How much work is done by a spring when its end is moved from one position x_i to another x_f? Because the force is not constant, the work equation becomes:

W = ∫ F dx, with x_i and x_f the limits on the integral. Substituting -kx for F gives:

W = - ∫ kx dx, which works out to:

W = - ½ kx² | with lower limit x_i and upper limit x_f

W = ½ kx_i² - ½ kx_f²

The potential energy of a spring is given by:

U = ½ kx²

Once again this can be derived by remembering that:

The change in potential energy is the negative of the work done, and:

The negative of the work done is ½ kx_f² - ½ kx_i²

A spring is a good example of a one-dimensional system where the force varies with position. In general, in one-dimension, the work done by a variable force F(x) is given by:

W = ∫ F(x) dx

Work is the area under the force vs. distance graph.