To solve the problem step by step, we will break it down into two parts: (a) finding the wavelengths of the fundamental and the first two overtones, and (b) calculating the frequencies of these standing waves.
### Part (a): Finding the Wavelengths
1. **Identify the Length of the Rope:**
- Given: Length of the rope \( L = 4 \, \text{m} \).
2. **Understand the Boundary Conditions:**
- The fixed end of the rope acts as a node (N), and the end tied to the light string acts as an antinode (A).
3. **Determine the Wavelength for the Fundamental Mode:**
- For the fundamental frequency (first harmonic), the relationship is:
\[
L = \frac{\lambda_1}{4}
\]
- Rearranging gives:
\[
\lambda_1 = 4L = 4 \times 4 = 16 \, \text{m}
\]
4. **Determine the Wavelength for the First Overtone:**
- For the first overtone (second harmonic), the relationship is:
\[
L = \frac{3\lambda_2}{4}
\]
- Rearranging gives:
\[
\lambda_2 = \frac{4L}{3} = \frac{4 \times 4}{3} = \frac{16}{3} \approx 5.33 \, \text{m}
\]
5. **Determine the Wavelength for the Second Overtone:**
- For the second overtone (third harmonic), the relationship is:
\[
L = \frac{5\lambda_3}{4}
\]
- Rearranging gives:
\[
\lambda_3 = \frac{4L}{5} = \frac{4 \times 4}{5} = \frac{16}{5} = 3.2 \, \text{m}
\]
### Summary of Wavelengths:
- Fundamental wavelength \( \lambda_1 = 16 \, \text{m} \)
- First overtone wavelength \( \lambda_2 \approx 5.33 \, \text{m} \)
- Second overtone wavelength \( \lambda_3 = 3.2 \, \text{m} \)
### Part (b): Finding the Frequencies
1. **Calculate the Mass per Unit Length (\( \mu \)):**
- Given: Mass of the rope \( m = 160 \, \text{g} = 0.16 \, \text{kg} \)
- The mass per unit length is:
\[
\mu = \frac{m}{L} = \frac{0.16 \, \text{kg}}{4 \, \text{m}} = 0.04 \, \text{kg/m}
\]
2. **Calculate the Wave Speed (\( v \)):**
- The wave speed is given by:
\[
v = \sqrt{\frac{T}{\mu}} = \sqrt{\frac{400 \, \text{N}}{0.04 \, \text{kg/m}}} = \sqrt{10000} = 100 \, \text{m/s}
\]
3. **Calculate the Frequencies:**
- **Fundamental Frequency (\( f_1 \)):**
\[
f_1 = \frac{v}{\lambda_1} = \frac{100 \, \text{m/s}}{16 \, \text{m}} = 6.25 \, \text{Hz}
\]
- **First Overtone Frequency (\( f_2 \)):**
\[
f_2 = \frac{v}{\lambda_2} = \frac{100 \, \text{m/s}}{5.33 \, \text{m}} \approx 18.75 \, \text{Hz}
\]
- **Second Overtone Frequency (\( f_3 \)):**
\[
f_3 = \frac{v}{\lambda_3} = \frac{100 \, \text{m/s}}{3.2 \, \text{m}} \approx 31.25 \, \text{Hz}
\]
### Summary of Frequencies:
- Fundamental frequency \( f_1 = 6.25 \, \text{Hz} \)
- First overtone frequency \( f_2 \approx 18.75 \, \text{Hz} \)
- Second overtone frequency \( f_3 \approx 31.25 \, \text{Hz} \)
