Question

The masses and radii of the earth and moon are M1R1 and M2R2respectively. 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?

Solution

Correct option is

 

Potential energy of m when it is midway between M1 and M2,

             

                 

And as potential energy at infinity is zero, so work required to shift m from the given position to infinity,

                

As this work is provided by initial kinetic energy,  

            

SIMILAR QUESTIONS

Q1

 

Imagine a planet whose diameter and mass are both one-half of those of earth. The day’s surface temperature of this planet reaches upto 800 K. Are oxygen molecules possible in the atmosphere of this planet? Give calculation. (Escape velocity on earth’s surface = 11.2 km s–1, Boltzmann’s constant

k = 1.38 × 10–23 JK–1, mass of oxygen molecule = 5.3 × 10–26 kg.)

Q2

A hollow spher is made of a lead of radius R such that its surface touches the outside surface of the lead sphere and passes through its centre. The mass of the lead sphere before hollowing was M. What is the force of attraction that this sphere would exert on a particle of mass which lies at a distance from the centre of the lead sphere on the straight line joining the centres of the sphere and the hollow (as shown in fig.)?

                                                                                          

Q3

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 –11MKS units.

Q4

In a certain region of space gravitational field is given by I = – (K/r). Taking the reference point to be at r = r0 with V = V0, find the potential.

Q5

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/s2.

Q6

Compute the mass and density of the moon if acceleration due to gravity on its surface is 1.62 m/s2 and its radius is 

Q7

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.

Q8

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 2d.

Q9

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/s2)

Q10

Distance between the centres of two stars is 10a. The masses of these stars are M and 16M and their radii a and 2a, 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 GM and a.