The distance between the centres of the Moon and the earth is `D`. The mass of the earth is `81` times the mass of the Moon. At what distance from the centre of the earth, the gravitational force will be zero?

Options

A. `(D)/(2)`
B. `(D)/(3)`
C. `(9D)/(10)`
D. `(D)/(4)`

Answer

C

Explanation

Suppose that the gravitational force becomes zero at a distance x from the centre of the earth.
Thus the gravitational force at P due to the earth `= (GM_(e ))/(x^(2))` and the gravitational force at P due to the moon
`= (GM_(m))/((D-x)^(2))`
`because (GM_(e ))/(x^(2))=(GM_(m))/((D-x)^(2))`
`therefore (M_(e ))/(M_(m))=(81)/(1)=(x^(2))/((D-x)^(2))`
`therefore` Taking square roots, we get
`9 = (x)/(D-x) " " therefore 9D-9x = x " " therefore 10 x = 9D " " therefore x = (9D)/(10)`