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,
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