Using the wave speed calculated from the period and wavelength, and the given tension, the linear density of the piano string is found to be 0.00867 kg/m.
Step-by-step explanation:
To find the linear density (μ) of the piano string, we will use the wave equation v = √(F/μ), where v is the wave speed, F is the tension in the string, and μ is the linear mass density of the string.
The wave speed (v) can also be calculated using the period (T) and the wavelength (λ) via the relationship v = λ/T. Given that the wavelength is 1.26 m and the period is 3.82 ms (0.00382 seconds), the wave speed is:
v = (1.26 m) / (0.00382 s) = 329.84 m/s
Now that we have the wave speed, we can solve for μ using the initial wave equation:
μ = F/v2 = 944 N / (329.84 m/s)2 = 944 N / (108875.71 m2/s2) = 0.00867 kg/m
Therefore, the linear density of the piano string is 0.00867 kg/m.