Halliday, Resnick & Walker — Potential Energy and Conservation of Energy: Problem 84_CALC

A certain spring is found not to conform to Hooke’s law. The force (in newtons) it exerts when stretched a distance \( x \) (in meters) is found to have magnitude \( 52.8x + 38.4x^2 \) in the direction opposing the stretch. (a) Compute the work required to stretch the spring from \( x = 0.500 \text{ m} \) to \( x = 1.00 \text{ m} \). (b) With one end of the spring fixed, a particle of mass \( 2.17 \text{ kg} \) is attached to the other end of the spring when it is stretched by an amount \( x = 1.00 \text{ m} \). If the particle is then released from rest, what is its speed at the instant the stretch in the spring is \( x = 0.500 \text{ m} \)? (c) Is the force exerted by the spring conservative or nonconservative? Explain.