Question The mass of the earth is times the mass of the Moon and the distance between the earth and the Moon is times the radius of earth. If R is the radius of the earth, then the distance between the Moon and the point on the line joining the Moon and the earth where the gravitational force becomes zero is

 Solution

Step 1: Given

The mass of the earth is $81$ times the mass of the moon.

The distance between the earth and the moon is $60$ times the radius of the earth ($R$).

Step 2: Formula used

Let the mass of the moon be $m$then the mass of the earth will be $81m$.

Formula: The gravitational force due to a body of mass $M$ on an object of mass $m$ will be $F = \frac{GMm}{R^2}$, where $G$ is the gravitational constant and $R$ is the distance between the two objects.

Step 3: Solution

Gravitational force is a vector quantity, The force of attraction is towards the object in the line that joins the two objects.

Let the required point be at a distance $x$ from the moon.

The force from both moon and earth has to be equal at this point to make the net force zero at that specific point as they both act in opposite directions.

⇒ $\frac{Gm}{x^2}=\frac{G\left(81m\right)}{{(60R - x)}^2}$

⇒ $81x^2 = {(60R - x)}^2$

⇒ $9x = 60R - x$

⇒ $10x = 60R$

⇒ $x = 6R$

The distance from the moon where the force due to both moon and earth will become zero is $6R$.