Rayleigh's Method

We can use the principle of conservation to obtain the natural frequency of a mass-spring system if the spring is linear. This can be useful because we don’t need to find the equation of motion first.

In simple harmonic motion, the kinetic energy is maximum and the potential energy is minimum at the equilibrium position $x=0$. When the displacement is maximum, the the potential energy is maximum but the kinetic energy is zero. Thus, we have:

$${\text{PE}}_{\text{max}}+{\text{KE}}_{\text{min}}={\text{PE}}_{\text{min}}+{\text{KE}}_{\text{max}}$$