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Question:

A thin smooth rod of length L and mass M is rotating freely with angular speed ω₀ about an axis perpendicular to the rod and passing through its center. Two beads of mass m and negligible size are at the center of the rod initially. The beads are free to slide along the rod. The angular speed of the system, when the beads reach the opposite ends of the rod, will be?

Mω₀M+2m

Mω₀M+6m

Mω₀M+3m

Mω₀M+m

Solution:

Correct option is D. Mω₀/(M+6m)
Applying angular momentum conservation, about axis of rotation
Li = Lf
(ML²/12)ω₀ = ((ML²/12) + m(L²/2) × 2)ω
=> ω = Mω₀/(M+6m).