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Question

A large spherical mass M is fixed at one position and two identical point masses m are kept on a line passing through the centre of M. The point masses are connected by a rigid massless rod of length l and this assembly is free to move along the line connecting them. All three masses interact only through their mutual gravitational interaction. When the point mass nearer to M is at a distance r=3l from M, the tension in the rod is zero for m=k(M288). The value of k is
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Solution

Force on left mass m F1=GMm(3l)2Gmm(l)2

Force on right mass m F2=GMm(4l)2+Gmm(l)2

When tension in the light rod is zero, F1=F2

GMm(3l)2Gmm(l)2=GMm(4l)2+Gmm(l)2

7GMm144l2=2Gm2l2

m=7M288

k=7


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