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((M)/(288))`. The value of k is

Answer

7

Explanation

From FBD of middle mass: `(GMm)/(9l^2)-(Gm^2)/l^2-T=ma`
Since T=0, `(GMm)/(9l^2) -(Gm^2)/l^2 =ma` ….(i)
Also , for the right most mass :
`(GMm)/(16l^2) + (Gm^2)/l^2 = ma` …(ii)
Equating (i) and (ii) , `(GMm)/(9l^2)-(GMm)/(16l^2)=(2GM^2)/(l^2) rArr m=(7M)/288`