Springs

Springs also exert reversible (or conservative) forces. We can therefore define a potential energy function for them. We will essentially always choose the zero of potential energy to be the equilibrium position of the spring - other choices are possible, but will only rarely be useful:

$$U(x) = -\int_{x_0}^x -k(x - x_0) dx$$ (3.38)

$$= \frac{1}{2}k x^2$$ (3.39)

or more generally:

$$U(x) = \frac{1}{2}k x^2 + U_0$$ (3.40)