Question

A larger 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 rigid massless rod of length l and this assembly is free to move along the line connecting them. All three masses interact only throght their mutual gravitational interaction. When the point mass nearer to M is at a distance r =3l form M, the tensin in the rod is zero for `m =k((M)/(288)).` The value of k is

Answer 1

Correct Answer - 7
At the point `P` the starts, let the gravitational field intensity be zero. So,
`(G(16M))/(x^(2))=(GM)/((10a-x)^(2))`
`rArr x/(10a-x)=4rArr x=-4x+40arArr x=8a`
If means that if the body crosses the point `P`, it is attracted by the other star. thus the critical velocity is the velocity of the body just to reach the point `P`, which can be given as
`(DeltaV)=1/2mv^(2)`, where `DeltaV`=potential differences between `A` and `P`
`rArr DeltaV=|"Potential at A-Potential at P"|`
`=((G(16M))/(2a)-(GM)/(8a))-((-G(16M))/(8a))+(GM)/(2a))`
`(45GM)/(8a), V=sqrt(2(DeltaV))=sqrt((90GM)/(8a))=sqrt((45GM)/(4a))`
`=3/2sqrt((GM)/a)=3xx2.414/2xxsqrt((GM)/a)`