The distance between the centre of moon and earth is \(D\) and mass of earth is 81 times the mass of moon.…
The distance between the centre of moon and earth is \(D\) and mass of earth is 81 times the mass of moon. At what distance from the centre of the earth, the gravitational force will be zero?
\(\frac{D}{2}\)
\(\frac{2 D}{3}\)
\(\frac{4 D}{3}\)
\(\frac{9 D}{10}\)
Solution
Let a unit mass \(m\) is present at a distance \(x\) from the earth, where gravitational force is zero.
\(\frac{G m M_e}{x^2}=\frac{G m M}{(D-x)^2}\)...(i)
where, \(M_e\) is mass of earth and \(M\) is mass of moon.
Given, \(M_e=81 \mathrm{M}\)
\(\therefore\) From Eq. (i), we have
\(\begin{aligned}
& \frac{G m 81 M}{x^2}=\frac{G m M}{(D-x)^2} \Rightarrow \frac{81}{x^2}=\frac{1}{(D-x)^2} \\
\Rightarrow & \left(\frac{9}{x}\right)^2=\frac{1}{(D-x)^2} \Rightarrow \frac{9}{x}=\frac{1}{D-x} \\
\Rightarrow & 9 D-9 x=x \Rightarrow 9 D=10 x \\
\Rightarrow & x= \frac{9 D}{10}
\end{aligned}\)