Question

The distance between earth and moon is $d$ and the mass of earth is $81$ times that of the moon. If the location of neutral point from the centre of the earth on the line joining the centres of the earth and moon is $\frac{n d}{10}$, then find $n$.

Answer 1

Let the neutral point be located at a distance $x$ from the centre of the earth on the line joining the centres of the earth and moon. If $M_{e}$ and $M_{m}$ are the masses of the earth and the moon, respectively, and $m$ the mass of the body placed at the neutral point, then the force exerted by $M_{e}$ on $m$ must be equal and opposite to that of $M_{m}$ on $m$.
$\frac{G M_{e} m}{x^{2}}=\frac{G M_{m} m}{(d-x)^{2}}$
$\therefore \frac{M_{e}}{M_{m}}=81=\frac{x^{2}}{(d-x)^{2}}$
Since $d >x$, there is only one solution.
$\frac{x}{d-x}=+9$
or $x=\frac{9}{10} d$