A simple pendulum of length L with a bob of mass m is deflected from its rest position by an angle heta and released, as shown in the figure. The string encounters a peg fixed at a distance x below the point of suspension, and the bob starts moving in a circle centred at the peg. Provided that the initial height of the bob is less than that of the peg, what is the maximum height reached by the bob?

Step-by-step solution

According to the law of conservation of mechanical energy, the total mechanical energy of the system remains constant since the only force doing work is gravity. The tension in the string does no work as it is always perpendicular to the instantaneous velocity of the bob. Let the lowest point of the bob's trajectory be the reference level for potential energy. When the bob is released from rest at an initial height h_i , its initial kinetic energy is zero. Thus, the total initial energy is purely potential: E_i = m