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 need to use the formula:
v = √(T/μ)
where:
v = velocity of the wave (in m/s)
T = tension in the wire (in N)
μ = linear density of the wire (mass per unit length, in kg/m)
First, let's calculate the velocity of the wave.
Given:
Frequency (f) = 27.5 Hz
∵ v = f * λ [where λ = wavelength]
∵ λ = 2 * (0.51 m) [as one-half wavelength occupies the wire]
∴ v = f * λ = 27.5 Hz * 2 * (0.51 m) = 27.5 Hz * 1.02 m = 28.05 m/s
Now, we can solve for μ using the formula:
μ = (T / v²)
Given:
Tension (T) = 305 N
Velocity (v) = 28.05 m/s
∴ μ = (305 N) / (28.05 m/s)² = 305 N / (789.18025 m²/s²) ≈ 0.000387 kg/m (rounded to 6 decimal places)
The linear density of the wire is approximately 0.000387 kg/m.
To find the mass of the wire, we need to use the wave equation which relates the speed of a wave to its frequency and wavelength.
The speed of a wave on a string is given by the equation:
v = √(T/μ)
where:
- v is the velocity of the wave on the string,
- T is the tension in the string,
- μ is the linear density of the string (mass per unit length).
From the given information, we have the frequency of the lowest A to be 27.5 Hz and the length of the wire to be 0.51 m. Since the frequency and length of the wire are related to the wavelength, we can determine the wavelength using the formula:
λ = 2L
where L is the length of the wire.
Since we have the frequency and wavelength, we can find the velocity of the wave using the formula:
v = fλ
Substituting the values we have:
v = 27.5 Hz * 2 * 0.51 m
Now, we can use the velocity of the wave and the tension in the wire to find the linear density (μ).
To find μ, rearrange the wave equation:
μ = T / v^2
Substituting the values we have:
μ = 305 N / (27.5 Hz * 2 * 0.51 m)^2
Now that we have the linear density, we can calculate the mass of the wire by multiplying the linear density by the length of the wire.
Mass (m) = μ * L
Substituting the values we have:
m = (305 N / (27.5 Hz * 2 * 0.51 m)^2) * 0.51 m
Simplifying the equation would give us the mass of the wire in units of kg.