Physics - SHM Question with Solution | TestHub

It is given that oscillator at rest at$t=0$i.e. at$t=0$,$v=0$. So, in option, we can check by putting$v=\frac{dx}{dt}=0$.

(1)$If x\propto \sin t+\frac{1}{2}\cos 2t$

$⟹v\propto \cos t+\frac{1}{2}\times 2 (-\sin 2t)$

$⟹ at t=0, v\propto 1-0\ne 0$

(2)$If x \propto \cos t-\frac{1}{2}\sin t$

$⟹ v\propto -\sin t-\frac{1}{2}\cos t$

$⟹at t=0, v\propto -\frac{1}{2}\ne 0$

(3)$If x\propto \sin t-\frac{1}{2} sin 2t$

$then \upsilon \propto \cos t-\frac{1}{2}\times 2\cos 2t$

$⟹at t=0, v\propto 1-1=0$

(4)$If x\propto \sin t+\frac{1}{2}\sin 2t$

$⟹v\propto \cos t+\frac{1}{2}\times 2\cos 2t$

$⟹at t=0, v\propto 1+1$

$⟹ v\propto 2\ne 0$

$\therefore in option \left(3\right) v=0 at t=0$