Energy of a Simple Harmonic Oscillator

Energy changes in according to displacement

The kinetic energy of a simple harmonic oscillator is

\begin{aligned} E_\text{kinetic} &= \frac{1}{2}m \omega^2(x_o^2 - x^2) \end{aligned}

In a SHM, the oscillator’s kinetic energy and potential energy always changes from maximum to zero throughout the oscillations. However, at all time, the total energy of the oscillator is constant. This value can be obtained by calculating the maximum kinetic energy of the system:

\begin{aligned} E_\text{total} &= \frac{1}{2}mv_o ^2\\&=\frac{1}{2}m \omega^2 x_o^2 \end{aligned}

To find out the potential energy of a simple harmonic oscillator,

\begin{aligned} E_\text{potential} &= E_\text{total} - E_\text{kinetic}\\&=\frac{1}{2}m \omega^2x_o^2 - \frac{1}{2}m \omega^2(x_o^2 - x^2)\\&=\frac{1}{2}m\omega^2 x^2\end{aligned}

The total energy is constant, but the kinetic and potential energy changes throughout the oscillator.
The total energy is constant, but the kinetic and potential energy changes throughout the oscillator.

The energy changes of an SHM oscillator changes in a sinusoidal pattern.

Energy changes according to time

From the velocity equation, the kinetic energy is

\begin{aligned} E_\text{kinetic}&=\frac{1}{2}mv^2\\&=\frac{1}{2}mv_o ^2 \cos^2{\omega t}\end{aligned}

The potential energy is the difference between the total energy and the kinetic energy,

\begin{aligned} E_\text{potential}&=\frac{1}{2}mv_o ^2 - \frac{1}{2}mv_o ^2 \cos^2{\omega t}\\&=\frac{1}{2}mv_o^2(1-\cos^2{\omega t})\\&=\frac{1}{2}mv_o \sin^2{\omega t} \end{aligned}