First slide

Rotational motion

Question

A thin rod of mass 0.9 kg and length 1 m is suspended, at rest, from one end so that it can freely oscillate in the vertical plane. A particle of mass 0.1 kg moving in a straight line with velocity 80 m/s hits the rod at its bottom most point and sticks to it (see figure). The angular speed (in rad/s) of the rod immediately after the collision will be____

Moderate

Solution

Applying conservation of angular momentum of particle + rod about the hinge
$\begin{array}{l} L_{i} = L_{f} \\ \Rightarrow m u L + I_{r o d} w_{i n i t i a l} = (I_{p a r t i c l e} + I_{r o d}) ω_{f i n a l} \\ \Rightarrow (m \times u \times L) + 0 = (m R^{2} + \frac{M R^{2}}{3}) ω \\ \Rightarrow (1 \times 0.1 \times 80) + 0 = (0.1 \times 1^{2} + \frac{0.9 \times 1^{2}}{3}) ω \\ \Rightarrow 8 = 0.4 ω \\ \Rightarrow ω = 20 r a d / s \end{array}$

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