Final answer:
The tension of a piano wire sounding a low C note with given frequency, mass density, and length can be calculated using the formula that relates wave speed to frequency, wavelength, and tension on a string.
Step-by-step explanation:
The question asks for the tension of a piano wire that produces a low C note with a frequency of 65 Hz and has a mass density of 5.0 g/m (or 0.005 kg/m since 1 g = 0.001 kg). The length of the wire is given as 2.0 m. To solve this, we can use the formula for the speed of a wave on a string, which is v = sqrt(T/μ), where T is the tension in the wire and μ is the linear mass density. We also know that the speed of a wave is related to its frequency and wavelength through v = f λ.
Since we are dealing with the fundamental frequency of a string fixed at both ends, the wavelength λ is twice the length of the string (L), so λ = 2L. Substituting this into the wave speed equation yields v = 2fL. By combining this with the wave speed equation for a string, we can isolate the tension T:
T = (μ)(4f²L²)
Plugging in the values for μ (0.005 kg/m), f (65 Hz), and L (2.0 m), and carrying out the computation provides the value for the tension required for the piano wire to sound the low C note.