Two waves executing simple harmonic motions travelling in the same direction with same amplitude and frequency are superimposed. The resultant amplitude is equal to the √3 times of amplitude of individual motions. The phase difference between the two motions is ___________ (degree).

Solution

Physics sir · 3 steps

Add the two SHMs like vectors at an angle, then set the resultant equal to √3A.

1. Resultant amplitude. Two SHMs of the same frequency combine like two vectors of lengths A₁ and A₂ with the phase difference φ as the angle between them.

   R² = A₁² + A₂² + 2A₁A₂cos φ

2. Equal amplitudes. With A₁ = A₂ = A and R = √3A, this becomes 3A² = 2A² + 2A²cos φ. So 2 cos φ = 1, which gives cos φ = 1/2.
3. So. cos φ = 1/2 gives φ = 60°. The phase difference is 60.

The common slip: 30. Using R = 2A cos φ/2 and stopping at φ/2 = 30° gives 30. The phase difference is twice that angle.

The idea to keep. For two equal amplitudes A, the resultant is R = 2A cos φ/2. Remember that this formula uses half the phase difference, so double the angle you get.

Answer from the NTA final answer key. Solution by PrepWiser.