The orbital velocity of a planet revolving close to earth's surface is 

(1) $\sqrt{2gR}$              

(2) $\sqrt{gR}$

(3) $\sqrt{\frac{2g}{R}}$               

(4) $\sqrt{\frac{g}{R}}$  

If the gravitational force between two objects were proportional to \(\frac{1}{R}\) (and not as\(\frac{1}{R^2}\)${\mathrm{}}^{}$) where \(R\) is the separation between them, then a particle in circular orbit under such a force would have its orbital speed \(v\) proportional to:
1. \(\frac{1}{R^2}\)
2. \(R^{0}\)
3. \(R^{1}\)
4. \(\frac{1}{R}\)

When a satellite going round the earth in a circular orbit of radius r and speed v loses some of its energy, then r and v change as 

(1) r and v both will increase

(2) r and v both will decrease

(3) r will decrease and v will increase

(4) r will decrease and v will decrease

Which of the following quantities does not depend upon the orbital radius of the satellite ?

(1) $\frac{T}{R}$                            

(2) $\frac{T^2}{R}$

(3) $\frac{T^2}{R^2}$                           

(4) $\frac{T^2}{R^3}$

A satellite moves round the earth in a circular orbit of radius R making one revolution per day. A second satellite moving in a circular orbit, moves round the earth once in 8 days. The radius of the orbit of the second satellite is -

(1) 8 R                           

(2) 4R

(3) 2R                             

(4) R

A satellite moves in a circle around the earth. The radius of this circle is equal to one half of the radius of the moon’s orbit. The satellite completes one revolution in

(1) $\frac{1}{2}$ lunar month                   

(2) $\frac{2}{3}$ lunar month

(3) $2^{\raisebox{1ex}{$-3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ lunar month               

(4) $2^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ lunar month

If the acceleration due to gravity at a height \(1\) km above the earth is similar to a depth \(d\) below the surface of the earth, then: 
1. \(d= 0.5\) km
2. \(d=1\) km
3. \(d=1.5\) km
4. \(d=2\) km

Two astronauts are floating in a gravity free space after having lost contact with their spaceship. The two will:

Starting from the centre of the earth having radius R, the variation of g (acceleration  due to gravity) is shown by:

(a)  

(b) 

(c) 

(d) 

At what height from the surface of earth the gravitation potential and the value of g are $-5.4\times {10}^{7}J$ $k{g}^{-2}$ and $6.0$ $m{s}^{-2}$ respectively? (Take, the radius of earth as 6400 km.)

(a) 1600 km             (b) 1400 km 

(c) 2000 km             (d) 2600 km