Randall D. Knight — Work and Kinetic Energy: Problem 51

Hooke’s law describes an ideal spring. Many real springs are better described by the restoring force \((F_{Sp})_s = -k \Delta s - q (\Delta s)^3\), where \(q\) is a constant. Consider a spring with \(k = 250\text{ N/m}\) and \(q = 800\text{ N/m}^3\). a. How much work must you do to compress this spring \(15\text{ cm}\)? Note that, by Newton’s third law, the work you do on the spring is the negative of the work done by the spring. Hint: Let the spring lie along the \(s\)-axis with the equilibrium position of the end of the spring at \(s = 0\). Then \(\Delta s = s\). b. By what percent has the cubic term increased the work over what would be needed to compress an ideal spring?