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 (see figure). 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 . The value of k is:

Engineering

Physics

Gravitational Force and Field

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 (see figure). 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 $(\frac{M}{288})$ . The value of k is:

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For tension in the rod to be zero.

Both masses should move with same acceleration, only due to gravitation alteraction

$\frac{GMm}{9 ℓ^{2}} - \frac{{Gm}^{2}}{ℓ^{2}} = \frac{GMm}{16 ℓ^{2}} + \frac{{Gm}^{2}}{ℓ^{2}}$

$\frac{GMm}{ℓ^{2}} {\frac{7}{9 \times 16}} = \frac{2 {Gm}^{2}}{ℓ^{2}}$

$m = (\frac{7 M}{288}) = K \frac{M}{288} So K = 7$