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### 4.1 Interchange between kinetic energy and potential energy

"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 brown curve(upside down curve),
What about the other(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:
1. 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.
2. This then implies that any loss in $E_{k}$ must always be associated with gain in potential energy $E_{p}$ and vice versa.
3. 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.