Simple Harmonic Oscillator

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.