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physics

mechanics

Easy

Question

A smooth uniform rod of length L and mass M has two identical beads (1 and 2) 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 $omega subscript a$ about an axis perpendicular to the rod and is passing through its midpoint. There are no external forces when the beads reach the ends of the rod, the angular velocity of the system is

$ω_{0}$
$\frac{M ω_{0}}{M + 6 m}$
$\frac{M ω_{0}}{M + 12 m}$
$\frac{M ω_{0}}{m}$

The correct answer is: $\frac{M ω_{0}}{M + 6 m}$

From the conservation of angular momentum
$table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em rowspacing 3 pt end attributes row cell fraction numerator M L squared over denominator 12 end fraction omega subscript 0 end cell cell equals left square bracket fraction numerator M L squared over denominator 12 end fraction plus 2 m left parenthesis L over 2 right parenthesis squared right square bracket omega end cell row omega cell equals left parenthesis fraction numerator M over denominator M plus 6 m end fraction right parenthesis omega subscript 0 end cell end table$

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