A point performs simple harmonic oscillation of period $T$ and the equation of motion is given by $x = a\sin{(w + \pi/6)}.$ After the elapse of what fraction of the time period the velocity of the point will be equal to half its maximum velocity

Question

Dr. Abubacker · Accepted Answer

(b) $x = a\sin\left( \omega t + \frac{\pi}{6} \right)$
$v = \frac{dx}{dt} = a\ \omega\cos\left( \omega t + \frac{\pi}{6} \right)$ …(i)
We know that $v_{\max} = a\omega$
So, by substituting $v = \frac{a\omega}{2}$ in equation (i) we get time $(t)$
$\frac{a\omega}{2} = a\omega\cos{\left( \omega t + \frac{\pi}{6} \right) \Rightarrow \frac{\pi}{3} = \omega t + \frac{\pi}{6} \Rightarrow \frac{\pi}{6} = \frac{2\pi}{T}.t}$
$\Rightarrow t = \frac{T}{12}$

Q. A point performs simple harmonic oscillation of period $T$ and the equation of motion is given by $x = a\sin{(w + \pi/6)}.$ After the elapse of what fraction of the time period the velocity of the point will be equal to half its maximum velocity

Explanation

Reviewed by Dr. Abubacker

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