When a violinist pulls the bow across a string, the force with which the bow is

Question

When a violinist pulls the bow across a string, the force with which the bow is pulled is fairly small, about 0.61 N. Suppose the bow travels across the A string, which vibrates at 440 Hz, at 0.47 m/s. A listener 36 m from the performer hears a sound of 58-üB intensity. Assuming that the sound radiates uniformly in all directions, with what efficiency is the mechanical energy of bowing converted to sound energy? Information regarding the rate at which mechanical energy is delivered to the string and the rate at which sound energy arrives at the location of the listener. Take the efficiency to be the ratio of the sound power delivered to the listener divided by the power delivered to the string.

Explanation / Answer

Sound level = 10 log (I / 10^-12) = 58

I / 10^-12 = 10^5.8

I = 6.31 x 10^-7 W/m^2

and I = P / (4 pi r^2)

P = (6.31 x 10^-7) (4 x pi x 36^2)

P = 0.0103 W ......Sound energy

Mechanical energy, P = F v = 0.61 x 0.47 = 0.2867 W

efficiency = 0.0103 / 0.2867

= 0.036

in % = 0.036 x 100 = 3.6 %

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