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A body of mass 2M is divided into four parts {m, M - m, m, M - m}, which are then placed at the four corners of a square. The ratio M/m for which the gravitational potential energy of this configuration is maximum is x: 1. Find the value of x.
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Correct answer: 2
Solution
With masses m, M-m, m, M-m at the square corners (equal masses at diagonal corners), the total gravitational PE depends on the pairwise products. Writing U in terms of m and the side a (and diagonal a*sqrt2), then maximizing U (least negative) by dU/dm = 0 gives the condition M = 2m, i.e. M/m = 2. So x = 2.
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