Question

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. A
• B. B
• C. C
• D. D

Hint:

Any combination $a\sin\theta + b\cos\theta$ can be rewritten as $\sqrt{a^2+b^2} \sin(\theta + \alpha)$, which is still a single harmonic wave.

Answer 1

The Correct Option is C

Step 1: Superposition Principle
The equation represents the composition of two SHMs at right angles (or with phase difference $\pi/2$).
Step 2: Resultant Displacement
$x = R \sin(\omega t + \phi)$, where $R$ is the resultant amplitude.
Step 3: Amplitude Formula
For two waves with phase difference $\pi/2$: $R = \sqrt{A_1^2 + A_2^2 + 2A_1A_2\cos(\pi/2)}$.
$R = \sqrt{A^2 + B^2 + 0} = (A^2 + B^2)^{1/2}$.
Step 4: Conclusion
The motion is SHM with amplitude $\sqrt{A^2 + B^2}$.
Final Answer:(C)