Question

Two identical spherical masses are kept at some distance. Potential energy when a mass 'm' is taken from the surface of one sphere to the other

• A. increases continuously
• B. decreases continuously
• C. first increases, then decreases
• D. first decreases, then increases

Answer 1

Answer: (C)

Gravitational potential energy between two masses (M) separated by a distance $R$ is $-GMM/R=-G{M}^2/R$.
When a small mass $m$ is taken out of mass $M$, and kept at a distance $x$ from $(M-m)$, the total $PE$ of the system becomes $-GMm/(R-x)-Gm(M-$ $m)/x-GM(M-m)/R$
As m moves towards M, the gravitational PE in the system will increase. Mass (M-m) will attract mass $m$ and external for will be required to move $m$ away from (M-m).
Once mass $m$ reaches a point $P$ where $R_1$ is the distance between mass $m$ and $M$ and $R_2$ is the distance between $m$ and $(M-m)$ and $R_1$ and $R_2$ related as $R_1/R_2=\sqrt{M/(M-m)}$, the potential energy on mass $m$ will be zero. At this point mass $m$ will not experience any force.
As mass m stars moving towards mass $M$, the PE will decrease as mass $M$ will attract mass $m$.
So, in this situation, PE of the total system will increase first and then decrease.