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Q.
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
Gravitation
Solution:
Gravitational potential energy between two masses (M) separated by a distance $R$ is $- GMM / R =- GM ^{2} / R$.
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.