To solve the problem step by step, we will break it down into parts (a) and (b) as outlined in the question.
### Part (a): Finding the Speed of the Transverse Wave on the Rope
1. **Identify the Tension in the Rope**:
The tension (T) in the rope is equal to the weight of the mass hanging from it. This can be calculated using the formula:
\[
T = m \cdot g
\]
where:
- \( m = 1.50 \, \text{kg} \) (mass)
- \( g = 9.81 \, \text{m/s}^2 \) (acceleration due to gravity)
Substituting the values:
\[
T = 1.50 \, \text{kg} \cdot 9.81 \, \text{m/s}^2 = 14.715 \, \text{N}
\]
2. **Calculate the Speed of the Wave**:
The speed (v) of the transverse wave on the rope can be calculated using the formula:
\[
v = \sqrt{\frac{T}{\mu}}
\]
where:
- \( \mu = 0.0550 \, \text{kg/m} \) (linear mass density)
Substituting the values:
\[
v = \sqrt{\frac{14.715 \, \text{N}}{0.0550 \, \text{kg/m}}}
\]
\[
v = \sqrt{267.545} \approx 16.35 \, \text{m/s}
\]
### Part (b): Finding the Wavelength
1. **Use the Wave Speed and Frequency**:
The wavelength (\(\lambda\)) can be calculated using the formula:
\[
\lambda = \frac{v}{f}
\]
where:
- \( f = 120 \, \text{Hz} \) (frequency)
Substituting the values:
\[
\lambda = \frac{16.35 \, \text{m/s}}{120 \, \text{Hz}} \approx 0.13625 \, \text{m}
\]
### Changes When the Mass is Increased to 3.00 kg
1. **Calculate New Tension**:
If the mass is increased to \( m = 3.00 \, \text{kg} \):
\[
T = 3.00 \, \text{kg} \cdot 9.81 \, \text{m/s}^2 = 29.43 \, \text{N}
\]
2. **Calculate New Speed**:
The new speed (\(v_2\)) can be calculated as:
\[
v_2 = \sqrt{\frac{T_2}{\mu}} = \sqrt{\frac{29.43 \, \text{N}}{0.0550 \, \text{kg/m}}}
\]
\[
v_2 = \sqrt{535.636} \approx 23.2 \, \text{m/s}
\]
3. **Calculate New Wavelength**:
The new wavelength (\(\lambda_2\)) can be calculated as:
\[
\lambda_2 = \frac{v_2}{f} = \frac{23.2 \, \text{m/s}}{120 \, \text{Hz}} \approx 0.1933 \, \text{m}
\]
### Summary of Results
- **Part (a)**: Speed of the wave with 1.5 kg mass: \( \approx 16.35 \, \text{m/s} \)
- **Part (b)**: Wavelength with 1.5 kg mass: \( \approx 0.136 \, \text{m} \)
- **With 3.0 kg mass**: Speed: \( \approx 23.2 \, \text{m/s} \), Wavelength: \( \approx 0.1933 \, \text{m} \)
