The D string on a banjo is.60 m long, has a mass of.30 g and vibrates at a funda

Question

The D string on a banjo is.60 m long, has a mass of.30 g and vibrates at a fundamental frequency of 587 Hz. Calculate the speed of the wave and the tension on the string. The speed of sound in air is dependent on the atmospheric pressure and the density of the air. If the pressure is 740 mm Hg and the density is 0.001181 gm/cm^3, find the speed of sound. The sound level at a distance of 3.00 m from a source is 140 dB. At what distance from the source is the sound level 70 dB and what is the power output of the source in watts.

Explanation / Answer

1)

Length , L = 0.60 m

m = 0.30 gm

m = 3 *10^-4 Kg

fo = 587 Hz

let the speed of wave is V

v = frequency * wavelength

v = 587 * (2 * 0.60)

v = 704.4 m/s

the speed of the wave is 704.4 m/s

let the tension in the cable is T

as v = sqrt(T/(m/L))

704.4 = sqrt(T * 0.60/(3 *10^-4))

solving for T

T = 248.1 N

the tension in the string is 248.1 N
2)

for v = sqrt(y * Po/p)

v = sqrt(1.402 * 740 * 10^-3 * 13600 * 9.8/(1.181))

v = 342.2 m/s

the soeed of sound is 342.2 m/s

3)

let the distance is d2

d1 = 3 m

140 -70 = 10 * log(3/d2)

d2 = 3.33 *10^7 m

the distance from the source is 3.33 *10^7 m

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