Energy and Amplitude in SHM

In simple harmonic motion, the amplitude $A$ is the maximum displacement from equilibrium. At this point, the mass momentarily stops before reversing direction. With velocity zero, kinetic energy vanishes. All mechanical energy resides in elastic potential energy.

For a system undergoing SHM, elastic potential energy at displacement $x$ from equilibrium is

$$U = \frac{1}{2}kx^2$$

where $k$ is the spring constant. At maximum displacement $x = \pm A$, potential energy reaches its peak:

$$U_{\text{max}} = \frac{1}{2}kA^2$$

Since mechanical energy is conserved in an ideal oscillator, this maximum potential energy equals the total energy $E$:

$$E = U_{\text{max}} = \frac{1}{2}kA^2$$

This connects amplitude directly to stored energy. Given $k$ and $A$, the maximum potential energy follows immediately. Given $E$ and $k$, the amplitude is

$$A = \sqrt{\frac{2E}{k}}$$

The diagram below shows a spring-mass system at maximum stretch, where all energy is stored in the spring.