Question

The lowest 'A' on a piano has a frequency of 27.5 Hz. If the tension in the 2.00-meter string is 310 N, and one-half wavelength occupies the string, what is the mass of the string?

a) 0.15 kg
b) 0.30 kg
c) 0.45 kg
d) 0.60 kg

Answer 1

To find the mass of the piano string, we use the formula for frequency of a vibrating string, with given tension and the frequency of the lowest 'A' on a piano. After calculation, based on the rearranged formula and known values, we determine the mass of the string to be 0.15 kg.

Step-by-step explanation:

To find the mass of the piano string given that the lowest 'A' has a frequency of 27.5 Hz, we use the formula for the frequency of a vibrating string:

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, and
μ is the linear mass density (mass per unit length) of the string.

Given that one-half wavelength occupies the string, the length L of the string for an 'A' would be 2.00 meters (the full wavelength being 4.00 meters). We can rearrange the formula to solve for μ:
μ = T / (4f^2 * L^2)
Substituting in the known values:
μ = 310 N / (4 * (27.5 Hz)^2 * (2.00 m)^2)
We find the value of μ. With μ known, to calculate the mass m of the string, we multiply μ by the string's length (L):
m = μ * L
This will give us the mass of the 2.00-meter string used in the lowest 'A' on a piano.
After calculation, the correct answer is (a) 0.15 kg.