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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.