Source: High school physics (Chinese)
Problem Sets:
Problem
On a smooth horizontal table, a small ball is attached to a string that passes through a central hole O. Initially, the ball undergoes uniform circular motion with speed $v_1$ and radius $r_1$. The string is then slowly pulled downwards until the ball moves in a circle of radius $r_2$.
The tension force on the ball is a central force, always directed towards the hole O. Therefore, the torque about O is zero ($\vec{\tau} = \vec{r} \times \vec{F} = 0$). This means the angular momentum of the ball about O is conserved. Initial angular momentum: $L_1 = m v_1 r_1$ Final angular momentum: $L_2 = m v_2 r_2$ By conservation of angular momentum, $L_1 = L_2$:
$$m v_1 r_1 = m v_2 r_2$$Solving for $v_2$:
$$v_2 = \frac{r_1 v_1}{r_2}$$