Gravitation 5 Question 10
11. A large spherical mass $M$ is fixed at one position and two identical 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 $r=3 l$ from $M$ the tension in the rod is zero for $m=k \frac{M}{288}$. The value of $k$ is
(2015 Adv.)
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Answer:
Correct Answer: 11. (7)
Solution:
- For point mass at distance $r=3 l$
$$ \frac{G M m}{(3 l)^{2}}-\frac{G m^{2}}{l^{2}}=m a $$
For point mass at distance $r=4 l$
$$ \frac{G M m}{(4 l)^{2}}+\frac{G m^{2}}{l^{2}}=m a $$
Equating the two equations, we have
$\frac{G M m}{9 l^{2}}-\frac{G m^{2}}{l^{2}}=\frac{G M m}{16 l^{2}}+\frac{G m^{2}}{l^{2}}$
$$ \begin{aligned} \frac{7 G M m}{144} & =\frac{2 G m^{2}}{l^{2}} \\ m & =\frac{7 M}{288} \end{aligned} $$
