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 [M/288] . The value of k is

Answer 1

Explanation:

Force on left mass m F

1

=

(3l)

2

GMm

−

(l)

2

Gmm

Force on right mass m F

2

=

(4l)

2

GMm

+

(l)

2

Gmm

When tension in the light rod is zero, F

1

=F

2

∴

(3l)

2

GMm

−

(l)

2

Gmm

=

(4l)

2

GMm

+

(l)

2

Gmm

⇒

144l

2

7GMm

=

l

2

2Gm

2

⇒m=

288

7M

⇒k=7

Answer 2

Answer:

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

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

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

∴(3l)2GMm−(l)2Gmm=(4l)2GMm+(l)2Gmm

⇒144l27GMm=l22Gm2

⇒m=2887M

⇒k=7