In planetary motion the areal velocity of position vector of a planet depends on angular velocity $\left(\omega \right)$ and the distance of the planet from Sun (r). If so the correct relation for areal velocity is -

1. $\frac{dA}{dt}\propto \omega r$                            

2. $\frac{dA}{dt}\propto {\omega }^{2}r$ 

3. $\frac{dA}{dt}\propto \omega r^2$                           

4. $\frac{dA}{dt}\propto \sqrt{\omega r}$ 

A sphere of mass M and radius R₂ has a concentric cavity of radius R₁ as shown in figure. The force F exerted by the sphere on a particle of mass m located at a distance r from the centre of sphere varies as $(0\le \mathrm{r}\le \infty )$. 

Which one of the following graphs represents correctly the variation of the gravitational field (I) with the distance (r) from the centre of a spherical shell of mass M and radius a ?

Which of the following graphs represents the motion of a planet moving about the sun ?

The correct graph representing the variation of total energy (${\mathrm{E}}_{\mathrm{t}}$), kinetic energy (${\mathrm{E}}_{\mathrm{k}}$) and potential energy (U) of a satellite with its distance from the centre of the earth is -

The diameters of two planets are in the ratio 4 : 1 and their mean densities in the ratio 1 : 2. The acceleration due to gravity on the planets will be in ratio

1.  1 : 2                             

2.  2 : 3

3.  2 : 1                           

4.  4 : 1

Weight of 1 kg becomes 1/6 kg on moon. If radius of moon is $1.768\times {10}^{6}m$, then the mass of moon will be -

1. $1.99\times {10}^{30}kg$                        

2. $7.56\times {10}^{22}kg$

3. $5.98\times {10}^{24}kg$                       

4. $7.65\times {10}^{22}kg$

Let g be the acceleration due to gravity at the earth's surface and K be the rotational kinetic energy of the earth. Suppose the earth's radius decreases by 2% keeping all other quantities the same, then

1.  g decreases by 2% and K decreases by 4%

2.  g decreases by 4% and K increases by 2%

3.  g increases by 4% and K increases by 4%

4.  g decreases by 4% and K increases by 4%

Distance of geostationary satellite from the center of the Earth $(R_e=6400 km)$ in terms of $R_e$ is-

1. $13.76 R_e$                        

2. $10.76 R_e$

3. $6.59 R_e$                            

4. $2.56 R_e$

Energy required to move a body of mass m from an orbit of radius 2R to 3R is

1.     $GMm/12R^2$               

2.    $GMm/3R^2$

3.     $GMm/8R$                 

4.     $GMm/6R$