Two sinusoidal waves of the same period, with amplitudes of 5.0 and 7.0 mm, travel in the same direction along a stretched string, they produce a resultant wave with an amplitude of 10.0 mm. The phase constant of the 5.0mm wave is 0. What is the phase constant of the 7.0mm wave?

Answer

To find the phase constant of the 7.0 mm wave, we can use the principle of superposition for waves. The resultant amplitude of two waves can be calculated using the formula for the resultant amplitude when two waves of different amplitudes and phase differences interfere.

### Step-by-step Solution:

1. **Identify Given Values**:
- Amplitude of Wave 1 (A1) = 5.0 mm
- Amplitude of Wave 2 (A2) = 7.0 mm
- Resultant Amplitude (R) = 10.0 mm
- Phase constant of Wave 1 (φ1) = 0

2. **Use the Resultant Amplitude Formula**:
The formula for the resultant amplitude when two waves interfere is given by:
\[ R = \sqrt{A_1^2 + A_2^2 + 2A_1A_2\cos(\phi)} \]
where φ is the phase difference between the two waves.

3. **Substituting Known Values**:
Substitute the known values into the formula:
\[ 10.0 = \sqrt{(5.0)^2 + (7.0)^2 + 2 \cdot (5.0) \cdot (7.0) \cdot \cos(\phi)} \]

4. **Square Both Sides**:
Squaring both sides to eliminate the square root gives:
\[ 100 = 25 + 49 + 70\cos(\phi) \]

5. **Simplify the Equation**:
Combine like terms:
\[ 100 = 74 + 70\cos(\phi) \]
Rearranging gives:
\[ 70\cos(\phi) = 100 - 74 \]
\[ 70\cos(\phi) = 26 \]

6. **Solve for cos(φ)**:
\[ \cos(\phi) = \frac{26}{70} \]

7. **Calculate φ**:
To find the phase constant φ, take the inverse cosine:
\[ \phi = \cos^{-1}\left(\frac{26}{70}\right) \]

8. **Final Calculation**:
Using a calculator, compute:
\[ \phi \approx \cos^{-1}(0.3714) \approx 68^\circ \]

### Conclusion:
The phase constant of the 7.0 mm wave is approximately **68 degrees**.

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