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 $$m=kM288$$. The value of k is

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  $$m=kM288$$.  The value of  k is

Infinity Learn · Accepted Answer

Net force on mass $$no.2$$,$$F2=F21$$−$$F23$$⇒$$F2=GMm(3l)2$$−$$Gmm(l)2Net$$ force on mass $$no.3$$,$$F3=F31+F32$$⇒$$F3=GMm(4I)2+Gmm(l)2$$ For tension in rod to be zero, $$F2=F3$$⇒$$GMm(3I)2$$−$$Gmm(I)2=GMm(4I)2+Gmm(l)2$$⇒$$M9$$−$$m=M16+m$$⇒$$m=7M288$$

Gravitational force and laws

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Gravitational force and laws

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Net force on mass no.2,F2=F21−F23⇒F2=GMm(3l)2−Gmm(l)2Net force on mass no.3,F3=F31+F32⇒F3=GMm(4I)2+Gmm(l)2 For tension in rod to be zero, F2=F3⇒GMm(3I)2−Gmm(I)2=GMm(4I)2+Gmm(l)2⇒M9−m=M16+m⇒m=7M288

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