Question

A pendulum, comprising a light string of length $L$ and a small sphere, swings in the vertical plane. The string hits a peg located a distance d below the point of suspension (Fig). If the pendulum is released from rest at the horizontal position $\left(\theta=90^{\circ}\right)$ and is to swing in a complete circle centered on the peg, the minimum value of d is.

• A. $\frac{L}{4}$
• B. $\frac{2L}{5}$
• C. $\frac{3L}{4}$
• D. $\frac{3L}{5}$

Answer 1

Answer: (D)

Energy is conserved in the swing of the pendulum, and the stationary peg does no work. So the ball’s speed does not change when the string hits or leaves the peg, and the ball swings equally high on both sides.
The ball will swing in a circle of radius $R =\left(L - d\right)$ about the peg.
If the ball is to travel in the circle, the minimum centripetal acceleration at the top of the circle must be that of gravity:
$\frac{m v^{2}}{R}=m g$
$\Rightarrow v^{2}=g(L-d)$
When the ball is released from rest, $U_{i} =$ mgL, and when it is at the top of the circle, $U_{i} = mg2\left(L - d\right),$ where height is measured from the bottom of the swing.
By energy conservation,
$m g L=m g 2(L-d)+\frac{1}{2} m v^{2}$
From this and the condition on $v^{2}$ we find $d=\frac{3 L}{5}$