A fixed solid cube of mass ‘M’, uniformly distributed, is having a very narrow tunnel along one of the body diagonals. Work done in slowly moving a point mass ‘m’ from centre to the corner is W and the work done in slowly moving the same particle from corner to a very large distance from the cube is $W'$ , then the ratio of W to $W'$ is given by ‘n’. Find n. (Consider every surface is smooth).

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Detailed Solution

If potential at the corner of solid cube is V
$V \propto \frac{m}{l}; m \propto l^{3} \Rightarrow V \propto l^{2}$ (l is side length of cube)
Potential at centre of cube = $(\frac{V}{4}) \times 8 = 2 V$
Work done in slowly moving a point mass from centre to corner
$= m (V_{c o r n e r} - V_{c e n t r e}) = m {V - (2 V)} = - m V = W$
Work done in slowly moving same particle from corner to infinity
$= m (V_{\inf i n i t y} - V_{c o r n e r}) = m (O - V) = - m V = W$