Question

A cubical block of mass $m$ and side '$a$' is moving horizontally on a smooth horizontal surface with a velocity $v_0$. The block collides with a ridge $P$ and starts rotating about $P$. Find the minimum value of $v_0$ needed to overcome the ridge.

Answer 1

Consider Initial angular momentum of the block

$L_i=Mv\times AC$

Angular momentum of the block after it hits the point O,

$L_f=I_c\omega$

From parallel axis theorem,

$I_o=I_c+MxO{C}^2=\frac{1}{6}M{a}^2+Mx\left(A{C}^2+A{O}^2\right)=\frac{1}{6}M{a}^2+\frac{1}{2}M{a}^2=\frac{2}{3}M{a}^2$

Using conservation of angular momentum,

$\frac{1}{2}Mva=\frac{2}{3}M{a}^2\omega$

$\omega =\frac{3v}{4a}$