Calculate the points along a line joining the centers of earth and moon where there is no gravitational force. ($M_e = 6 * 10^{24}\;kg, M_m = 7.4 * 10^{22}\;kg, d = 3.8 * 10^8\;m$).

Given, (need figure)

Mass of the earth $(M_e) = 6 * 10^{24}\;kg$

Mass of the Moon $(M_m) = 7.4 * 10^{22}\;kg$

Distance between centers of earth and the moon $(d) = 3.8 * 10^8\;m$

Let $P$ be the point at which gravitational force between earth and the moon is zero, then

$\frac{G\;*\;M_e\;*\;1}{x^2} = \frac{G\;*\;M_m\;*\;1}{(d-x)^2}$

$\frac{6\;*\;10^{24}}{x^2} = \frac{7.4\;*\;10^{22}}{(d\;-\;x)^2}$

$\frac{600}{x^2} = \frac{7.4}{(d-x)^2}$

$\left ( \frac{24.5}{x} \right )^2 = \left ( \frac{2.72}{d-x} \right )^2$

$\frac{24.5}{x} = \frac{2.72}{(d-x)}$

$27.22\;*\;x = 24.5\;*\;d$

$x = \frac{24.5\;*\;3.8\;*\;10^8}{27.22} = 3.42\;*\;10^8\;m$

∴ The distance from the earth center is $3.42\;*\;10^8\;m$

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