A simple pendulum of length L with a bob of mass m is deflected from its equilibrium position by an angle heta = 90^ and released. The string strikes a peg fixed at a distance x = L/2 directly below the suspension point, causing the bob to travel in a circle centred at the peg. Find the maximum height attained by the bob above its lowest position before the string becomes slack.

Step-by-step solution

Let the lowest point of the pendulum be the reference level for potential energy. The initial height of the bob when released from = 90^ is L . By conservation of mechanical energy, the velocity v₀ of the bob at the lowest point is given by: 1 2 m v₀^2 = mgL v₀ = 2gL When the string strikes the peg at a distance x = L/2 below the suspension point, the bob starts moving in a new circle of radius R = L - x = L/2 centered at the peg. Let the string become slack when it makes an angle with the upward vertical. The heig