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Q.
The velocity and acceleration of a particle performing simple harmonic motion have a steady phase relationship. The acceleration shows a phase lead over the velocity in radians of
Let displacement equation of a particle executing $SHM$ is given as
$x=a \sin \omega t$
then velocity $(v)=\frac{d x}{d t}=a \omega \cos \omega t\,..(i)$
and acceleration $(a)=\frac{d^{2} x}{d t^{2}}=\frac{d v}{d t}=-a \omega^{2} \sin \omega t $
$a=a \omega^{2} \cos \left(\omega t +\frac{\pi}{2}\right) \,...(ii)$
$\left(\because \cos \left(\theta+\frac{\pi}{2}\right)=-\sin \theta\right)$
Now, comparing Eqs (i) and (ii), we can conclude that the acceleration lead the velocity by a phase of $+\frac{\pi}{2}$ radians.