Power is to be transmitted along a taut string by means of transverse harmonic w

Question

Power is to be transmitted along a taut string by means of transverse harmonic waves. The wave speed is 9 m/s and the linear mass density of the string is 0.009 kg/m. The power source oscillates with an amplitude of 0.46 mm.
(a) What average power is transmitted along the string if the frequency is 420 Hz?


? mW

(b) The power transmitted can be increased by increasing the tension in the string, the frequency of the source, or the amplitude of the waves. By how much would each of these quantities have to increase to cause an increase in power by a factor of 100 if it is the only quantity changed?
tension = ?
frequency = ?
amplitude = ?

Explanation / Answer

The wave speed v = 9m/s the linear mass density = 0.009 kg/m The amplitude of the oscillation y = 0.46*10^-3 m The frequency f = 420Hz Power is to be transmitted along a taut string by means of transverse harmonic waves         P = 1/2 v^2 y^2            = (0.5)(0.009)(9)(2*420)^2 (0.46*10^-3)^2            = 0.0596 W   or 59.6 mW (b) If the power increased by P' = 100P If the tension is varied            P = 1/2 v^2 y^2               = (0.5) (T) ^2 y^2         then tension T is square of the power so the tension is has to increse by a factor 100^2 = 10000 from the expression f is proportional to the square root of the power           f P
then the frequency is increased by 10 times now amplitude is directly proportional to the power so the amplitude is increased by a factor 100              

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