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Energy equations of SHM in terms of time

The kinetic energy $$(E_{\text{k}})$$ and the potential energy $$(E_{\text{p}})$$ of an object undergoing simple harmonic motion in terms of time $$(t)$$ are given by different sets of equations.

When the motion begins from the equilibrium point (i.e. when $$t=0$$, $$x=0$$), $$v=\omega x_{0}\cos\omega t$$. When the motion begins from an extreme position (i.e. when $$t=0$$, $$x=\pm x_{0}$$), $$v=\mp\omega x_{0}\sin\omega t$$.

The expression for kinetic energy is obtained by plugging the equation for the velocity of a simple harmonic oscillator with respect to time in the kinetic energy equation (i.e. $$E_{\text{k}}=mv^{2}/2$$).

The expression for potential energy $$E_{p}$$ is determined by taking the total energy minus the kinetic energy (i.e. $$E_{\text{p}}=E_{\text{T}}-E_{\text{k}}$$).

Equilibrium position Positive extreme position Negative extreme position
$$v$$ $$\omega x_{0}\cos\omega t$$ $$-\omega x_{0}\sin\omega t$$ $$\omega x_{0}\sin\omega t$$
$$E_{\text{k}}$$ $$\frac{1}{2}m\omega^{2}x_{0}^{2}\cos^{2}\omega t$$ $$\frac{1}{2}m\omega^{2}x_{0}^{2}\sin^{2}\omega t$$ $$\frac{1}{2}m\omega^{2}x_{0}^{2}\sin^{2}\omega t$$
$$E_{\text{p}}$$ $$\frac{1}{2}m\omega^{2}x_{0}^{2}\sin^{2}\omega t$$ $$\frac{1}{2}m\omega^{2}x_{0}^{2}\cos^{2}\omega t$$ $$\frac{1}{2}m\omega^{2}x_{0}^{2}\cos^{2}\omega t$$

The potential energy of a simple harmonic oscillator is made up of different types of energy or combinations thereof:

Elastic potential energy in the case of a spring-mass system

Gravitational potential energy in the case of a simple pendulum

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