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  4. A particle executing SHM of amplitude ' a ' has a displacement a / 2 at t=T / 4 and a negative velocity. The epoch of the particle is

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Q. A particle executing SHM of amplitude ' $a$ ' has a displacement $a / 2$ at $t=T / 4$ and a negative velocity. The epoch of the particle is

Oscillations

Solution:

Equation of SHM $=y \sin (\omega t+\phi)$
When $y=\frac{a}{2}, t=\frac{T}{4}=\frac{2 \pi}{4 \omega}=\frac{\pi}{2 \omega}$
$v=a \omega \cos (\omega t+\phi)$,
velocity is negative
$\frac{a}{2}=a \sin (\omega t+\phi) $
$ \Rightarrow \sin (\omega t+\phi)=\frac{1}{2}$
$\left(\frac{\pi}{2}+\phi\right)=\frac{5 \pi}{6}$
Substituting in the above equation, we get $\phi=\pi / 3$.