A guitar string is 97 cm long and has a mass of 3.6 g. The length L from the bridge to the support post is 61 cm, and the string is under a tension of 570 N. What is the frequency of the fundamental tone?
string length = 0.97 meter
m = 0.0036 kg
u = mass/length = .00371
half a wavelength = 0.61 meter
speed = sqrt (570/.0031) = 429 m/s
time to go half a wavelength
= .61/429 = .00142 seconds
time for a wavelength = .00284 s
f = 1/time = 351 Hz
To find the frequency of the fundamental tone of a guitar string, we can use the formula:
frequency = (1/2L) * sqrt(T/u)
Where:
L is the length of the vibrating portion of the guitar string
T is the tension in the string
u is the linear mass density of the string
First, let's find the linear mass density of the string. The linear mass density u is defined as the mass per unit length, and can be calculated by dividing the mass of the string by its length:
u = mass / length
Given:
mass = 3.6 g
length = 97 cm
Converting the mass to kg:
mass = 3.6 g / 1000 = 0.0036 kg
Converting the length to m:
length = 97 cm / 100 = 0.97 m
Now we can calculate the linear mass density:
u = 0.0036 kg / 0.97 m ≈ 0.0037 kg/m
Next, we substitute the values into the frequency formula:
frequency = (1/2 * 0.61 m) * sqrt(570 N / 0.0037 kg/m)
Simplifying the formula:
frequency = 0.61 * sqrt(154054.05)
Using a calculator, calculate the square root:
sqrt(154054.05) ≈ 392.50
Substituting the value back into the formula:
frequency ≈ 0.61 * 392.50
Multiplying the values:
frequency ≈ 239.23 Hz
Therefore, the fundamental frequency of the guitar string is approximately 239.23 Hz.