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

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

Oscillations

Solution:

$x=A \sin \omega t$
$v=\frac{d x}{d t}=A \omega \cos \omega t=\omega \sqrt{A^{2}-x^{2}}$
$a=\frac{d v}{d t}=-A \omega^{2} \sin \omega t=\frac{d v}{d t}=-\omega^{2} x$
But $x=-\frac{a}{\omega^{2}}$
$\therefore v=\omega \sqrt{A^{2}-\frac{a^{2}}{\omega^{4}}} $
or $ v^{2}=\omega^{2}\left(A^{2}-\frac{a^{2}}{\omega^{4}}\right)$