Hooke's Law - The Spring Equation

The Spring Equation

The most commonly encountered form of Hooke's law is probably the spring equation, which relates the force exerted by a spring to the distance it is stretched by a spring constant, k, measured in force per length.

The negative sign indicates that the force exerted by the spring is in direct opposition to the direction of displacement. It is called a "restoring force", as it tends to restore the system to equilibrium. The potential energy stored in a spring is given by

which comes from adding up the energy it takes to incrementally compress the spring. That is, the integral of force over displacement. (Note that potential energy of a spring is always non-negative.)

This potential can be visualized as a parabola on the U-x plane. As the spring is stretched in the positive x-direction, the potential energy increases (the same thing happens as the spring is compressed). The corresponding point on the potential energy curve is higher than that corresponding to the equilibrium position (x = 0). The tendency for the spring is to therefore decrease its potential energy by returning to its equilibrium (unstretched) position, just as a ball rolls downhill to decrease its gravitational potential energy.

If a mass m is attached to the end of such a spring, the system becomes a harmonic oscillator. It will oscillate with a natural frequency given either as an angular frequency

or as a natural frequency

This idealized description of spring mechanics works as long as the mass of the spring is very small compared to the mass m, there is no significant friction on the system, and the spring is not overextended beyond its natural range (which can deform it permanently).