A steel piano string is 47.1 inches long. The diameter of thestring 0.0421 inche

Question

A steel piano string is 47.1 inches long. The diameter of thestring

0.0421 inches. When struck, this string produces the musical note Bwhich has the pitch

of 123 Hz. (a) Find the tension of this string. Give the answer inpounds. The density of

steel is 7.86 g/cm3 . (b) There are 228 strings in a piano. (Somenotes use more than one

string.) If we assume that the tension in every string is the same,?nd the total force (in

tons, 1 ton = 2000 lb) exerted on the frame of the piano. (Addtogether the tensions of

all of the strings.) Piano frames are made of steel so that theycan withstand this kind of

force.

Explanation / Answer

v = (F / )   where v is the speed ofthe wave in the string, F tension, the m / L of thestring = 2 L for the fundamental frequency = 94.2 in = 2.39 m f = 123   so v = 123 * 2.39 = 294 m/s = 7.86 g / m3 = 7860 kg / m3 = V / L = A L / L = A = 7860 * * 2.86 * 10E-7  (convert area tom2) = .00706 kg / m = V / L = A L / L = A = 7860 * * 2.86 * 10E-7  (convert area tom2) = .00706 kg / m So F = v2 * .00706= 610  N 610 * .225 lb / N = 137 lb Total force = 228 * 137 = 31,300 lbs

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