Question

Two SHM's \(x_{1}=a\sin \left(\omega t+\frac{3\pi }{2}\right)\) and \(x_{2}=a\sin (\omega t+\pi )\) are superimposed on each other. The resultant amplitude of motion is \((\sin \pi /2=1\) and \(\cos \pi /2=0)\)

• A. \(2a\)
• B. \(2a^{2}\)
• C. \(a\sqrt{2}\)
• D. \(0\)

Hint:

Find the phase difference between the two motions, then use the resultant amplitude formula for equal amplitudes.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
Two SHMs of the same frequency add up to another SHM. The resultant amplitude is \(A=\sqrt{a_{1}^{2}+a_{2}^{2}+2a_{1}a_{2}\cos\phi}\).

Step 2: Phase difference
\(\phi=\dfrac{3\pi}{2}-\pi=\dfrac{\pi}{2}\), and \(\cos\dfrac{\pi}{2}=0\).

Step 3: Calculation
With \(a_{1}=a_{2}=a\): \(A=\sqrt{a^{2}+a^{2}+0}=a\sqrt{2}\).

Final Answer:
The resultant amplitude is root 2 times a. \[ \boxed{\text{(C) }a\sqrt{2}} \]