Question

What is the frequency of the fundamental mode of vibration of a steel piano wire stretched to a tension of 440 N? The wire is 0.600 m long and has a mass of 5.60grams.


234 Hz
517 Hz
312 Hz
366 Hz
295 Hz

i tried using this equation- v = √tension / √mass per unit length

so v = √440 / √5.6/0.6 = 6.87 m/s

then i plugged that into f1 = 1/2(L) * vsof1 = 1/2(0.6) * 6.87m/s = 5.721... which is way wrong. what am i doing wrong?

Answer 1

mass per unit length is .0056kg/.6m = 0.009333kg/m
So, v =√440/√(.009333) = 217.12m/s

Then, the first node has frequency f1 = v/(2L)

f1 = (217.12)/(2*.6) = 181 Hz.

Answer 2

Given:Tension in the string T = 440 NLength of the wire L = 0.60 mmass of the string m = 5.60 grams= 0.0056 kg.................................................................................................................................wave velocity in vibrating string is given by v = √ T/(m/L) ...... (1)= √(440 N) / [( 0.0056 kg)/( 0.60 m)]= 217.12 m/secfundamental frequency f = v/4L ...... (2)=(217.12 m/sec) / 4 ( 0.60 m)=90.4Hz.