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Energy equations of SHM in terms of displacement

The kinetic energy $$(K)$$ and the potential energy $$(U)$$ in terms of displacement $$(x)$$ are derived from the equation for the velocity in terms of displacement ($$v=\pm\sqrt{x_{0}^{2}-x^{2}}$$). This is given by:$$$\begin{align*}E_{\text{k}}&= \frac{1}{2}mv^{2}\\&=\frac{1}{2}m\omega^{2}(x_{0}^{2}-x^{2})\\E_{\text{p}}&=E_{\text{T}}- E_{\text{k}}\\&=\frac{1}{2}m\omega^{2}x^{2}\end{align*}$$$

Note that both energies are positive since $$x$$ is always smaller than $$x_{0}$$.

$$m=$$mass of the object; $$x_{0}=$$maximum displacement; $$\omega=$$angular frequency.