1.A uniform rod of length $L$ and mass $M$ pivots smoothly at one end. It is released horizontally from rest. Find its angular velocity when it passes the vertical position and reaction force at pivot.
Sol: $\frac{1}{2} I \omega^2 = Mg \frac{L}{2} \implies \frac{1}{2} \left(\frac{1}{3}ML^2\right) \omega^2 = \frac{MgL}{2} \implies \omega = \sqrt{\frac{3g}{L}}$. Vertical reaction at pivot $N - Mg = M \omega^2 \frac{L}{2} = M \left(\frac{3g}{L}\right) \frac{L}{2} = \frac{3Mg}{2} \implies N = \frac{5Mg}{2}$.
2.A cue strikes a billiard ball of radius $R$ horizontally at height $h$ above center. Find $h$ such that the ball rolls purely without sliding immediately after impact.
Sol: Angular impulse = Linear impulse $\times h \implies I \omega = (m v) h \implies \left(\frac{2}{5}m R^2\right) \left(\frac{v}{R}\right) = m v h \implies h = \frac{2}{5} R$.
