A simple harmonic oscillator of angular frequency 2 ,rad ,s^ -1 i… - Vidyadip

MCQ

A simple harmonic oscillator of angular frequency $2\,rad\,s^{-1}$ is acted upon by an external force $F = sin\,t\,N .$ If the oscillator is at rest in its equilibrium position at $t = 0,$ its position at later times is proportional to

A
$\sin \,t\, + \,\frac{1}{2}\,\cos \,2t$
B
$\cos \,t\, - \,\frac{1}{2}\,\sin \,2t$
✓
$\sin \,t\, - \,\frac{1}{2}\,\sin \,2t$
D
$\sin \,t\, + \,\frac{1}{2}\,\sin \,2t$

✓

Answer

Correct option: C.

$\sin \,t\, - \,\frac{1}{2}\,\sin \,2t$

c
As we know,

$F=m a \Rightarrow a \propto F$
or, $a \propto \sin t$
$\Rightarrow \frac{d v}{d t} \propto \sin t$
$\Rightarrow \int_{0}^{0} d V \propto \int_{0}^{t} \sin t d t$
$V \propto-\cos t+1$
$\int_{0}^{x} d x=\int_{0}^{t}(-\cos t+1) d t$
$x=\sin t-\frac{1}{2} \sin 2 t$

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