A uniform stick of mass $m = 18 k g$ and length 3 meters spins around on a frictionless horizontal plane, with its centre of mass stationary. A particle of mass $M$ is placed on the plane, and the stick collides elastically with it, as shown (with the contact point being the end of the stick). What should $M (i n k g)$ be so that after the collision the stick has translational motion, but no rotational motion?

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answer is 9.

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Detailed Solution

i) $L_{i} = L_{f}; \frac{m l^{2}}{12} ω = M V \times \frac{l}{2} \Rightarrow V = \frac{m ω l}{6 M}$
ii) $P_{i} = P_{f}; 0 = M V - m V_{1} \Rightarrow V = \frac{m V_{1}}{M} \Rightarrow \frac{ω l}{6} = V_{1}$
iii) $e = 1 = \frac{V - (- V_{1})}{ω \frac{l}{2} - 0} \Rightarrow V + V_{1} = \frac{ω l}{2}; \frac{m ω l}{6 M} + \frac{ω l}{6} = \frac{ω l}{2}; (\frac{m}{M} + 1) = 3$
$\Rightarrow \frac{m}{M} = 2 \Rightarrow M = \frac{m}{2} = 9 k g .$
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