Question

Two simple harmonic waves having equal amplitudes of 8 cm and equal frequency of 10 Hz are moving along the same direction. The resultant amplitude is also 8 cm. The phase difference between the individual waves is _____ degree.

Hint:

For SHM superposition:

• Use the formula: \[ A_{\text{resultant}} = \sqrt{A_1^2 + A_2^2 + 2A_1A_2\cos\phi}. \]
• Substitute known values and solve for \(\cos\phi\) to find the phase difference.

Answer 1

Correct Answer: 120

1. Resultant Amplitude Formula: - The resultant amplitude is given by: \[ A_{\text{resultant}} = \sqrt{A_1^2 + A_2^2 + 2A_1A_2\cos\phi}. \]
2. Given Data: - \(A_1 = A_2 = 8 \, \text{cm}, \, A_{\text{resultant}} = 8 \, \text{cm}.\)
3. Substitute Values: \[ 8 = \sqrt{8^2 + 8^2 + 2 \cdot 8 \cdot 8 \cos\phi}. \]

Simplify:
\[ 64 = \sqrt{128 + 128 \cos\phi}. \] Square both sides: \[ 64 = 128(1 + \cos\phi). \] Solve for \(\cos\phi\): \[ \cos\phi = -\frac{1}{2}. \]

1. Phase Difference: - The phase difference is: \[ \phi = 120^\circ. \]

Final Answer: \(120^\circ\)