Two particles of masses m₁ and m₂ are connected by a light spring and rotate on a smooth horizontal plane about their common centre of mass with angular speed ω. If their instantaneous separation is L, the spring tension is:
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Question

Two particles of masses m₁ and m₂ are connected by a light spring and rotate on a smooth horizontal plane about their common centre of mass with angular speed ω. If their instantaneous separation is L, the spring tension is:

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G1 Physics · Accepted Answer

Let the distances of m₁ and m₂ from the centre of mass be r₁ and r₂. Then r₁+r₂=L and m₁r₁=m₂r₂. Hence r₁ = m₂L/(m₁+m₂). The spring tension provides centripetal force for m₁: T = m₁ω²r₁ = m₁ω²[m₂L/(m₁+m₂)]. Therefore T = [(m₁m₂)/(m₁+m₂)]ω²L. The factor m₁m₂/(m₁+m₂) is the reduced mass.