To solve this, we need to apply the formula for the frequency of a standing wave on a string, which is given by:

$f = \frac{n}{2L} \cdot \sqrt{\frac{T}{\mu}}$

Where:

• $f$ is the frequency (in Hz),
• $n$ is the number of segments (antinodes),
• $L$ is the length of the string (in meters),
• $T$ is the tension in the string (in Newtons),
• $\mu$ is the mass per unit length (in kg/m).

Given:

• Mass per unit length $\mu = 0.0035 \, \text{g/cm} = 0.00035 \, \text{kg/m}$,
• Frequency $f = 108 \, \text{Hz}$,
• Number of segments $n = 4$,
• Tension $T = 0.42 \, \text{kg} = 0.42 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 4.116 \, \text{N}$.

We are asked to find the length of the string $L$.

Step 1: Rearrange the formula to solve for $L$:

$L = \frac{n}{2f} \cdot \sqrt{\frac{T}{\mu}}$

Step 2: Substitute the given values:

$L = \frac{4}{2 \times 108} \cdot \sqrt{\frac{4.116}{0.00035}}$

Let's calculate this.The length of the string is approximately $200.82 \, \text{cm}$.

Would you like further details or clarification on any part of this solution?

Here are 5 related questions to deepen your understanding:

1. What is the relationship between wave speed, frequency, and wavelength in a standing wave?
2. How does the tension in the string affect the frequency of the waves?
3. How would the length of the string change if the number of segments increased?
4. What effect would increasing the mass per unit length have on the wavelength of the wave?
5. How is energy transferred along the string in wave motion?

Tip: In standing waves, the number of segments (antinodes) corresponds to the harmonic number of the wave.