What tension would you need to make a middle c (261.6 hz) fundamental mode on a 1 m string (for example, on a harp)? the linear mass density is 0.02 g/cm?
The frequency of middle C on a string is f = 261.6 Hz.
The given linear density is ρ = 0.02 g/cm = (0.02 x 10⁻³ kg)/(10⁻² m) = 0.002 kg/m
The length of the string is L = 1 m.
Let T = the tension in the string (N). The velocity of the standing wave is [tex]v= \sqrt{ \frac{T}{\rho} } [/tex]
In the fundamental mode, the wavelength, λ, is equal to the length, L. That is Because v = fλ, therefore [tex] \sqrt{ \frac{T}{\rho} } =f \lambda = fL \\\\ \frac{T}{\rho} = (fL)^{2} \\\\ T = \rho (fL)^{2}[/tex]
From given information, obtain T = (0.002 kg/m)*(261.6 1/s)²*(1 m)² = 136.87 N
