Consider two identical rings of mass M and radius R fixed on a horizontal axis in a gravity-free space. A particle of mass m is held at rest at the centre of 1 st ring and projected along the axis tow

Question

Consider two identical rings of mass M and radius R fixed on a horizontal axis in a gravity-free space. A particle of mass m is held at rest at the centre of 1 st ring and projected along the axis towards 2 nd ring. The minimum speed of projection so that the particle reaches the centre of 2 nd ring is

TestHub · Accepted Answer

The particle should be projected such that it can reach halfway because after midpoint it will move under the attraction of 2^ndring and will reach the center of the ring.

${PE}_{i} + {KE}_{i} = {KE}_{f} + {PE}_{f}$

$\frac{- GMm}{R} - \frac{GMm}{\sqrt{5} R} + \frac{1}{2} {mv}^{2} = - \frac{GMm}{\sqrt{2} R} \times 2$

$v = \sqrt{\frac{2 G M}{R} [1 + \frac{1}{\sqrt{5}} - \sqrt{2}]}$

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