Question

A particle is subjected to two simple harmonic motions in the same direction having equal amplitudes and equal frequency. If the resultant amplitude is equal to the amplitude of the individual motion, the phase difference $(\delta)$ between the two motions is

• A. δ =π /3
• B. δ =2π/3
• C. δ =π
• D. δ = π/2

Answer 1

The Correct Option is B

1. Given: Two simple harmonic motions (SHMs) in the same direction with:
   • Equal amplitude \( A \)
   • Equal frequency
   • Resultant amplitude = \( A \)
2. Formula for resultant amplitude when combining two SHMs:
   \[ R = \sqrt{A^2 + A^2 + 2A^2\cos\delta} = A\sqrt{2(1 + \cos\delta)} \]
3. Given: \( R = A \)
   So, \[ A = A\sqrt{2(1 + \cos\delta)} \Rightarrow \sqrt{2(1 + \cos\delta)} = 1 \Rightarrow 2(1 + \cos\delta) = 1 \Rightarrow \cos\delta = -\frac{1}{2} \]
4. Therefore, \[ \delta = \cos^{-1}\left(-\frac{1}{2}\right) = \frac{2\pi}{3} \]
5. Final Answer: \( \boxed{\delta = \frac{2\pi}{3}} \)