Calculating a Spring’s Potential & Kinetic Energy

W = Fs

As you stretch or compress a spring, the force varies, but it varies in a linear way (because in Hooke’s law, force is proportional to the displacement).

The distance (or displacement), s, is just the difference in position, x_f – x_i, and the average force is (1/2)(F_f + F_i). Therefore, you can rewrite the equation as follows:

Hooke’s law says that F = –kx. Therefore, you can substitute –kx_f and –kx_i for F_f and F_i:

Distributing and simplifying the equation gives you the equation for work in terms of the spring constant and position:

The work done on the spring changes the potential energy stored in the spring. Here’s how you give that potential energy, or the elastic potential energy:

For example, suppose a spring is elastic and has a spring constant, k, of

and you compress the spring by 10.0 centimeters. You store the following amount of energy in it:

You can also note that when you let the spring go with a mass on the end of it, the mechanical energy (the sum of potential and kinetic energy) is conserved:

PE₁ + KE₁ = PE₂ + KE₂

When you compress the spring 10.0 centimeters, you know that you have

of energy stored up. When the moving mass reaches the equilibrium point and no force from the spring is acting on the mass, you have maximum velocity and therefore maximum kinetic energy — at that point, the kinetic energy is

by the conservation of mechanical energy.