Tension and Frequency Question-I am confused please walk me through the problem.

Question

A piano tuner stretches a steel piano wire with a tension of 765 N. The steel wire has a length of 0.700 m and a mass of 5.25 g.

1. What is the frequency of the string's fundamental mode of vibration?
2. What is the number n of the highest harmonic that could be heard by a person who is capable of hearing frequencies up to f = 16 kHz?

Answer 1

First, we have to use the equation that relates the speed of a wave on the wire to its tension.

This equation is v = sq root (T/ µ) where v = velocity of the wave, T the tension, and µ the mass per unit length.

So, using the information we were given v = sq.rt (765 N / 0.00525 kg / 0.7m) = 319.4 m/s.  

F1 = 319.4 m/s / (2 x 0.7 m) =  228.1 s^-1  ( F1 = v / 2L)

The nth harmonic is given by Fn = n x F1 or n = Fn / F1.

 n = 16000 Hz / 228.1 s^-1 = 70

Truth be told, I didn't remember, the equation relating speed to wire tension, but this can be found on the internet or in a textbook. The key is to recall the relationships between speed, wavelength, and frequency, the fundamentals of standing waves, and a little algebraic manipulation.

always check the math.