Troubleshooting Wave Velocity on a Piano Wire

[ac7597]

Homework Statement

Joe Edison has a great idea for an invention: he'll transmit power across the city without electricity -by using pianos. If a piano wire stretches from each apartment to the power station, then a worker can play a piano in the station, sending a disturbance along the wire and into each home. A simple device can then convert the disturbance into useful energy ...

To test his idea, Joe sets up a miniature model. He starts with a piece of piano wire of length L=1.6 m and mass m=7 grams. He stretches the wire on a truss so that it has a tension T=280 Newtons. Joe then repeatedly whacks one end of the wire, creating a wave of amplitude 2 mm in height and angular frequency 50 radians per second.

What is the speed at which the wave runs down the wire?

What is the rate at which energy is transmitted down the wire? Express your answer in Joules per second.

Relevant Equations

power= (1/2)(velocity)(mass length)(angular frequency)^2 (amplitude)^2

angular frequency= 50 rad/s= 2*pi*frequency
frequency= 7.96 Hz

k=2*pi/wavelength
k=2*pi/(2*1.6m) = 1.96
velocity=angular frequency/ k
velocity=50/ 1.96 = 25.5 m/s
For some reason this velocity is wrong

 

[Cutter Ketch]

That would all be correct IF the wave was the fundamental oscillation of the wire. (your assumption that the wavelength is 2*L) It is not. You need to find the velocity based on the given properties of the wire. You will find the velocity is much higher and the wavelength is much shorter.