Two bodies of masses m and M are placed a distance d apart. The gravitational potential at the position where the gravitational field due to them is zero is V.
Answer & explanation
Correct answer: option 4
Let gravitational field be zero at a point lying at distance x from M. Then,
$\frac{G M}{x^2}=\frac{G m}{(d-x)^2}$
$\Rightarrow \frac{d-x}{x}=\sqrt{\frac{m}{M}}$
$\Rightarrow \frac{d}{x}-1=\sqrt{\frac{m}{M}}$
$\Rightarrow x=\left(\frac{\sqrt{M}}{\sqrt{M}+\sqrt{m}}\right) d$ ......(1)
$\Rightarrow(d-x)=\left(\frac{\sqrt{M}}{\sqrt{M}+\sqrt{m}}\right) d$ ......(2)
Since, $V_p=-\frac{G m}{d-x}-\frac{G m}{x}$ ......(3)
Substituting (1) and (2) in (3), we get
$V_p=-\frac{G}{d}(\sqrt{m}+\sqrt{M})^2$