\(1\) kg of sugar has maximum weight:
1. at the pole.
2. at the equator.
3. at a latitude of \(45^{\circ}.\)
4. in India.

A body is thrown vertically upwards with an initial speed \(\sqrt{gR}\), where \(R\) is the radius of the earth. The maximum height reached by the body from the surface of the earth is:
1. \(\frac{R}{2}\)
2. \(\frac{3R}{2}\)
3. \(R\)
4. \(\frac{R}{4}\)

A particle is located midway between two point masses each of mass \(M\) kept at a separation \(2d.\) The escape speed of the particle is: (neglect the effect of any other gravitational effect)
1. \(\sqrt{\frac{2 GM}{d}}\)
2. \(2 \sqrt{\frac{GM}{d}}\)
3. \(\sqrt{\frac{3 GM}{d}}\)
4. \(\sqrt{\frac{GM}{2 d}}\)

Three identical particles each of mass \(M\) are located at the vertices of an equilateral triangle of side \(a\). The escape speed of one particle will be:
1. \(\sqrt{\frac{4 GM}{a}}\)
2. \(\sqrt{\frac{3 GM}{a}}\)
3. \(\sqrt{\frac{2 GM}{a}}\)
4. \(\sqrt{\frac{GM}{a}}\)

Two identical hollow spheres of negligible thickness are placed in contact with each other. The force of gravitation between the spheres will be proportional to (\(R\) = radius of each sphere):
1. \(R\)
2. \(R^2\)
3. \(R^4\)
4. \(R^3\)

A planet is revolving around a massive star in a circular orbit of radius R. If the gravitational force of attraction between the planet and the star is inversely proportional to ${\mathrm{R}}^3$, then the time period of revolution T is proportional to:

1. ${\mathrm{R}}^5$

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

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

4. R

When a planet revolves around the sun in an elliptical orbit, then which of the following remains constant?