Question

Identify the equation that DOES NOT represent oscillatory motion.
(\(A\) is amplitude of motion)

• A. \(y=A\sin \omega t+A\cos 2\omega t\)
• B. \(x=A\cos \left(\omega t-\frac{\pi }{4}\right)\)
• C. \(y=A\sin \omega t+A\cos (\omega t+\pi /2)\)
• D. \(y=A\sin \left(\omega t-\frac{\pi }{4}\right)\)

Hint:

Simplify each expression; one of them reduces to zero for all time.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
Motion is oscillatory if the displacement varies periodically with time. A displacement that stays constant at zero is not oscillation.

Step 2: Check options with different forms
\(y=A\sin\omega t+A\cos2\omega t\) is a sum of two periodic terms and repeats, so it is oscillatory. The forms \(A\cos\left(\omega t-\dfrac{\pi}{4}\right)\) and \(A\sin\left(\omega t-\dfrac{\pi}{4}\right)\) are simple harmonic.

Step 3: Simplify option (C)
\(\cos\left(\omega t+\dfrac{\pi}{2}\right)=-\sin\omega t\).
So \(y=A\sin\omega t-A\sin\omega t=0\) for all \(t\). The particle stays at rest.

Final Answer:
The equation \(y=A\sin\omega t+A\cos(\omega t+\pi/2)\) gives no oscillation. \[ \boxed{\text{(C) }y=A\sin \omega t+A\cos (\omega t+\pi /2)} \]