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

The value of acceleration due to gravity at a height of 800 km from the surface of the earth (radius of the earth is 6400 km and value of acceleration due to gravity on the earth's surface is 981 cm/${\mathrm{s}}^2$) is:

A satellite of mass m revolving around the earth in a circular orbit of radius r has its angular momentum equal to L about the centre of the earth. The potential energy of the satellite is: 

If R is the radius of the orbit of a planet and T is the time period of the planet, then which of the following graphs correctly shows the motion of a planet revolving around the sun?

1.		2.
3.		4.

The figure shows a planet in an elliptical orbit around the sun (S). The ratio of the momentum of the planet at point A to that at point B is:

Three identical point masses, each of mass 1 kg lie at three points (0, 0), (0, 0.2 m), (0.2 m, 0). The net gravitational force on the mass at the origin is:
1. \(6.67\times 10^{-9}(\hat i +\hat j)~\text{N}\)
2. \(1.67\times 10^{-9}(\hat i +\hat j) ~\text{N}\)
3. \(1.67\times 10^{-9}(\hat i -\hat j) ~\text{N}\)
4. \(1.67\times 10^{-9}(-\hat i -\hat j) ~\text{N}\)

Two particles of mass \(\mathrm{m}\) and \(\mathrm{4m}\) are separated by a distance \(\mathrm{r}.\) Their neutral point is at:

1.  $\frac{\mathrm{r}}{2}$ $\mathrm{from}$ $\mathrm{m}$

2.  $\frac{\mathrm{r}}{3}$ $\mathrm{from}$ $4\mathrm{m}$

3.  $\frac{\mathrm{r}}{3}$ $\mathrm{from}$ $\mathrm{m}$

4.  $\frac{\mathrm{r}}{4}$ $\mathrm{from}$ $4\mathrm{m}$

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

1.  $\sqrt{\frac{2\mathrm{GM}}{\mathrm{d}}}$

2.  $2\sqrt{\frac{\mathrm{GM}}{\mathrm{d}}}$

3.  $\sqrt{\frac{3\mathrm{GM}}{\mathrm{d}}}$

4.  $\sqrt{\frac{\mathrm{GM}}{2\mathrm{d}}}$

1 kg of sugar has maximum weight:
1. at the pole.
2. at the equator.
3. at a latitude of 45$°$.
4. in India.

Radii and densities of two planets are ${\mathrm{R}}_1, {\mathrm{R}}_2$ and ${\mathrm{\rho }}_1, {\mathrm{\rho }}_2$ respectively. The ratio of accelerations due to gravity on their surfaces is:

1.  $\frac{{\mathrm{\rho }}_1}{{\mathrm{R}}_1} : \frac{{\mathrm{\rho }}_2}{{\mathrm{R}}_2}$

2.  $\frac{{\mathrm{\rho }}_1}{{\mathrm{R}}_1^2} : \frac{{\mathrm{\rho }}_2}{{\mathrm{R}}_2^2}$

3.  ${\mathrm{\rho }}_1{\mathrm{R}}_1 : {\mathrm{\rho }}_2{\mathrm{R}}_2$

4.  $\frac{1}{{\mathrm{\rho }}_1{\mathrm{R}}_1} : \frac{1}{{\mathrm{\rho }}_2{\mathrm{R}}_2}$