A tuning fork of frequency 440 Hz is attached to a long string of linear mass density `0.01 kg m^-1` kept under a tension of 49 N. The fork produces transverse waves of amplitude 0.50 mm on the string. (a) Find the wave speed and the wavelength of the waves. (b) Find the maximum speed and acceleration of a particle of the string. (c) At what average rate is the tuning fork transmitting energy to the string ?

Answer

70 m/s, 15.9

Explanation

Wave speed
`v = sqrt((T)/(mu)) = sqrt((49)/(0.01)) = 70 m//s`
`lambda = (v)/(n) = (70)/(440) = 0.159m`
Maximum particle speed is
`v_(max) = Aomega = 0.5 xx 10^(-3) xx 2pi xx 440` = 1.381 m/s
Maximum particle acceleration is
`a_(max) = Aomega^(2) = 0.5 xx 10^(-3)xx (880pi)`
= 3872 m/s
Wave power is `P = 2pi^(2)n^(2)A^(2)muv`
`= 2 xx 10 xx (440)^(2) xx (0.5 xx 10^(-3)) xx (0.01) xx 70`
= 0.667 W