Question

If the mass of the moon is $\frac{M}{81}$, where $M$ is the mass of the earth, find the distance of the point where the gravitational field due to earth and moon cancel each other, from the centre of the moon. Given that the distance between the centres of the earth and moon is $60R$ where $R$ is the radius of earth.

• A. $4R$
• B. $8R$
• C. $12R$
• D. $6R$

Answer 1

The correct option is D $6R$

Given that,
Mass of the earth $=M$
Mass of the moon $=\frac{M}{81}$
Radius of earth $=R$
Distance between centre of earth and moon $=60R$

For a large spherical object like a planet, the gravitational field at a point outside the planet can be calculated by assuming that the entire mass is concentrated at its centre.

The gravitational field intensity $\left(E\right)$ at a distance $r$ from a planet having mass $M$ is given by
$E=\frac{GM}{r^2}$

Let the gravitational field due to the moon be cancelled by the gravitational field due to the earth at a distance $r^{\prime }$ from the centre of the moon.
$E_{\text{moon}}=E_{\text{earth}}$
$\Rightarrow \frac{GM/81}{r^{\prime 2}}$ =$\frac{GM}{(60R-r^{\prime }{)}^2}$
Taking square root on both sides.
$\Rightarrow \frac{1}{9r^{\prime }}=\frac{1}{60R-r^{\prime }}$
$\Rightarrow 60R-r^{\prime }=9r^{\prime }$
$\Rightarrow r^{\prime }=6R$

Hence, option (d) is correct.

Why this question? To make students familiar with the concept of gravitational field intensity and that it is a vector quantity. Key Concept: Between two masses, there will be a point where the gravitational field of one mass will be equal and opposite to the gravitational field of the other mass.