A smooth uniform rod of length L and mass M has two identical beads of negligible size, each of mass m , which can slide freely along the rod. Initially the two beads are at the centre of the rod and

Question

A smooth uniform rod of length L and mass M has two identical beads of negligible size, each of mass m , which can slide freely along the rod. Initially the two beads are at the centre of the rod and the system is rotating with angular velocity ω 0 about its axis perpendicular to the rod and passing through its mid point (see figure). There are no external forces. When the beads reach the ends of the rod, the angular velocity of the system is

TestHub · Accepted Answer

From the principle of conservation of angular momentum, $I_{0} ω_{0} = I ω$ , where $I_{0}$ and $ω_{0}$ are the moment of inertia and angular velocity when the beads are at the centre of the rod and $I$ and $ω$ those when the beads are at the ends of the rod.

$I_{0} = \frac{M L^{2}}{12}$ and $I = \frac{M L^{2}}{12} + \frac{m L^{2}}{4} + \frac{m L^{2}}{4} = \frac{L^{2}}{12} (M + 6 m)$

$∴ \frac{M L^{2}}{12} ω_{0} = \frac{(M + 6 m) ω L^{2}}{12}$

Or $ω = \frac{M ω_{0}}{M + 6 m}$

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