Question

A simple pendulum of length L having a bob of mass m is deflected from its rest position by an angle 9 and released (figure 8-E16). The string hits a peg which is fixed at a distance x below the point of suspension and the bob starts going in a circle centred at the peg. (a) Assuming that initially the bob has a height less than the peg, show that the maximum height reached by the bob equals its initial height. (b) If the pendulum is released with θ = 90° and x=L/2 find the maximum height reached by the bob above its lowest position before the string becomes slack. (c) Find the minimum value of x/L for which the bob goes in a complete circle about the peg when the pendulum is released from θ = 90°.

Answer 1

a) When the bob has an initial height less than the peg and then released from rest (figure 1), let body travels from A to B. Since, Total energy at A = Total energy at B,

So, the maximum height reached by the bob is equal to initial height.
b) When the pendulum is released with θ = 90° and x = L/2, (figure 2) the path of the particle is shown in the figure 2. At point C, the string will become slack and so the particle will start making projectile motion. (Refer Q.No. 56)
(1/2)mvc2 – 0 = mg (L/2) (1 – cos θ)

distance between A nd C in the vertical direction is L/2 (1 – cos α)

Again, applying energy principle between A and C,