The lowest A on a piano has a frequency of 27.5 Hz.
Assume: The tension in the A piano wire (of length 0.51 m) is 305 N, and one-half wavelength occupies the wire.
What is the mass of m the wire? Answer in units of kg.
To find the mass of the wire, we can use the formula for the velocity of a wave on a string:
v = √(T/μ)
Where:
v = velocity of the wave
T = tension in the string
μ = mass per unit length (linear mass density) of the string
We are given:
Frequency = 27.5 Hz
Length of the wire = 0.51 m
Tension in the wire = 305 N
One-half wavelength occupies the wire
First, let's find the velocity of the wave using the frequency and wavelength.
Since one-half wavelength occupies the wire, the full wavelength is twice the length of the wire, which is 2 * 0.51 = 1.02 m.
The velocity of the wave is given by the formula:
v = λ * f
Where:
v = velocity of the wave
λ = wavelength
f = frequency
Substituting the values, we get:
v = 1.02 m * 27.5 Hz = 28.05 m/s
Now, let's find the linear mass density (μ) of the string.
μ = T / (v²)
Substituting the values, we get:
μ = 305 N / (28.05 m/s)² = 0.394 kg/m
Since the mass per unit length is given in kg/m, we can find the mass of the wire by multiplying the linear mass density by the length of the wire.
Mass of the wire (m) = μ * length of the wire
= 0.394 kg/m * 0.51 m
= 0.201 kg
Therefore, the mass of the wire is 0.201 kg.