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Question

A smooth light horizontal rod AB can rotate about a vertical axis passing through its end A. The rod is fitted with a small sleeve of mass m attached to the end A by a weightless spring of length l0 and stiffness κ. What work must be performed to slowly get this system going and reaching the angular velocity ω?

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Solution

Let the deformation in the spring be Δl, when the rod AB has attained the angular velocity ω. From the second law of motion in projection form Fn=mwn.
κΔl=mω2(l0+Δl) or, Δl=mω2l0κmω2
From the energy equation, Aext=12mv2+12κΔl2
=12mω2(l0+Δl)2+12κΔl2
=12mω2(l0+mω2l0κmω2)2+12κ(mω2l0κmω2)2
On solving Aext=κ2l20η(1+η)(1η)2, where η=mω2κ

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