Gravitation

Question :

Two bodies of masses M₁ and M₂ are placed at a distance d apart .What is the potential at the position where the gravitational field due to them is zero?

a)$-\frac{G}{d}(M_1+M_2)$

b)$-\frac{G}{d}(M_1+M_2+2\sqrt{M_1{M}_2})$

c)$\frac{G}{d}(M_1-M_2)$

d) None of these

Solution :

Let the gravtiational field be zero at a point distant x from M₁

$\frac{G{M}_1}{x^2}=\frac{G{M}_2}{{(d-x)}^2}$

$\frac{x}{d-x}=\sqrt{\frac{M_1}{M_2}}$

$x\sqrt{M_2}=\sqrt{M_1}.d-x\sqrt{M_1}$

$x[\sqrt{M_1}$+$\sqrt{M_2}]=\sqrt{M_1}d$

$x=\frac{d\sqrt{M_1}}{\sqrt{M_1}+\sqrt{M_2}}$

$d-x=\frac{\sqrt{M_2}}{\sqrt{M_1}+\sqrt{M_2}}$

Potentail at this point due to both the masses will be

$=-\frac{G{M}_1}{x}-\frac{G{M}_2}{(d-x)}=G[\frac{M_1(\sqrt{M_1}+\sqrt{M_2})}{d\sqrt{M_1}}+\frac{M_2(\sqrt{M_1}+\sqrt{M_2})}{d\sqrt{M_2}}]$

$=-\frac{G}{d}(M_1+M_2+2\sqrt{M_1{M}_2})$