Question

The distance between earth and moon is about $3.8\times {10}^5$ km. At what points will the net gravitational force of earth-moon system be zero? [Given the mass of the earth is 81 times the moon's mass]. (Hint: Assume a body of unit mass at the null point)

Answer 1

Given $M_{earth}=81m_{moon}$
Let p be a null point
Let x m be the distance from the moon to the null point.
Distance of null point from earth is $(3.8\times {10}^8-x)m$
Force due to earth on null point $=$ Force due to moon on null point
$\frac{G{m}_{earth}\left(1\right)}{(3.8\times {10}^8-x{)}^2}=\frac{G{m}_{moon}\left(1\right)}{x^2}$
$\frac{81m_{moon}}{(3.8\times {10}^8-x{)}^2}=\frac{m_{moon}}{x^2}$
$\frac{9}{3.8\times {10}^8-x}=\frac{1}{x}$

$9x=3.8\times {10}^8-x$
$x=3.8\times {10}^{7}m$
The net is zero at $3.8\times {10}^7$ from moon.