Chapter 4.3, Problem 57E

Problems 57–58 use the fact that a physical system is in stable equilibrium if the total energy, E, is a local minimum.⁷

A mass m hanging on the end of a spring extends its length by y. See Figure 4.55. For g, the acceleration due to gravity, and positive constant k, the total energy is

$E=\frac{1}{2}k{y}^2-mgy.$

Is there a length that gives a stable equilibrium position for a constant mass m? If so, what is it?

Figure 4.55