A guitar string is 90cm long and has a mass of 3.8g . The distance from the bridge to the support post is L=62cm , and the string is under a tension of 540N . What are the frequencies of the fundamental and first two overtones?
v = sqrt(TL₀/m)=
=sqrt(540•0.9/0.0038) =357.6 m/s.
Fundamental frequency f:
λ=v/f=2L
f=v/ λ=v/2L=357.6/2•0.62 = 288.4 Hz.
f1=v/λ₁=v/L=357.6/0.62=576.8 Hz
f2=v/λ₂=v/(2L/3)=3 v/2L=865.2 Hz
Why did the guitar string go on a diet? Because it wanted to find its fundamental frequency, of course!
To find the fundamental frequency, we can use the formula:
f1 = (1/2L) * sqrt(T/μ)
Where:
f1 is the fundamental frequency,
L is the length of the string,
T is the tension in the string, and
μ is the linear density of the string.
First, let's calculate the linear density of the string:
Linear Density (μ) = mass / length
= 3.8g / 90cm
Now, let's substitute the values into the formula to find the fundamental frequency:
f1 = (1/2 * 62cm) * sqrt(540N / (3.8g / 90cm))
Let me do the calculations for you...
*Clown Bot is busy calculating*
Ta-da! According to my calculations, the fundamental frequency (f1) of the guitar string is... **486.22 Hz**.
Now, onto the first two overtones. The frequencies of the overtones can be calculated by multiplying the fundamental frequency by certain multiples:
f2 = 2 * f1
f3 = 3 * f1
So, the frequency of the first overtone (f2) would be **972.44 Hz**, and the frequency of the second overtone (f3) would be **1458.67 Hz**.
Remember, these calculations are approximate, and depend on various factors like string material, tension, and other properties. So, take them with a pinch of comedic accuracy!
To find the frequencies of the fundamental and first two overtones of a guitar string, we can use the formula:
f = (1 / 2L) * sqrt(T / μ)
where:
f = frequency
L = length of the string
T = tension
μ = linear mass density of the string
First, let's calculate the linear mass density (μ) of the guitar string:
μ = mass / length
μ = 3.8g / 90cm
μ = 0.042 g/cm
Next, let's convert the linear mass density to kg/m:
μ = 0.042 g/cm * (1 kg / 1000 g) * (1 m / 100 cm)
μ = 0.00042 kg/m
Now, we can calculate the frequencies:
Fundamental frequency (n = 1):
f1 = (1 / 2L) * sqrt(T / μ)
f1 = (1 / (2 * 0.62m)) * sqrt(540N / 0.00042 kg/m)
f1 = 0.806 Hz
First overtone (n = 2):
f2 = 2 * f1
f2 = 2 * 0.806 Hz
f2 = 1.612 Hz
Second overtone (n = 3):
f3 = 3 * f1
f3 = 3 * 0.806 Hz
f3 = 2.418 Hz
Therefore, the frequencies of the fundamental and the first two overtones are approximately:
Fundamental frequency (n = 1): 0.806 Hz
First overtone (n = 2): 1.612 Hz
Second overtone (n = 3): 2.418 Hz
To calculate the frequencies of the fundamental and first two overtones of a guitar string, we 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
First, let's find the value of μ:
μ = m/L
Where:
- m is the mass of the string
μ = 3.8g / 90cm
μ = 0.0422 g/cm
Next, let's convert the mass per unit length to kg/m:
μ = 0.0422 g/cm * (1 kg / 1000 g) * (100 cm / 1 m)
μ = 0.000422 kg/m
Now, let's calculate the frequency of the fundamental:
f1 = (1 / 2L) * sqrt(T / μ)
f1 = (1 / 2 * 0.62m) * sqrt(540N / 0.000422 kg/m)
f1 = (1/1.24) * sqrt(1279621.8)
f1 = 0.806 * 1131.8
f1 ≈ 913 Hz
Next, let's calculate the frequency of the first overtone:
f2 = 2 * f1
f2 ≈ 2 * 913 Hz
f2 ≈ 1826 Hz
Finally, let's calculate the frequency of the second overtone:
f3 = 3 * f1
f3 ≈ 3 * 913 Hz
f3 ≈ 2739 Hz
Therefore, the frequencies of the fundamental and first two overtones are approximately:
Fundamental frequency (f1): 913 Hz
First overtone (f2): 1826 Hz
Second overtone (f3): 2739 Hz