Three uniform spheres each having a mass ‘M’, and radius ‘a’ are kept in such a way that each touches the other two. Find the magnitude of gravitational force on any of spheres due to the other two.

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$\sqrt{2} \frac{G M}{4 a^{2}}$

$\sqrt{2} \frac{G M}{2 a^{2}}$

$\sqrt{2} \frac{G M}{3 a^{2}}$

$\sqrt{3} \frac{G M}{4 a^{2}}$

answer is D.

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Detailed Solution

$F o r c e o f a t t r a c t i o n b e t w e e n t w o m a s s e s = \frac{G M^{} m}{r^{2}} F o r c e o n a n y s p h e r e i s F = \frac{G M^{2}}{4 a^{2}}; u n i v e r s a l g r a v i t a t i o n c o n s \tan t = G; m a s s = m; d i s \tan c e b e t w e e n t w o c e n t r e s o f s p h e r e s = r = 2 a; T h e r e s u l \tan t f o r c e o n a n y s p h e r e i s$

$f r o m p a r a l l e \log r a m l a w o f a d d i t i o n o f v e c o t o r s F^{1} = \sqrt{F_{1}^{2} + F_{2}^{2} + 2 F_{1} F_{2} \cos 60^{0}} h e r e F_{1} = F_{2} = F F^{1} = \sqrt{3} F F^{1} = \sqrt{3} \frac{G M}{4 a^{2}} = f o r c e o n e a c h s p h e r e$