A steel piano string for the note middle C is about 0.68 m long and is under a tension of 400 N. If the fundamental frequency is 226 Hz, what is the string’s diameter?
To find the string's diameter, we can use the formula for the fundamental frequency of a vibrating string:
f = (1/2L) * sqrt(T/μ)
Where:
- f is the fundamental frequency (226 Hz)
- L is the length of the string (0.68 m)
- T is the tension in the string (400 N)
- μ is the linear mass density of the string
The linear mass density of the string (μ) is computed using the formula:
μ = (π/4) * (d^2) * ρ
Where:
- d is the diameter of the string (what we're trying to find)
- ρ is the density of the material (for steel, it's typically 7850 kg/m^3)
We can substitute the expression for μ into the formula for the frequency:
f = (1/2L) * sqrt(T / [(π/4) * (d^2) * ρ])
To find d, we can rearrange the equation:
d^2 = (4 * T * L^2) / [(π * f^2 * ρ)]
Taking the square root, we get:
d = sqrt((4 * T * L^2) / [(π * f^2 * ρ)])
Now, we can plug in the given values and calculate the diameter.