Question

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

• A.
• B.
• C.
• D.

Answer 1

Answer: (D)

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