*"Energy can neither be created nor destroyed but can transform from one form to another.."*

From the previous section, we know the velocity is given by the following expression,

$$v=v_{0}cos(\omega t)=\pm \omega \sqrt{{x_{0}}^{2}-{x}^{2}}$$

Therefore, the kinetic energy of a body performing simple harmonic motion is equal to,

\begin{align*}
E_{k}&=\frac{1}{2}mv^{2}\\ &=\frac{1}{2}m\omega^{2}({x_{0}}^2-x^{2})\end{align*}

The plot then of kinetic energy for some amplitude $x_{0}$ and angular frequency $\omega$, is shown below as the

What about the other(**brown**curve(upside down curve),**orange**) curve? The orange curve is the potential energy stored in the sytem by virtue of its position. $E_{p}=\frac{1}{2}kx^{2}$

The Total Energy for the spring system(for e.g) is $E_{t}=E_{k}+ E_{p}$, which is simply a constant as shown below,

$$E_{t}=\frac {1}{2}m\omega^{2}{x_{0}}^{2}$$

**Deduction:**

**In absence of frictional and viscous forces(as in the cases for simple pendulum and spring), the total energy is always constant, or in other words energy is always conserved.****This then implies that any loss in $E_{k}$ must always be associated with gain in potential energy $E_{p}$ and vice versa.****Therefore, we can see in the plot that when either $E_{k}$ is maximum the $E_{p}$ is zero and likewise the reverse, when $E_{p}$ is maximum the $E_{k}$ is minimum.**

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