The mass of the earth is 81 times the mass of the Moon and the distance between the earth and the Moon is 60 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

 Consider that the moon’s mass is M. As a result, the earth’s mass is 81M. The radius of the earth is R once more.

As a result, r=60R is the distance between the moon and the earth. At a location on the line between the moon to the earth, the gravitational force is zero.

Let’s say the point is x miles away from the moon. As a result, the point’s distance from the earth is 60Rx.

Because the gravitational pull at that position is zero, putting a mass m there will have no effect. The gravitational force exerted by the earth and moon on mass m will be balanced, i.e., the two forces will be equal and opposite. Let’s say the force exerted by the earth on a particle of mass m is Fe. So, Fe So,

Fe = G × 81M × m(60R−x)2

Again, let the force on the particle of mass m due to moon is, Fm

So, Fm = GMm/x2 For the gravitational force to be zero the force Fe and Fm should be equal and opposite. So, we can write,

Fm=Fe

G × 81Mm/(60R−x)2 = GMm/x2

81/(60R−x)2 = 1/x2

(60R−x)2 = 81x2

60R − x = 9x

10x = 60R

x = 6R

So, the point at which the gravitational force is zero will be 6R