A thin horizontal rod AB of length $L$ falls freely from a height $H$ above the surface of the table, in such a way that the end A of the rod just hits the edge of the table. This collision is instantaneous and totally elastic. The angular velocity of the rod after the collision is $\frac{\sqrt{n g H}}{L}$ . Find the value of ‘ $n$ ’.

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Detailed Solution

From the conservation of angular momentum
$m \sqrt{2 g H} \frac{L}{2} = \frac{m L^{2} ω}{12} + m v \frac{L}{2}$
$\sqrt{2 g H} = - v + \frac{ω L}{2}$ [This is the equation of e is applied only at the point of collision]
$∴ v = \sqrt{\frac{g H}{2}}$
$ω = \frac{3 \sqrt{2 g h}}{L}$

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