To solve this problem, we need to relate the given parameters to the wavelength of the standing waves produced in the string.

Key Information:

• Mass per unit length, $\mu = 0.0026 \, \text{g/cm} = 0.0026 \times 10^{-3} \, \text{kg/m} = 2.6 \times 10^{-3} \, \text{kg/m}$
• Frequency of vibration, $f = 85 \, \text{cps}$ (cycles per second or Hertz)
• Tension, $T = 0.54 \, \text{kg} = 0.54 \times 9.8 \, \text{N} = 5.292 \, \text{N}$
• Number of segments, $n = 4$ (This represents the number of half-wavelengths on the string.)

Step 1: Determine the wave speed $v$

The wave speed on a string under tension is given by the formula: $v = \sqrt{\frac{T}{\mu}}$ Substituting the given values: $v = \sqrt{\frac{5.292 \, \text{N}}{2.6 \times 10^{-3} \, \text{kg/m}}}$ Let's calculate this.

Step 2: Relationship between wave speed, frequency, and wavelength

For a wave, the speed $v$, frequency $f$, and wavelength $\lambda$ are related by: $v = f \lambda$ For a string with $n$ segments (or half-wavelengths), the length of the string $L$ is related to the wavelength by: $L = \frac{n\lambda}{2}$ Using the wave speed $v = f \lambda$, we can find $\lambda$, and then $L$.

Let's perform the necessary calculations.The length of the string is approximately $1.06 \, \text{m}$, which is equivalent to $106.15 \, \text{cm}$.

Would you like any further details or explanations?

Here are five related questions for further practice:

1. How would the length change if the tension were doubled?
2. What happens to the number of segments if the frequency increases to 100 Hz?
3. How would the wave speed change if the mass per unit length were reduced by half?
4. How is the length of the string affected if the number of segments is increased to 6?
5. What is the fundamental frequency of the string with the current length and tension?

Tip: The speed of a wave on a string increases with tension but decreases with increasing mass per unit length.