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Physics Question on Gravitational Potential Energy

Two small and heavy spheres, each of mass MM, are placed a distance rr apart on a horizontal surface. The gravitational potential at the mid-point on the line joining the centre of the spheres is

A

Zero

B

GMr\frac{GM}{r}

C

−2GMr-\frac{2GM}{r}

D

−4 GMr\frac{-4\,GM}{r}

Answer

−4 GMr\frac{-4\,GM}{r}

Explanation

Solution

Gravitation potential at OO due to the mass MM at
A,VA=−GMr/2A, V_{A}=\frac{-G M}{r / 2}

Similarly, for the mass MM at BB,
gravitation potential VB=−GMr/2V_{B}=-\frac{G M}{r / 2}
Total potential VA+VB=−GMr/2−GMr/2V_{A}+V_{B}=-\frac{G M}{r / 2}-\frac{G M}{r / 2}
=−4GMr=-\frac{4 G M}{r}