A pendulum comprising a light string of length `L` and small sphere, swings in the vertical plane . The string hits a peg located a distance of bellow the point of suspension (figure). If the pendulum is released from rest at the horizontal possition `(theta = 90^(@))` and is to swing in a complete circule centered on the peg, the minimum value of `d` is

Options

A. `(L)/(4)`
B. `(2L)/(4)`
C. `(3L)/(4)`
D. `(3L)/(5)`

Answer

D

Explanation

Energy is conserved in the swing of the pendulum , and the stationary peg does no work . So the ball's speed does nolt change when the string hits of leaveles
the peg and the ball swings equally high on both sides.
The ball will awing in a circule of radius `R = (L -d)` about the peg . If the ball is to travel in the circule, the minimum centripental acceleration at the top of the circule must be that of gravity:
`(mv^(2))/( R) = mg implies v^(2) = g(L -d)`
When the ball is relaased from rest , `U_(i) = mgL`, and when it is at the top of the circle. `U_(i) = mg2 (L - d)`, where height is measered from the bottom of the swing. By energy conservation,
`mgL = mg2(L - d) + (1)/(2) mv^(2)`
From this and the condition on `v^(2)` we final `d = (3L)/(5)`