The guitar is a musical instrument of the chordophone family, being a stringed instrument played by plucking. The tone of an acoustic guitar is produced by the vibration of the strings. Guitar spring (made out of steel) has length L = 0.6 m. The elastic limit for the tension (maximum tensile stress) of steel wire, Smax = 2.7*10^8 Pa. The density of steel is 7.9*10^3 kg/m3. Find the wavelength of the fundamental SOUND wave produced by plucking the string in its middle point. Hint 1: start with finding the frequency of the fundamental wave produced on the string. Hint 2: rearrange T/mu using following formula: density = mass/volume, volume = length x cross-sectional area. Hint 3: use formula for speed of the wave to find the wavelength of the sound wave. Wavelength of the fundamental sound wave is NOT equal to wavelength of the fundamental wave on a string!

To find the wavelength of the fundamental sound wave produced by plucking the string at its middle point, we need to follow these steps:

Step 1: Find the frequency of the fundamental wave produced on the string.
Step 2: Use the formula: density = mass/volume to find the mass of the steel wire.
Step 3: Calculate the tension in the string using the formula: T = Smax * A, where A is the cross-sectional area of the wire.
Step 4: Use the formula for the speed of the wave to find the wavelength of the sound wave.

Let's go through each step in detail:

Step 1: Find the frequency of the fundamental wave produced on the string.
The fundamental frequency of a vibrating string is given by the formula:
f = 1 / (2L) * sqrt(T / μ), where L is the length of the string, T is the tension in the string, and μ is the linear mass density of the string.

Since we are plucking the string at its middle point, L = 0.6/2 = 0.3 m.

Step 2: Use the formula: density = mass/volume to find the mass of the steel wire.
The linear mass density (μ) is given by the formula:
μ = M / L, where M is the total mass of the wire and L is the length of the wire.

Since the density of steel is 7.9*10^3 kg/m^3 and the length of the wire is 0.6m, we can rearrange the formula to find the mass (M):
M = μ * L

Step 3: Calculate the tension in the string using the formula: T = Smax * A, where A is the cross-sectional area of the wire.
The cross-sectional area (A) of the wire can be calculated using the formula:
A = (π * d^2) / 4, where d is the diameter of the wire.

The diameter of the wire is not given, so we cannot calculate the cross-sectional area directly.

Step 4: Use the formula for the speed of the wave to find the wavelength of the sound wave.
The speed of the wave (v) can be calculated using the formula:
v = f * λ, where f is the frequency of the wave and λ is the wavelength of the wave.

Based on the given information, we cannot calculate the wavelength of the sound wave without knowing the tension in the string (T).

To find the wavelength of the fundamental SOUND wave produced by plucking the guitar string, we can follow the hints given.

Hint 1: Start with finding the frequency of the fundamental wave produced on the string.

The fundamental frequency of a vibrating string is inversely proportional to its length. Since the string is plucked in its middle, the length of each half is L/2 = 0.3 m.

The formula for the fundamental frequency of a vibrating string is:
f = (1/2L) * sqrt(Tension/μ)

Hint 2: Rearrange T/μ using the following formula: density = mass/volume, volume = length x cross-sectional area.

Rearranging the formula gives:
Tension/μ = density * (cross-sectional area/length)

We need to find the cross-sectional area. Since the string is made of steel wire, we can assume it is circular in shape. The cross-sectional area of a circular wire is given by:
cross-sectional area = π * (radius)^2

However, we need to find the radius of the wire. We can do this by knowing the density and the length of the wire:
density = mass/volume
mass = density * volume
mass = density * (cross-sectional area * length)
mass = density * (π * (radius^2) * length)

Rearranging this equation for the radius:
radius = sqrt((mass/(density * π * length))

Substituting this value of the radius back into the cross-sectional area formula, we get:
cross-sectional area = π * (sqrt((mass/(density * π * length)))^2

Hence, we can rewrite Tension/μ as:
Tension/μ = density * (π * (sqrt((mass/(density * π * length)))^2 / length)

Hint 3: Use the formula for the speed of the wave to find the wavelength of the sound wave.

The speed of a wave on a string is given by the equation:
v = sqrt(Tension/μ)

Now, we have found both the frequency and the speed of the wave produced by plucking the string. The wavelength can be determined using the formula for the speed of the wave:
v = λ * f

Hence, rearranging the equation, we get:
λ = v/f

Substituting the values we have calculated, the length of the string (L), and the elastic limit for tension (Smax = 2.7*10^8 Pa), the density of steel (7.9*10^3 kg/m^3), we can calculate the wavelength of the fundamental sound wave.