1 2G m m a l               B)  1 2  2 1 2G m m a l               C)   2 1 2 2 1 2G m m a l               D)   2 1 2 1 1 2G m m a l               16. The density of the core of a planet is 1 and that of the outer shell is 2 . The radii of the core and that the planet are R and 2R re- spectively. Gravitational acceleration at the surface of the planet is same as at a depth R. Find the ratio 1 2   .  2R R 1 2 A) 1 2 3 7 P P  B) 1 2 4 9 P P  C) 1 2 7 3 P P  D) 1 2 3 8 P P 
NISHITH Multimedia India (Pvt.) Ltd., 7 9 GRAVITATION NISHITH Multimedia India (Pvt.) Ltd., JEE - ADVANCED - VOL-III 17. A projectile of mass m is fired from the sur- face of the earth at an angle 0   60 from the vertical. The initial speed 0 v is equal to e e GM R . How high does the projectile rise? Neglect air resistance and the earth’s rota- tion. Re max r  V0 A) Re 2 B) Re 5 C) Re 4 D) Re 8 18. Find the velocity of a satellite travelling in an elliptical orbit, when it reaches point C on the end of the semiminor axis.  a b c c V A) c a V R g  B) 2 c g V R a  C) c g V R a  D) c 2 g V R a  MULTI ANSWER QUESTIONS 19. A cannon shell is fired to hit a target at a horizontal distance R. However it breaks into two equal parts at its highest point. One partd A returns to the cannon. The other part A) Will fall at a distance R beyond the target B) Will fall at a distance 3R beyond the target C) Will hit the target D) Have nine times the kinetic energy of A 20. A particle mooving with kinetic energy 3 J makes an elastic collison (head - on) with a stationary particle which has twice its mass, During impact : A) The minimum kinetic energy potential of sys- tem is 1 J B) The minimum elastic potential energy of the system is 2 J C) Momentum and total energy are conserved at energy instant D) The ratio of kinetic energy to potential en- ergy of the system first decreases and then in- creases. 21. Consider a thin spherical shell of uniformly density of mass M and radius R : A) The gravitational field inside the shell will be zero B) The gravitational self energy of shell is 2 2 G M R C) Attractive force experience by unit area of the shell pull the other half is 2 2 2 GM R D) Net gravitational force with which one hemi- sphere of the shell arrracts other, is 2 2 8 GM R 22. A satellite moves in an elliptical orbit about the earth. The minimum and maximum dis- tance of the satellite from the centre of earth are 7000 km and 8750 km respectively. For this situation mark the correct statement(s). [Take 24 6 10 M kg e   ] A) The maximum speed of the satellite during its motion is 5.64 km/s B) The minimum speed of the satellite during its motion is 4.51 km/s C) The length of major axis of orbit is 15750 km D) None of the above 23. The gravitational potential changes uni- formly from -20J/kg to -40J/kg as one moves along x -axis from x m  1 to x m  1 . Mark the correct statement about gravita- tional field intensity of origin. A) The gravitational field intensity at x  0must be equal to 10N/kg. B) The gravitational field intensity at x  0may be equal to 10N/kg. C) The gravitational field intensity at x  0may be greater than 10N/kg. D) The gravitational field intensity at x  0must not be less than 10N/kg.

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