# 💯An Ingelious Way on Oscillations Summary

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## Oscillations Summary

Here’s an “ingelious” way on JC A Level H2 Physics Oscillations summary for Simple Harmonic Motion.

Let’s proceed to derive all the equations.

$x = x_0cos\omega t$
$cos\omega t = \frac{x}{x_0}$

$v = \frac{dx}{dt} = -x_0\omega^2sin\omega t$
$= \mp x_0\omega \sqrt{1-cos^2\omega t}$
$= \mp x_0\omega \sqrt{1-\frac{x^2}{x_o^2}}$
$= \mp \omega\sqrt{x_0^2-x^2}$
$a = \frac{dv}{dt} = -x_0\omega^2cos\omega t$
$= -\omega^2x_0$

Total energy
$KE = \frac{1}{2}mv^2 = \frac{1}{2}m\omega^2(x_0^2-x^2)$
$TE = \frac{1}{2}mv^2 = \frac{1}{2}m\omega^2(x_0^2)$
$PE = \frac{1}{2}mv^2 = \frac{1}{2}m\omega^2(x^2)$

Define right as positive and track displacement x, velocity v and acceleration a as the pendulum completes a period.

The movement of the pendulum can be illustrated on a a-x graph as shown:

The movement of the pendulum can also be illustrated on a v-x graph as shown:

The total energy is always constant.
The kinetic energy is maximum at the amplitude, minimum at the equilibrium.
The potential energy is minimum at the amplitude, maximum at the equilibrium.