To find the frequency of a wave in the second rope, we can use the wave equation:
\[ v = f \lambda \]
where:
- \( v \) is the wave speed,
- \( f \) is the frequency,
- \( \lambda \) is the wavelength.
First, we will calculate the speed of the wave in the first rope using the given frequency and wavelength:
For the first rope:
- Frequency (\( f_1 \)) = 1.5 Hz
- Wavelength (\( \lambda_1 \)) = 6 m
Calculating the wave speed (\( v \)):
\[ v = f_1 \lambda_1 = 1.5 , \text{Hz} \times 6 , \text{m} = 9 , \text{m/s} \]
Now, we will use this speed to find the frequency (\( f_2 \)) of the wave in the second rope, where the wavelength (\( \lambda_2 \)) is 2 m.
Using the wave speed calculated from the first rope:
\[ v = f_2 \lambda_2 \] \[ 9 , \text{m/s} = f_2 \times 2 , \text{m} \]
Solving for \( f_2 \):
\[ f_2 = \frac{9 , \text{m/s}}{2 , \text{m}} = 4.5 , \text{Hz} \]
Thus, the frequency of the wave in the second rope is 4.5 hertz.