A rod of mass M and length l stands along z-axis. Its lower end is hinged at the centre of a disc of same mass M and radius R. The whole arrangement is rotating freely about z-axis with an angular velocity ω₀. The rod falls on disc & rotates with disc. Find the angular speed of the arrangement.
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Question

A rod of mass M and length l stands along z-axis. Its lower end is hinged at the centre of a disc of same mass M and radius R. The whole arrangement is rotating freely about z-axis with an angular velocity ω₀. The rod falls on disc & rotates with disc. Find the angular speed of the arrangement.

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G1 Physics · Accepted Answer

Write translation ΣF=Ma and rotation Στ about CM=I about CMα, together with the no-slip relation v of CM=ωR and a of CM=αR whenever pure rolling applies. Solving the equations for the shown geometry gives 3ω₀R²/(2ℓ²+3R²).