There are two spheres one of mass M, radius R and another of has mass 4 M and radius 2R. The centre-to-centre distance between them is 12R. Find the distance from the centre of smaller sphere where net gravitational force on an object of mass m is zero.

Let mass m be placed at point A which is at a distance x from the smaller sphere such that the net

force = 0

Gravitational force of $1^{st}$ sphere

= G.F of $2^{nd}$ sphere

$\frac{\cancel{G\cancel{M}}\cancel{m}}{x^2}=\frac{4\cancel{G\cancel{M}}\cancel{m}}{{(12R-x)}^2}$                   

$\frac{1}{4}=\frac{x^2}{{\left(12R-x\right)}^2}$   

${\left(\frac{1}{2}\right)}^{\cancel{2}}={\left(\frac{x}{12R-x}\right)}^{\cancel{2}}$

$\frac{1}{2}=\frac{x}{12R-x}$        

$\text{12R}-x=2x$

$12\text{R}=2x+x$

$12\text{R}=3x$

$x=\frac{\cancel{{ }^{4}12}R}{\cancel{3}}$

$x=4\text{R}$
Therefore the distance of mass m from the smaller sphere is 4R.