A simple pendulum of length L with a bob of mass m is displaced from its rest position by an angle heta = 90^ and released. The string hits a peg located at a distance x below the point of suspension, and the bob begins to move in a circle centred around the peg. Calculate the minimum value of x/L required for the bob to execute a complete circle about the peg.

Step-by-step solution

Let the point of suspension be O . The bob is released from a horizontal position ( = 90^ ), so it falls through a vertical height L before reaching its lowest point. By conservation of mechanical energy, the velocity v of the bob at the lowest point is given by: 1 2 mv^2 = mgL v = 2gL When the string is vertical, it hits the peg located at a distance x below O . The bob then begins to move in a new vertical circle centered at the peg. The radius of this new circle is R = L - x . For the bob to just complete a full