nd the period of $x(t)$, we should determine the smallest positive $T$ such that $x(t+T)=$ $x(t)$.

Since both $\sin (\omega t)$ and $\cos (\omega t)$ have a period of $\frac{2 \pi}{\omega}$, we need to see how this affects $x(t)$. and since, $\sin (\omega t)$ has a period of $\frac{2 \pi}{\omega}$, squaring it reduces the period by half to $\frac{\pi}{\omega}$.

IAT (IISER) 2024 Physics - Simple Harmonic Motion Question 4 English Explanation 1

$\cos ^3(\omega t)$ retains the period $\frac{2 \pi}{\omega}$ since the cubic power does not change the period.

IAT (IISER) 2024 Physics - Simple Harmonic Motion Question 4 English Explanation 2

The smallest common period of both $\sin ^2(\omega t)$ and $\cos ^3(\omega t)$ is the least common multiple (LCM) of $\frac{\pi}{\omega}$ and $\frac{2 \pi}{\omega}$.

Since $\frac{2 \pi}{\omega}$ is exactly twice $\frac{\pi}{\omega}$, the LCM is $\frac{2 \pi}{\omega}$.

Therefore, the time period $T$ of the motion described by $x(t)=\sin ^2(\omega t) \cos ^3(\omega t)$ is:

$$ T=\frac{2 \pi}{\omega} $$

Simple Harmonic Motion

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