Gravitation

A particle of mass m, and angular momentum is moving in a circular orbit of radius r0 under the influence of an attractive force F(r)=- kr2 r. Keeping its angular momentum unchanged, the particle is displaced radially by a small distance r r0, due to which its radial distance varies periodically. The corresponding time period is:

1. 2 3mk2
2. 2 mk
3. 2 33mk2
4. 2 35mk2

Consider a star of mass m2 kg revolving in a circular orbit around another star of mass m1 kg with m1 m2. The heavier star slowly acquires mass from the lighter star at a constant rate of kg/s. In this transfer process, there is no other loss of mass. If the separation between the centers of the stars is r, then its relative rate of change 1r drdt (in s-1) is given by:

1. - 3 2m2
2. - 2 m2
3. - 2 m1
4. - 3 2m1

A geostationary satellite above the equator is orbiting around the earth at a fixed distance r1 from the center of the earth. A second satellite is orbiting in the equatorial plane in the opposite direction to the earth's rotation, at a distance r2 from the center of the earth, such that r1=1.21 r2. The time period of the second satellite as measured from the geostationary satellite is 24p hours. The value of p is

A particle of mass m is under the influence of the gravitational field of a body of mass M ( m). The particle is moving in a circular orbit of radius r0 with time period T0 around the mass M. Then, the particle is subjected to an additional central force, corresponding to the potential energy Vc(r) = m /r3, where is a positive constant of suitable dimensions and r is the distance from the center of the orbit. If the particle moves in the same circular orbit of radius r0 in the combined gravitational potential due to M and Vc(r), but with a new time period T1, then (T12 - T02)/T12 is given by [G is the gravitational constant.]

1. 3 GMr02
2. 2GMr02
3. GMr02
4. 2 GMr02