A large spherical mass is fixed at one position and two identical point masses are kept on a line passing through the centre of (see figure). [2015]

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 $r = 3 ℓ$ from $M$ , the tension in the rod is zero for $m = k (\frac{M}{288})$ . The value of $k$ is _____ . [2015]

Smart Achievers · Accepted Answer

(7)

For point mass at distance $m$

$\frac{G M m}{{(4 l)}^{2}} + \frac{G m^{2}}{l^{2}} = m a$

For point mass at distance $r = 3 l$

$\frac{G M m}{{(3 l)}^{2}} - \frac{G m^{2}}{l^{2}} = m a$

$∴$ $\frac{G M m}{{(4 l)}^{2}} + \frac{G m m}{l^{2}} = \frac{G M m}{{(3 l)}^{2}} - \frac{G m m}{l^{2}}$

$∴$ $2 m = M [\frac{1}{9} - \frac{1}{16}]$

$\Rightarrow m = \frac{7 M}{288}$

$∴$ $K = 7$

Solution

Want Us to Email you About Special Offers & Updates?

Solution

Ans. (7) For point mass at distance r=4l GMm(4l)2+Gm2l2=ma For point mass at distance r=3l GMm(3l)2-Gm2l2=ma ∴ GMm(4l)2+Gmml2=GMm(3l)2-Gmml2 ∴ 2m=M[19-116] ⇒m=7M288 ∴ K=7

Want Us to Email you About Special Offers & Updates?