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?
I have tried it several times and continue to get an incorrect answer.
To calculate the string's diameter, we can use the formula for the fundamental frequency of a vibrating string:
f = (1 / 2L) * √(T / μ)
Where:
f = fundamental frequency (Hz)
L = length of the string (m)
T = tension in the string (N)
μ = linear mass density (kg/m)
Rearranging the formula, we can solve for the linear mass density:
μ = (T / (4L²)) * (1 / f²)
To find the string's diameter, we can calculate the linear mass density using the formula above and then use it to find the mass per unit length (μ') of the string. Finally, we can calculate the string's diameter (d) using the equation for the mass of a cylindrical object:
m = μ' * π * (d / 2)²
Rearranging the formula, we get:
d = √((4m) / (π * μ'))
Now, let's calculate step by step:
Given:
L = 0.68 m (length of the string)
T = 400 N (tension in the string)
f = 226 Hz (fundamental frequency)
1. Calculate the linear mass density (μ):
μ = (T / (4L²)) * (1 / f²)
= (400 / (4 * 0.68²)) * (1 / 226²)
= 0.29706 kg/m
2. Calculate the mass per unit length (μ'):
μ' = μ / L
= 0.29706 / 0.68
= 0.4362 kg/m
3. Calculate the string's diameter (d):
d = √((4m) / (π * μ'))
= √((4 * 0.4362) / (π * 0.29706))
= √(1.7448 / 0.931027)
= √1.8727
= 1.369 mm (approximately)
Therefore, the string's diameter for the note middle C is approximately 1.369 mm.