Gravitation

Three identical spheres of mass m are placed at the vertices of an equilateral triangle of length a. When released, they interact only through gravitational force and collide after a time T = 4 seconds. If the sides of the triangle are increased to length 2a and also the masses of the spheres are made 2 m, then they will collide after seconds.

A satellite of mass 1000 kg is launched to revolve around the earth in an orbit at a height of 270 km from the earth's surface. Kinetic energy of the satellite in this orbit is 1010 J. (Mass of earth = 6 1024 kg, Radius of earth = 6.4 106 m, Gravitational constant = 6.67 10-11 N m2 kg-2)

Two planets, A and B are orbiting a common star in circular orbits of radii RA and RB, respectively, with RB=2RA. The planet B is 4 2 times more massive than planet A. The ratio ( LBLA ) of angular momentum (LB) of planet B to that of planet A (LA) is closest to integer .

Earth has mass 8 times and radius 2 times that of a planet. If the escape velocity from the earth is 11.2 km/s, the escape velocity in km/s from the planet will be:

1. 8.4
2. 11.2
3. 5.6
4. 2.8

A satellite is launched into a circular orbit of radius 'R' around the earth. A second satellite is launched into an orbit of radius 1.03 R. The time period of revolution of the second satellite is larger than the first one approximately by

1. 3%
2. 2.5%
3. 4.5%
4. 9%

Acceleration due to gravity on the surface of earth is 'g'. If the diameter of earth is reduced to one third of its original value and mass remains unchanged, then the acceleration due to gravity on the surface of earth is g.

If a satellite orbiting the Earth is 9 times closer to the Earth than the Moon, what is the time period of rotation of the satellite? Given rotational time period of Moon = 27 days and gravitational attraction between the satellite and the moon is neglected.

1. 3 days
2. 27 days
3. 81 days
4. 1 day

A satellite of mass M2 is revolving around earth in a circular orbit at a height of R3 from earth surface. The angular momentum of the satellite is M GMRx. The value of x is , where M and R are the mass and radius of earth, respectively. ( G is the gravitational constant)

A small point of mass m is placed at a distance 2R from the centre ' O ' of a big uniform solid sphere of mass M and radius R . The gravitational force on ' m ' due to M is F1. A spherical part of radius R/3 is removed from the big sphere as shown in the figure and the gravitational force on m due to remaining part of M is found to be F2. The value of ratio F1:F2 is [figure]

1. 11 : 10
2. 12 : 11
3. 16 : 9
4. 12 : 9

An astronaut takes a ball of mass m from earth to space. He throws the ball into a circular orbit about earth at an altitude of 318.5 km. From earth's surface to the orbit, the change in total mechanical energy of the ball is x GMe m21Re. The value of x is (take Re = 6370 km) :

1. 12
2. 11
3. 9
4. 10