MHT CET · Physics · Oscillations

The motion of the particle is given by the equation \(x=A \sin \omega t+B \cos \omega t\).
The motion of the particle is

A simple harmonic with amplitude \((A+B)\)
B simple harmonic with amplitude \((A-B)\)
C simple harmonic with amplitude \(\left(A^2+B^2\right)^{\frac{1}{2}}\)
D not simple harmonic

Verified Solution

Answer & Solution

Correct Answer

(C) simple harmonic with amplitude \(\left(A^2+B^2\right)^{\frac{1}{2}}\)

Step-by-step Solution

Detailed explanation

Let \(A = R \cos \phi\) and \(B = R \sin \phi\). \(x = R \cos \phi \sin \omega t + R \sin \phi \cos \omega t = R \sin(\omega t + \phi)\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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The motion of the particle is given by the equation \(x=A \sin \omega t+B \cos \omega t\).The motion of the particle is

Step-by-step Solution

The motion of the particle is given by the equation \(x=A \sin \omega t+B \cos \omega t\).
The motion of the particle is