Physics SHM Periodicity Analysis

This solution analyzes the periodicity and Simple Harmonic Motion (SHM) properties of the given functions by matching List - I with List - II.

Function A Analysis: $\sin^{2} \omega t$

Using the trigonometric identity $\sin^{2} \theta = \frac{1 - \cos(2\theta)}{2}$, we rewrite the function:

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

This expression involves $\cos(2\omega t)$, indicating an angular frequency of $2\omega$. The time period ($T$) is determined by this frequency:

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

The function oscillates around a mean value of $1/2$. The deviation from this mean oscillates harmonically with the period $T=\frac{\pi}{\omega}$. This matches the description 'Periodic with time period $T=\frac{\pi}{\omega}$ and SHM' (List - III).

Function B Analysis: $\sin^{3} (2\omega t)$

Using the identity $\sin^3\theta = \frac{3\sin\theta - \sin(3\theta)}{4}$, we get:

$\sin^{3} (2\omega t) = \frac{3\sin(2\omega t) - \sin(6\omega t)}{4}$

This function is a sum of two sinusoidal terms with different angular frequencies, $2\omega$ and $6\omega$. The corresponding periods are $T_1 = \frac{2\pi}{2\omega} = \frac{\pi}{\omega}$ and $T_2 = \frac{2\pi}{6\omega} = \frac{\pi}{3\omega}$.

The time period of the combined function is the least common multiple (LCM) of $T_1$ and $T_2$:

$T = \text{LCM}(\frac{\pi}{\omega}, \frac{\pi}{3\omega}) = \frac{\pi}{\omega}$

Since the function is a sum of sinusoids with different frequencies, it is periodic but not SHM. This matches 'Periodic with time period $T=\frac{\pi}{\omega}$ but not SHM' (List - I).

Function C Analysis: $\sin (\omega t) + \cos(\pi \omega t)$

The function is a sum of two sinusoidal terms:

• Term 1: $\sin(\omega t)$ has angular frequency $\omega_1 = \omega$.
• Term 2: $\cos(\pi \omega t)$ has angular frequency $\omega_2 = \pi \omega$.

The ratio of their angular frequencies is $\frac{\omega_1}{\omega_2} = \frac{\omega}{\pi \omega} = \frac{1}{\pi}$. Since this ratio is irrational, the sum of these two terms is non-periodic. This matches 'Non-periodic' (List - IV).

Function D Analysis: $\cos \omega t + \cos 2\omega t$

The function is a sum of two sinusoidal terms:

• Term 1: $\cos(\omega t)$ has angular frequency $\omega_1 = \omega$. Its period is $T_1 = \frac{2\pi}{\omega}$.
• Term 2: $\cos(2\omega t)$ has angular frequency $\omega_2 = 2\omega$. Its period is $T_2 = \frac{2\pi}{2\omega} = \frac{\pi}{\omega}$.

The ratio of frequencies is $\frac{\omega_1}{\omega_2} = \frac{1}{2}$, which is rational.

The time period of the sum is the LCM of the individual periods:

$T = \text{LCM}(\frac{2\pi}{\omega}, \frac{\pi}{\omega}) = \frac{2\pi}{\omega}$

As it's a sum of sinusoids with different frequencies, it is periodic but not SHM. This matches 'Periodic with time period $T=\frac{2\pi}{\omega}$ but Not SHM' (List - II).

Final Matching

Based on the analysis:

• A matches List - III
• B matches List - I
• C matches List - IV
• D matches List - II

The correct combination is A-III, B-I, C-IV, D-II.