Energy in Simple Harmonic Oscillators

In simple harmonic motion, the total mechanical energy $E$ is the sum of kinetic energy and potential energy.

Kinetic Energy

The kinetic energy of an oscillating mass is:

$$K = \frac{1}{2}mv^2$$

where $m$ is the mass and $v$ is the instantaneous velocity.

Potential Energy in SHM

For a system in SHM subject to a linear restoring force, the potential energy is elastic potential energy stored:

$$U = \frac{1}{2}kx^2=\frac{1}{2}m\omega^2x^2$$

where $k=m\omega^2$ is the 'spring' constant (or stiffness) and $x$ is the displacement from equilibrium. This energy increases as the mass travels away from equilibrium.

Total Mechanical Energy

...