You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard X

Physics

Gravitational Force

A large spher...

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 $r = 3 l$ from $M$ , the tension in the rod is zero for $m = k (\frac{M}{288})$ . The value of $k$ is

Open in App

Solution

Force on left mass $m$ $F_{1} = \frac{G M m}{(3 l)^{2}} - \frac{G m m}{(l)^{2}}$

Force on right mass $m$ $F_{2} = \frac{G M m}{(4 l)^{2}} + \frac{G m m}{(l)^{2}}$

When tension in the light rod is zero, $F_{1} = F_{2}$
$∴ \frac{G M m}{(3 l)^{2}} - \frac{G m m}{(l)^{2}} = \frac{G M m}{(4 l)^{2}} + \frac{G m m}{(l)^{2}}$
$\Rightarrow \frac{7 G M m}{144 l^{2}} = \frac{2 G m^{2}}{l^{2}}$
$\Rightarrow m = \frac{7 M}{288}$
$\Rightarrow k = 7$

Suggest Corrections

Similar questions

Q. 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 = 3 l

from

M

, the tension in the rod is zero for

m = k (\frac{M}{288})

. The value of

k

Q. 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

as shown in the 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 distance

r = 3 l

from

M

, the tension in the rod is zero for

m = k (\frac{M}{288})

. The value of

k

Q. 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 interact only

through their mutual gravitational interaction. When the point mass m 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

Q. Find the potential energy of the gravitational interaction of a mass point of mass

m

and a thin uniform rod of mass

M

and length

l

, if they are located along a straight line at a distance

a

from each other; also find the force of their interaction.

Q. Two masses

m

and

\frac{m}{2}

are connected at the two ends of a massless rigid rod of length

l

. The rod is suspended by a thin wire of torsional constant

k

at the centre of mass of the rod-mass system (see figure). Because of the torsional constant

k

, the restoring torque is

τ = k θ

for angular displacement

θ

. If the rod is rotated by

θ_{0}

and released, the tension in it when it passes through its mean position will be:

Join BYJU'S Learning Program

Force on left mass m F1=GMm(3l)2−Gmm(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)2−Gmm(l)2=GMm(4l)2+Gmm(l)2 ⇒7GMm144l2=2Gm2l2 ⇒m=7M288 ⇒k=7