Find the fundamental frequency and the next three frequencies that could cause a standing wave pattern on a string that is 30.0 m long, has a mass per unit length of `9.00 xx 10^(-3) kg//m` and is stretched to a tension of 20.0 N.

Explanation

To find the fundamental frequency and the next three frequencies that could cause a standing wave pattern on a string, we will follow these steps:

### Step 1: Calculate the speed of the wave on the string
The speed \( v \) of a wave on a string can be calculated using the formula:

\[ v = \sqrt{\frac{T}{\mu}} \]

where:
- \( T \) is the tension in the string (20.0 N)
- \( \mu \) is the mass per unit length of the string (9.00 x \( 10^{-3} \) kg/m)

Substituting the values:

\[ v = \sqrt{\frac{20.0 \, \text{N}}{9.00 \times 10^{-3} \, \text{kg/m}}} \]

Calculating the value:

\[ v = \sqrt{\frac{20.0}{0.009}} = \sqrt{2222.22} \approx 47.14 \, \text{m/s} \]

### Step 2: Calculate the fundamental frequency
The fundamental frequency \( f_1 \) can be calculated using the formula:

\[ f_1 = \frac{v}{2L} \]

where \( L \) is the length of the string (30.0 m). Substituting the values:

\[ f_1 = \frac{47.14 \, \text{m/s}}{2 \times 30.0 \, \text{m}} = \frac{47.14}{60} \approx 0.786 \, \text{Hz} \]

### Step 3: Calculate the next three frequencies
The next three frequencies can be found using the harmonic series. The frequencies are given by:

- \( f_2 = 2f_1 \)
- \( f_3 = 3f_1 \)
- \( f_4 = 4f_1 \)

Calculating these:

1. \( f_2 = 2 \times 0.786 \approx 1.572 \, \text{Hz} \)
2. \( f_3 = 3 \times 0.786 \approx 2.358 \, \text{Hz} \)
3. \( f_4 = 4 \times 0.786 \approx 3.144 \, \text{Hz} \)

### Final Results
- Fundamental frequency \( f_1 \approx 0.786 \, \text{Hz} \)
- Second harmonic \( f_2 \approx 1.572 \, \text{Hz} \)
- Third harmonic \( f_3 \approx 2.358 \, \text{Hz} \)
- Fourth harmonic \( f_4 \approx 3.144 \, \text{Hz} \)