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 ?

Suppose, the acceleration due to gravity at the earth’s surface is 10 $\mathrm{m}/{\mathrm{s}}^2$ and at the surface of Mars, it is 4.0 $\mathrm{m}/{\mathrm{s}}^2$. A 60 kg passenger goes from the earth to Mars in a spaceship moving with a constant velocity. Neglect all other objects in the sky. Which part of the figure best represents the weight (net gravitational force) of the passenger as a function of time ?

(a) A                              (b) B

(c) C                              (d) D

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 -

What is the intensity of gravitational field of the centre of a spherical shell?

(1)     $Gm/r^2$           

(2)    g

(3)      Zero             

(4)    None of these

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 the ratio of:

1. 1 : 2                           
2. 2 : 3
3. 2 : 1                      
4. 4 : 1

If the angular speed of the earth is doubled, the value of acceleration due to gravity (g) at the north pole

(1) Doubles                   

(2)  Becomes half

(3)  Remains same       

(4)   Becomes zero

Escape velocity of a body of 1 kg mass on a planet is 100 m/sec. Gravitational Potential energy of the body at the planet is -

(1)   – 5000 J                     

(2)  – 1000 J

(3)   – 2400 J                      

(4)  5000 J