A guitar string is 92 cm long and has a mass of 3.7 g. The length L from the bridge to the support post is 65 cm, and the string is under a tension of 570 N. What is the frequency of the fundamental tone

thanks!

Velocity in the stretched string is

v = sqrt(TL₀/m) =
=sqrt(570•0.92/0.0037) = 376 m/s,
λ =2L =2•0.65 = 1.3 m,
λ = v/f =>
f=v/λ =376/1.3=289 Hz.

To calculate the frequency of the fundamental tone of a guitar string, you can use the formula:

f = (1 / 2L) * sqrt(T / μ)

Where:
- f is the frequency
- L is the length of the string
- T is the tension in the string
- μ is the mass per unit length of the string

Let's calculate it step by step:

1. Convert the length of the string from centimeters to meters:
L = 65 cm = 0.65 m

2. Convert the mass of the string from grams to kilograms:
m = 3.7 g = 0.0037 kg

3. Calculate the mass per unit length:
μ = m / L

μ = 0.0037 kg / 0.65 m

4. Calculate the square root of the tension:
sqrt(T) = sqrt(570 N)

5. Calculate the frequency:
f = (1 / 2L) * sqrt(T / μ)

Substitute the values:
f = (1 / 2 * 0.65) * sqrt(570 / (0.0037 / 0.65))

Now, calculate the square root separately and substitute the value:
sqrt(570 / (0.0037 / 0.65)) = 488.23

Finally, substitute this value back into the frequency equation to get the frequency of the fundamental tone:

f = (1 / 2 * 0.65) * 488.23

Simplify:
f = 375.56 Hz

Therefore, the frequency of the fundamental tone of the guitar string is approximately 375.56 Hz.

To find the frequency of the fundamental tone of a guitar string, we can use the formula:

f = (1 / 2L) * sqrt(T / μ)

where:
f = frequency of the fundamental tone
L = length of the string between the bridge and the support post
T = tension in the string
μ = linear mass density of the string

First, let's calculate the linear mass density (μ) of the string:

μ = mass / length
= 3.7 g / 92 cm
= 0.0377 g/cm

Next, let's convert the length (L) and linear mass density (μ) to SI units:

L = 65 cm / 100 cm/m (1 m/100 cm) = 0.65 m
μ = 0.0377 g/cm / 1000 g/kg (1 kg/1000 g) / 0.92 m = 0.00004 kg/m

Now, let's substitute the values into the formula:

f = (1 / 2L) * sqrt(T / μ)
= (1 / 2 * 0.65 m) * sqrt(570 N / 0.00004 kg/m)
= (1 / 1.3) * sqrt(14,250,000 N/kg)
= 0.769 * sqrt(14,250,000)
= 0.769 * 3,775
≈ 2907 Hz

Therefore, the frequency of the fundamental tone of the guitar string is approximately 2907 Hz.