"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:
- 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.
No comments:
Post a Comment