## Question

A mass (= mass of earth) is to be compressed in a sphere in such a way that the escape velocity from its surface is 3 × 10^{8} m/s (equal to that of light). What should be the radius of the sphere?

### Solution

#### SIMILAR QUESTIONS

What are the values of gravitational attraction and potential at the surface of earth referred to zero potential at infinite distance? Given that the mass of the earth is the radius of earth is 6400 km and *G* = 6.67 ×10^{ –11}*MKS* units.

In a certain region of space gravitational field is given by *I* = – (*K*/*r*). Taking the reference point to be at *r* = *r*_{0} with *V* = V_{0}, find the potential.

Calculate the acceleration due to gravity at the surface of Mars if its diameter is 6760 km and mass one-tenth that of earth. The diameter of earth is 12742 km and acceleration due to gravity on earth is 9.8 m/s^{2}.

Compute the mass and density of the moon if acceleration due to gravity on its surface is 1.62 m/s^{2} and its radius is

Two equal masses *m* and *m* are hung from a balance whose scale pans differ in vertical height by *h*. Calculate the error in weighing, if any, in terms of density of earth ρ.

Three particles each of mass *m* are placed at the corners of an equilateral triangle of side *d*. Calculate (*a*) the potential energy of the system, (*b*) work done on this system if the side of the triangle is changed from *d* to 2*d*.

What will be the acceleration due to gravity on the surface of the moon if its radius were (1/4)th the radius of earth and its mass (1/80)th the mass of earth? What will be the escape velocity on the surface of moon if it is 11.2 km/s on the surface of the earth? (Given that *g* = 9.8 m/s^{2})

The masses and radii of the earth and moon are *M*_{1}, *R*_{1} and *M*_{2}, *R*_{2}respectively. Their centres are at distance *d* apart. What is the minimum speed with which a particle of mass *m* should be projected from a point midway between the two centres so as to escape to infinity?

Distance between the centres of two stars is 10*a*. The masses of these stars are *M* and 16*M* and their radii *a* and 2*a*, respectively. A body of mass *m* is fired straight from the surface of the larger star towards the smaller star. What should be its minimum initial speed to reach the surface of the smaller star? Obtain the expression in terms of *G*, *M* and *a*.

A projectile is fired vertically upward from the surface of earth with a velocity *Kv _{e}* where

*v*is the escape velocity and

_{e}*K*< 1. Neglecting air resistance, show that the maximum height to which it will rise measured from the centre of earth is

*R/*(1 –

*K*

^{2}) where

*R*is the radius of the earth.