The lowest A on a piano has a frequency of 27.5 Hz.
Assume: The tension in the A piano wire (of length 1.8 m) is 305 N, and one-half wave- length occupies the wire.
What is the mass of m the wire? Answer in units of kg.
To find the mass (m) of the wire, we can use the relationship between frequency (f), tension (T), and linear mass density (μ) of the wire. The equation is given by:
f = (1/2L) * sqrt(T/μ)
Where:
f = frequency of the wave
L = length of the wire
T = tension in the wire
μ = linear mass density of the wire
First, we need to rearrange the equation to solve for μ:
μ = (T/4L^2) * (1/f^2)
Given values:
f = 27.5 Hz
L = 1.8 m
T = 305 N
Now we can substitute the values into the equation to find the linear mass density (μ):
μ = (305 N / 4 * (1.8 m)^2) * (1 / (27.5 Hz)^2)
Calculating this expression:
μ ≈ 0.00118 kg/m
Therefore, the mass of the wire (m) is equal to the linear mass density (μ) multiplied by the length (L):
m = μ * L
m = 0.00118 kg/m * 1.8 m
m ≈ 0.00212 kg
So, the mass of the wire is approximately 0.00212 kg.
To find the mass of the wire, we need to use the wave equation:
v = √(T/μ)
Where:
v = velocity of the wave (m/s),
T = tension in the wire (N), and
μ = linear mass density of the wire (kg/m).
We can rearrange the equation to solve for μ:
μ = T / v^2
First, let's calculate the velocity of the wave. Since the wavelength is one-half the length of the wire, we can find the velocity by using the formula:
v = fλ
Where:
v = velocity of the wave (m/s),
f = frequency (Hz), and
λ = wavelength (m).
Given:
Frequency (f) = 27.5 Hz
Wavelength (λ) = 1.8 m / 2 = 0.9 m
Using the formula, we have:
v = 27.5 Hz * 0.9 m = 24.75 m/s
Now we can calculate the linear mass density (μ):
μ = 305 N / (24.75 m/s)^2
μ = 305 N / 610.5625 m^2/s^2
μ ≈ 0.5006 kg/m
Hence, the mass of the wire is approximately 0.5006 kg.