Gravitation

Kepler's third law states that square of period of revolution (T) of a planet around the sun, is proportional to third power of average distance r between sun and planet i.e. T2 = Kr3 here K is constant. If the masses of sun and planet are M and m respectively then as per Newton's law of gravitation force of attraction between them is F = GMmr2, here G is gravitational constant. The relation between G and K is described as

1. K = G
2. K = 1G
3. GK = 4 2
4. GMK = 4 2

Two spherical bodies of mass M and 5M and radii R and 2R are released in free space with initial separation between their centres equal to 12R. If they attract each other due to gravitational force only, then the distance covered by the smaller body before collision is :

1. 7.5R
2. 1.5R
3. 2.5R
4. 4.5R

A black hole is an object whose gravitational field is so strong that even light cannot escape from it. To what approximate radius would earth (mass = 5.98 1024 kg) have to be compressed to be a black hole?

1. 10-9 m
2. 10-6 m
3. 10-2 m
4. 100 m

Dependence of intensity of gravitational field (E) of earth with distance (r) from centre of earth is correctly represented by

1. [figure]
2. [figure]
3. [figure]
4. [figure]

Infinite number of bodies, each of mass 2 kg are situated on x-axis at distance 1 m, 2 m, 4 m, 8 m, . . . , respectively, from the origin. The resulting gravitational potential due to this system at the origin will be

1. - 43 G
2. - 4G
3. - G
4. - 83 G

A body of mass 'm' is taken from the earth's surface to the height equal to twice the radius (R) of the earth. The change in potential energy of body will be

1. 3mgR
2. 13mgR
3. mg2R
4. 23mgR

The radius of a planet is twice the radius of earth. Both have almost equal average mass densities. VP and VE are escape velocities of the planet and the earth, respectively, then

1. VP = 1.5 VE
2. VP = 2 VE
3. VE = 3 VP
4. VE = 1.5 VP

A particle of mass 'm' is kept at rest at a height 3R from the surface of earth, where 'R' is radius of earth and 'M' is mass of earth. The minimum speed with which it should be projected, so that it does not return back, is (g is acceleration due to gravity on the surface of earth)

1. ( GM2R )1/2
2. ( gR4 )1/2
3. ( 2gR )1/2
4. ( GMR )1/2

If ve is escape velocity and vo is orbital velocity of a satellite for orbit close to the earth's surface, then these are related by

1. vo = 2ve
2. vo = ve
3. ve = 2vo
4. ve = 2 vo

Which one of the following plots represents the variation of gravitational field on a particle with distance r due to a thin spherical shell of radius R? (r is measured from the centre of the spherical shell)

1. [figure]
2. [figure]
3. [figure]
4. [figure]

The height at which the weight of a body becomes ( 116 )th of its weight on the surface of earth (radius R), is

1. 5R
2. 15R
3. 3R
4. 4R

A spherical planet has a mass MP and diameter DP. A particle of mass m falling freely near the surface of this planet will experience an acceleration due to gravity, equal to

1. 4GMPDP2
2. GMP mDP2
3. GMPDP2
4. 4GMP mDP2

A geostationary satellite is orbiting the earth at a height of 5R above the surface of the earth, R being the radius of the earth. The time period of another satellite in hours at a height of 2R from the surface of the earth is

1. 5
2. 10
3. 6 2
4. 6 2

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

Imagine a new planet having the same density as that of earth but it is 3 times bigger than the earth in size. If the acceleration due to gravity on the surface of earth is g and that on the surface of the new planet is g', then

1. g' = g/9
2. g' = 27g
3. g' = 9g
4. g' = 3g

For a satellite moving in an orbit around the earth, the ratio of kinetic energy to potential energy is

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

The density of a newly discovered planet is twice that of earth. The acceleration due to gravity at the surface of the planet is equal to that at the surface of the earth. If the radius of the earth is R, the radius of the planet would be

1. 2R
2. 4R
3. 14R
4. 12R

Two spheres of masses m and M are situated in air and the gravitational force between them is F. The space around the masses is now filled with a liquid of specific gravity 3. The gravitational force will now be

1. 3F
2. F
3. F/3
4. F/9

The acceleration due to gravity on the planet A is 9 times the acceleration due to gravity on planet B. A man jumps to a height of 2 m on the surface of A. What is the height of jump by the same person on the planet B?

1. (2/9) m
2. 18 m
3. 6 m
4. (2/3) m

A body of mass m is placed on earth's surface which is taken from earth surface to a height of h = 3R, then change in gravitational potential energy is

1. mgR4
2. 23mgR
3. 34mgR
4. mgR2

With what velocity should a particle be projected so that its height becomes equal to radius of earth?

1. ( GMR )1/2
2. ( 8GMR )1/2
3. ( 2GMR )1/2
4. ( 4GMR )1/2