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(A) Let the point $O$ be at a distance $d$ from mass $M$ and at a distance $(r-d)$ from mass $4M$.At point $O$,the gravitational field intensity is ze

Vedclass · Accepted Answer

$\frac{-9 G M}{r}$

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(A) Let the point $O$ be at a distance $d$ from mass $M$ and at a distance $(r-d)$ from mass $4M$.At point $O$,the gravitational field intensity is zero,so the magnitudes of the gravitational fields due to both masses must be equal:$\frac{G M}{d^2} = \frac{G (4 M)}{(r-d)^2}$$\Rightarrow \frac{(r-d)^2}{d^2} = 4$Taking the square root on both sides:$\frac{r-d}{d} = 2$ (since $d$ must be between the masses,we take the positive root)$r-d = 2d \Rightarrow 3d = r \Rightarrow d = \frac{r}{3}$Thus,the distance from mass $4M$ is $r - d = r - \frac{r}{3} = \frac{2r}{3}$.Now,calculate the gravitational potential $V$ at point $O$:$V = V_1 + V_2 = -\frac{G M}{d} - \frac{G (4 M)}{(r-d)}$$V = -\frac{G M}{r/3} - \frac{4 G M}{2r/3}$$V = -\frac{3 G M}{r} - \frac{6 G M}{r} = -\frac{9 G M}{r}$

Two point masses having mass $M$ and $4 M$ are placed at a distance $r$. The gravitational potential at a point,where the gravitational field intensity is zero,is ..............

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