Oscillations - Result Question 10

13. The displacement of a particle along the $x$-axis is given by $x=a \sin ^{2} \omega t$. The motion of the particle corresponds to:

[2010]

(a) simple harmonic motion of frequency $\omega / \pi$

(b) simple harmonic motion of frequency $3 \omega / 2 \pi$

(c) non simple harmonic motion

(d) simple harmonic motion of frequency $\omega / 2 \pi$

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Answer:

Correct Answer: 13. (a)

Solution:

  1. (a) $x=a \sin ^{2} \omega t=\frac{a}{2}(1-\cos 2 \omega t)$

$\frac{d x}{d t}=\frac{a}{2} 2 \omega \sin 2 \omega t$

$\Rightarrow \frac{d^{2} x}{d t^{2}}=\frac{4 \omega^{2} a}{2} \cdot \cos 2 \omega t$

This represents an S. H. M. of frequency

$=\frac{\omega}{\pi}$