Work done by a spring

Consider a block on a horizontal, frictionless surface connected to a spring. If the spring is either stretched or compressed a small distance from its unstretched position, the force exerted by the spring is

\[F_{\mathrm{s}} = -k x\]

where \(x\) is the distance from the unstretched position and \(k\) is called the force constant of the spring (the ‘stiffness’ of the spring).

Let us calculate the work done by the spring as the block moves from a position \(x_0\) to the position \(x_1\). Applying the formal definition:
\[W = \int_{x_0}^{x_1} F_x(x) \mathrm{d}x = \int_{x_0}^{x_1}-k x\mathrm{d}x = -k \left[\frac{x^2}{2}\right]_{x_0}^{x_1} = -k \left(\frac{x_1^2}{2} - \frac{x_0^2}{2}\right) = \frac{1}{2} k x_0^2 - \frac12 k x_1^2.\]