A smooth uniform rod of length 'L' and mass 'M' has two identical beads of negli

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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 an axis perpendicular to the rod and passing through the mid point of the rod (see figure). There are no external forces. When the beads reach the ends of the rod, the angular velocity of the system is

A Smooth Uniform Rod Of Length 'L' And Mass 'M' Has Two Iden...

Options:

A . ω0

B . Mω0M+12m

C . Mω0M+2m

D . Mω0M+6m

Answer: Option D
:
D
Since there are no external forces therefore the angular momentum of the system remains constant.
Initially when the beads are at the centre of the rod angular momentum

L_{1} = (\frac{M L^{2}}{12}) ω_{0}

.....(i)
When beads reach the ends of the rod then angular momentum

= (m {(\frac{L}{2})}^{2} + m {(\frac{L}{2})}^{2} + \frac{M L^{2}}{12}) ω^{'}

..(ii) Equating (i) and (ii)

\frac{M L^{2}}{12} ω_{0} = (\frac{m L^{2}}{2} + \frac{M L^{2}}{12}) ω^{'} \Rightarrow ω^{'} = \frac{M ω_{0}}{M + 6 m} .

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