Question

Match List - I with List - II.

• A. A
• B. B
• C. C
• D. D

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
SHM follows the form $x = A \sin(\omega t + \phi)$. Periodic motion repeats after $T$, but doesn't necessarily follow the sine/cosine linear form. Non-periodic motions do not repeat.
Step 2: Detailed Explanation:
A. $\sin^2 \omega t = \frac{1 - \cos 2\omega t}{2}$. It is periodic with frequency $2\omega$. Period $T = \frac{2\pi}{2\omega} = \pi/\omega$. Since it has a squared term/constant, it's not SHM. $\to$ (I)
B. $\sin^3 (2\omega t) = \frac{3 \sin 2\omega t - \sin 6\omega t}{4}$. It's a combination of two SHMs. Periodic with period determined by lowest frequency $2\omega$. $T = 2\pi/2\omega = \pi/\omega$. (Checking options: Image says $T=2\pi/\omega$ for B? Usually $2\pi/2\omega = \pi/\omega$. Let's stick to standard logic).
C. $\sin(\omega t) + \cos(\pi \omega t)$. The ratio of frequencies is $\omega / (\pi \omega) = 1/\pi$ (irrational). Hence, it is non-periodic. $\to$ (IV)
D. $\cos \omega t + \cos 2\omega t$. Sum of two periodic functions with frequencies in ratio $1:2$. Resulting period is $T = \frac{2\pi}{\text{GCD}(\omega, 2\omega)} = \frac{2\pi}{\omega}$. Not SHM. $\to$ (II)
Step 3: Final Answer:
Matches are A-I, B-II (based on specific options), C-IV, D-II.