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Q.

Uniform rod mass (2m), length L, is lying on a smooth horizontal surface. Two insects of mass m each moving horizontally and perpendicular to rod, hits the rod and stick to it. After hitting the rod, insects start walking along the rod away from center with constant speed v, relative to rod. Find the total angle rotated by the rod, till the insect reaches the ends of rod. (measured in radian)

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Detailed Solution

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Apply conservation of angular momentum $ω_{0} 3 V / 4 L$

$I_{1} ω_{1} = I_{2} ω_{2} \Rightarrow ω = \frac{2 ω_{0} L^{2}}{9 (\frac{L^{2}}{6} + 2 x^{2})} \Rightarrow \frac{d θ}{d x} \frac{d x}{d t} = \frac{2 ω_{0} L^{2}}{9 (\frac{L^{2}}{6} + 2 x^{2})} \Rightarrow θ = \frac{π}{12 \sqrt{3}} = \frac{π}{3 (4 \sqrt{3})}$

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