Gravitation

A particle of mass m is thrown upwards from the surface of the earth, with a velocity u. The mass and the radius of the earth are, respectively, M and R. G is gravitational constant and g is acceleration due to gravity on the surface of the earth. The minimum value of u so that the particle does not return back to earth, is

1. 2GMR2
2. 2GMR
3. 2gMR2
4. 2gR2

A particle of mass M is situated at the centre of a spherical shell of same mass and radius a. The magnitude of the gravitational potential at a point situated at a2 distance from the centre, will be :

1. GMa
2. 2GMa
3. 3GMa
4. 4GMa

A planet moving along an elliptical orbit is closest to the sun at a distance r1 and farthest away at a distance of r2. If v1 and v2 are the linear velocities at these points respectively, then the ratio v1v2 is

1. (r1/r2)2
2. r2/r1
3. (r2/r1)2
4. r1/r2

The additional kinetic energy to be provided to a satellite of mass m revolving around a planet of mass M, to transfer it from a circular orbit of radius R1 to another of radius R2 (R2 > R1) is

1. GmM ( 1R12- 1R22 )
2. GmM ( 1R1- 1R2 )
3. 2GmM ( 1R1- 1R2 )
4. 12GmM ( 1R1- 1R2 )

The dependence of acceleration due to gravity g on the distance r from the centre of the earth, assumed to be a sphere of radius R of uniform density is as shown in figures below [figure] The correct figure is

1. (4)
2. (1)
3. (2)
4. (3)

A particle of mass M is situated at the centre of a spherical shell of same mass and radius a. The magnitude of the gravitational potential at a point situated at a/2 distance from the centre, will be :

1. GMa
2. 2GMa
3. 3GMa
4. 4GMa

The radii of circular orbits of two satellites A and B of the earth, are 4R and R, respectively. If the speed of satellite A is 3V, then the speed of satellite B will be

1. 3V4
2. 6V
3. 12V
4. 3V2

The figure shows elliptical orbit of a planet m about the sun S. The shaded area SCD is twice the shaded area SAB. If t1 is the time for the planet to move from C to D and t2 is the time to move from A to B then [figure]

1. t1 = 4t2
2. t1 = 2t2
3. t1 = t2
4. t1 > t2

Two satellites of earth, S1 and S2 are moving in the same orbit. The mass of S1 is four times the mass of S2. Which one of the following statements is true?

1. The potential energies of earth and satellite in the two cases are equal.
2. S1 and S2 are moving with the same speed.
3. The kinetic energies of the two satellites are equal.
4. The time period of S1 is four times that of S2.

The earth is assumed to be a sphere of radius R. A platform is arranged at a height R from the surface of the earth. The escape velocity of a body from this platform is fv, where v is its escape velocity from the surface of the Earth. The value of f is

1. 1/2
2. 2
3. 1/ 2
4. 1/3