Question
Two bodies of masses $$m$$ and $$4 \,m$$ are placed at a distance $$r.$$ The gravitational potential at a point on the line joining them where the gravitational field is zero is:
A.
$$ - \frac{{4Gm}}{r}$$
B.
$$ - \frac{{6Gm}}{r}$$
C.
$$ - \frac{{9Gm}}{r}$$
D.
$$Zero$$
Answer :
$$ - \frac{{9Gm}}{r}$$
Solution :
Let the gravitational field at $$P,$$ distant $$x$$ from mass $$m,$$ be zero.
$$\therefore \frac{{GM}}{{{x^2}}} = \frac{{4Gm}}{{{{\left( {r - x} \right)}^2}}}\,\,\, \Rightarrow x = \frac{r}{3}$$
Gravitational potential a $$P,\,\,V = - \frac{{Gm}}{{\frac{r}{3}}} - \frac{{4Gm}}{{\frac{{2r}}{3}}} = - \frac{{9Gm}}{r}$$
Let the gravitational field at $$P,$$ distant $$x$$ from mass $$m,$$ be zero.
$$\therefore \frac{{GM}}{{{x^2}}} = \frac{{4Gm}}{{{{\left( {r - x} \right)}^2}}}\,\,\, \Rightarrow x = \frac{r}{3}$$
Gravitational potential a $$P,\,\,V = - \frac{{Gm}}{{\frac{r}{3}}} - \frac{{4Gm}}{{\frac{{2r}}{3}}} = - \frac{{9Gm}}{r}$$