If orbital velocity of planet is given by $v=G^a{M}^b{R}^c$, then -

(1)$a=1/3, b=1/3, c=‐1/3$      

(2)$a=1/2, b=1/2, c=‐1/2$      

(3)$a=1/2, b=‐1/2, c=1/2$   

(4)$a=1/2, b=‐1/2, c=‐1/2$          

The mass of a planet that has a moon whose time period and orbital radius are T and R respectively can be written as -

(1)$4{\mathrm{\pi }}^2{\mathrm{R}}^3{\mathrm{G}}^{‐1}{\mathrm{T}}^{‐2}$                                

(2)$8{\mathrm{\pi }}^2{\mathrm{R}}^3{\mathrm{G}}^{‐1}{\mathrm{T}}^{‐2}$  

(3)$12{\mathrm{\pi }}^2{\mathrm{R}}^3{\mathrm{G}}^{‐1}{\mathrm{T}}^{‐2}$                               

(4)$16{\mathrm{\pi }}^2{\mathrm{R}}^3{\mathrm{G}}^{‐1}{\mathrm{T}}^{‐2}$  

The eccentricity of earth's orbit is 0.0167. The ratio of its maximum speed in its orbit to its minimum speed is

(a)   2.507                                             (b)   1.033

(c)   8.324                                             (d)   1.000

Assuming the earth to have a constant density, point out which of the following curves show the variation of acceleration due to gravity from the centre of earth to the points far away from the surface of earth -

The radius of orbit of a planet is two times that of the earth. The time period of the planet is:

(1)   4.2 years                          

(2)   2.8 years

(3)   5.6 years                          

(4)   8.4 years

The diagram showing the variation of gravitational potential of earth with distance r from the centre of earth is -

The condition for a uniform spherical mass m of radius r to be a black hole is [G= gravitational constant and g= acceleration due to gravity] .

(1) ${\left(2Gm/r\right)}^{1/2}\le c$                

(2) ${\left(2Gm/r\right)}^{1/2}=c$ 

(3) ${\left(2Gm/r\right)}^{1/2}\ge c$                

(4)  ${\left(gm/r\right)}^{1/2}\ge c$

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 ?