First slide

Simple harmonic motion

Question

A graph of the square of the velocity against the square of the acceleration of a given simple harmonic motion is

Moderate

A
B
C
D

Solution

$\begin{array}{l} x = A \sin ω t \\ \begin{aligned} v & = \frac{d x}{d t} = A ω \cos ω t = ω \sqrt{A^{2} - x^{2}} \\ a & = \frac{d v}{d t} = - A ω^{2} \sin ω t = \frac{d v}{d t} = - ω^{2} x \end{aligned} \end{array}$
$But x = - \frac{a}{ω^{2}}$
$∴ v = ω \sqrt{A^{2} - \frac{a^{2}}{ω^{4}}} or v^{2} = ω^{2} (A^{2} - \frac{a^{2}}{ω^{4}})$

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