In the arrangement depicted in the figure, the system is released from rest while the spring is initially unstretched. Assuming the pulley is frictionless, determine the maximum elongation of the spring.

Step-by-step solution

Let the maximum elongation of the spring be x . At the position of maximum elongation, the velocity of the block is zero. Therefore, the change in kinetic energy of the system is zero ( K = 0 ). As the block of mass m moves downwards by a distance x , the string pulls the top end of the spring upwards by the same distance x . Since the bottom end of the spring is fixed to the ground, the spring gets stretched by x . The loss in gravitational potential energy of the block is: U_g = -mgx The gain in elastic potential