Two-point masses m and 4m are separated by a distance d on a line. A third point mass m₀ is to be placed at a point on the line such that the net gravitational force on it is equal to zero.

The distance of that point from the mass m is:

1. $\frac{\mathrm{d}}{2}$

2. $\frac{\mathrm{d}}{4}$

3. $\frac{\mathrm{d}}{3}$

4. $\frac{\mathrm{d}}{5}$

Three particles A, B and C, each of mass m are lying at the corners of an equilateral triangle of side L. If particle A is released keeping the particles B and C fixed, the magnitude of the instantaneous acceleration of A is:

1. $\sqrt{3}\frac{{\mathrm{Gm}}^2}{{\mathrm{L}}^2}$

2. $\sqrt{2}\frac{{\mathrm{Gm}}^2}{{\mathrm{L}}^2}$

3. $\sqrt{2}\frac{\mathrm{Gm}}{{\mathrm{L}}^2}$

4. $\sqrt{3}\frac{\mathrm{Gm}}{{\mathrm{L}}^2}$

A uniform sphere of mass M and radius R is surrounded by a concentric spherical shell of the same mass but radius 2R. A point mass m is kept at a distance x (>R) in the region bounded by the spheres as shown in the figure. The net gravitational force on that particle is:

1. $\frac{\mathrm{GMm}}{{\mathrm{x}}^2}$

2. $\frac{\mathrm{GMmx}}{{\mathrm{R}}^3}$

3. $\frac{\mathrm{G}(\mathrm{M}+\mathrm{m})}{{\mathrm{x}}^2}$

4. Zero

The gravitational constant depends upon:

1. Size of the bodies

2. Gravitational masses

3. Distance between the bodies

4. None of these

Two planets have the same density but different radii. The acceleration due to gravity would be:

1. same on both planets.

2. greater on the smaller planet.

3. greater on the larger planet.

4. dependent on the distance of the planet from the sun.

If the radius of the earth shrinks by 1.5% (mass remaining the same), then the value of the gravitational acceleration changes by:

1. 2%

2. -2%

3. 3%

4. -3%

If the density of a planet is double that of the earth and the radius is 1.5 times that of the earth, the acceleration due to gravity on the surface of the planet is:

1. $\frac{3}{4}$ times that on the surface of the earth.

2. 3 times that on the surface of the earth.

3. $\frac{4}{3}$ times that on the surface of the earth.

4. 6 times that on the surface of the earth.

The gravitational force on a body of mass 1.5 kg situated at a point is 45 N. The gravitational field intensity at that point is:

1. 30 N/kg

2. 67.5 N/kg

3. 46.5 N/kg

4. 43.5 N/kg

Two-point masses having mass m and 2m are placed at distance d. The point on the line joining point masses, where the gravitational field intensity is zero will be at a distance:

1. $\frac{2\mathrm{d}}{\sqrt{3}+1}$ from point mass "2m".

2. $\frac{2\mathrm{d}}{\sqrt{3}-1}$ from point mass "2m".

3. $\frac{\mathrm{d}}{1+\sqrt{2}}$ from point mass "m".

4. $\frac{\mathrm{d}}{1-\sqrt{2}}$ from point mass "m".

At what height above the surface of the earth, the value of "g" decreases by 2%?

[radius of the earth is 6400 km]

1. 32 km

2. 64 km

3. 128 km

4. 1600 km