Question

Which motion is simple harmonic motion?

• A. \(y=ae^{\omega t}\)
• B. \(y=3t^2+at\)
• C. \(y=4t^3+2t^2+at\)
• D. \(y=a\cos\omega t+b\sin\omega t\)

Hint:

Simple harmonic motion is always represented by sine or cosine functions of time.

Answer 1

The Correct Option is D

Concept:
For simple harmonic motion, displacement must be a sinusoidal function of time. The general form of SHM is: \[ y=A\cos\omega t+B\sin\omega t \]

Step 1: Check option (A). \[ y=ae^{\omega t} \] This is an exponential function, not periodic. Therefore, it is not SHM.

Step 2: Check option (B). \[ y=3t^2+at \] This is a polynomial in \(t\), not a sinusoidal function. Hence it is not SHM.

Step 3: Check option (C). \[ y=4t^3+2t^2+at \] This is also a polynomial function of time, so it cannot represent SHM.

Step 4: Check option (D). \[ y=a\cos\omega t+b\sin\omega t \] This is the general sinusoidal solution of SHM. \[ \therefore \text{Correct Answer is (D)} \]