Out of the following functions, which represents SHM?
I. $y = \sin \omega t - \cos \omega t$
II. $y = \sin^3 \

Question

Out of the following functions, which represents SHM?
I. $y = \sin \omega t - \cos \omega t$
II. $y = \sin^3 \omega t$
III. $y = 5 \cos\left(\frac{3\pi}{4} - 3\omega t\right)$
IV. $y = 1 + \omega t + \omega^2 t^2$

Accepted Answer

1.  **Condition for SHM:** A function represents Simple Harmonic Motion (SHM) if it can be written in the form $y = A \sin(\Omega t + \phi)$ or $y = A \cos(\Omega t + \phi)$, involving a single frequency.
2.  **Analysis of Function I:** $y = \sin \omega t - \cos \omega t$. Multiply and divide by $\sqrt{2}$:
    $$ y = \sqrt{2} \left( \frac{1}{\sqrt{2}} \sin \omega t - \frac{1}{\sqrt{2}} \cos \omega t ight) = \sqrt{2} \sin\left(\omega t - \frac{\pi}{4}ight) $$
    This is in the standard SHM form with amplitude $\sqrt{2}$ and frequency $\omega$. So, **I is SHM**.
3.  **Analysis of Function II:** $y = \sin^3 \omega t$. Using the identity $\sin 3	heta = 3 \sin 	heta - 4 \sin^3 	heta$, we get:
    $$ y = \frac{1}{4} (3 \sin \omega t - \sin 3\omega t) $$
    This represents the superposition of two SHMs with different frequencies ($\omega$ and $3\omega$). While the motion is **periodic**, it is **not SHM** because it involves more than one frequency.
4.  **Analysis of Function III:** $y = 5 \cos\left(\frac{3\pi}{4} - 3\omega tight)$. Using $\cos(-	heta) = \cos(	heta)$:
    $$ y = 5 \cos\left(3\omega t - \frac{3\pi}{4}ight) $$
    This is in the standard SHM form with amplitude $5$ and angular frequency $3\omega$. So, **III is SHM**.
5.  **Analysis of Function IV:** $y = 1 + \omega t + \omega^2 t^2$. This function increases indefinitely with time (parabolic growth) and never repeats. It is **non-periodic** and thus not SHM.
6.  **Conclusion:** Only functions I and III represent SHM.

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