🔥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) $

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Define right as positive and track displacement x, velocity v and acceleration a as the pendulum completes a period.
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The movement of the pendulum can be illustrated on a a-x graph as shown:
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The movement of the pendulum can also be illustrated on a v-x graph as shown:
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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.
Oscillations summary 9

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