A string fixed at both ends is 7.50 m long and has a mass of 0.120 kg. It is subjected to a tension of 96.0 N and set oscillating (a) What is the speed of the waves on the string? (b) What is thc longest possible wavelength for a standing wave? (c) Give the frequency of that wave.

Answer

To solve the problem step by step, we will address each part of the question systematically.

### Given Data:
- Length of the string (L) = 7.50 m
- Mass of the string (m) = 0.120 kg
- Tension in the string (T) = 96.0 N

### (a) Speed of the waves on the string

1. **Calculate the mass per unit length (μ)**:
\[ \mu = \frac{m}{L} = \frac{0.120 \, \text{kg}}{7.50 \, \text{m}} = 0.016 \, \text{kg/m} \]

2. **Use the formula for wave speed (v)**:
\[ v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{96.0 \, \text{N}}{0.016 \, \text{kg/m}}} \]

3. **Calculate the speed**:
\[ v = \sqrt{6000} \approx 77.46 \, \text{m/s} \]

### (b) Longest possible wavelength for a standing wave

1. **Identify the relationship for the longest wavelength**:
The longest wavelength (λ) occurs when the string vibrates in its fundamental mode, which is when the length of the string is half of the wavelength:
\[ \frac{\lambda}{2} = L \implies \lambda = 2L \]

2. **Calculate the longest wavelength**:
\[ \lambda = 2 \times 7.50 \, \text{m} = 15.0 \, \text{m} \]

### (c) Frequency of that wave

1. **Use the wave equation**:
The relationship between speed (v), wavelength (λ), and frequency (f) is given by:
\[ v = f \cdot \lambda \]

2. **Rearrange to find frequency**:
\[ f = \frac{v}{\lambda} \]

3. **Substitute the values**:
\[ f = \frac{77.46 \, \text{m/s}}{15.0 \, \text{m}} \approx 5.16 \, \text{Hz} \]

### Summary of Answers:
- (a) Speed of the waves on the string: **77.46 m/s**
- (b) Longest possible wavelength for a standing wave: **15.0 m**
- (c) Frequency of that wave: **5.16 Hz**

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