Question

Find the gravitational potential at a point where the gravitational field intensity is zero due to two particles of masses $m_1=1kg$ and $m_2=4kg$ separated through a distance $l=3m$?

Answer 1

Let the gravitational field intensity due to the point masses $m_1$ and $m_2$ be zero at $P$.
$\Rightarrow g_P=0\Rightarrow |g_1+g_2|=0$
$\Rightarrow g_1=g_2$ (numerically)
$\Rightarrow \frac{G{m}_1}{x^2}=\frac{G{m}_2}{(l-x{)}^2}$
$\Rightarrow \frac{l}{x^2}=\frac{4}{(3-x{)}^2}$
$\Rightarrow \frac{3-x}{x}=\pm 2$
$\Rightarrow 3-x=-2x$ or, $3-x=+2x$
$\Rightarrow x=-3\left(irrelevant\right)$ or $x=1$
$V_P=(-\frac{G{m}_1}{x})+(-\frac{G{m}_2}{l-x})$
$=-G[\frac{m_1}{x}+\frac{m_2}{l-x}]$
Putting $G=6.67\times {10}^{-11}N-m^2/k{g}^2$
$m_1=1kg,m_2=4kg,l=3m,x=1m$, we obtain,
$V_P=-6.67\times {10}^{-11}(\frac{1}{1}+\frac{4}{3-1})J/kg$
$=-20.01\times {10}^{-11}J/kg$.