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  4. The displacement of simple harmonic oscillator after 3 seconds starting from its mean position is equal to half of its amplitude. The time period of harmonic motion is :

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Q. The displacement of simple harmonic oscillator after $3$ seconds starting from its mean position is equal to half of its amplitude. The time period of harmonic motion is :

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Solution:

$X=A \sin \omega t\left(t=3, X=\frac{A}{2}\right)$
$\Rightarrow \frac{A}{2}=A \sin 3\, \omega$
$\Rightarrow \sin 3 \omega=\frac{1}{2}$
$\Rightarrow 3 \omega=\frac{\pi}{6}$
$\Rightarrow \omega=\frac{\pi}{18}=\frac{2 \pi}{T}$
$\Rightarrow T=36 \,s$