Consider a mass m situated at the end of a spring of length L with negligible mass. Hooke's Law states states that the reaction of the spring is proportional to its displacement x from the equilibrium point , Fspring = -k x , where k is a constant which depends on the physical characteristics of the spring. Applying Newton's Second Law leads to the following second order differential equation
d2x / dt2 = -(k/m) x .
This equation occurs frequently through the sciences and produces simple harmonic motion. Because the solution depends only on the ratio k/m and because this ratio is always a positive number, the differential equation is usually recast in terms of an angular frequency parameter ω = (k/m)1/2 .
d2x / dt2 = - ω2 x
The analytic solution to this differential equation is well know and can be expressed in terms of a sinusoidal function
x(t) = A sin( ωt + φ )
where the amplitude A is the maximum displacement and the phase angle φ is determines that location of mass at time t=0. The angular frequency is measured in radians per second and related to the frequency of oscillation by ω= 2 π f.