TUNING YOUR PIANO

While playing your piano, you notice that your A note sounds out of key. You decide to tune the piano yourself and restore the A note back to its original frequency of 440Hz with the help of your tuning fork. When you strike the tuning fork and the piano string, you hear a beat frequency of 4 beats per second. Since you have taken Physics 101, you know how to approach tuning your piano with a few simple measurements. You measure the string length to be 27cm and estimate from online sources the linear mass density to be 1.2g/m.

a) Based on the beat frequency, what could your piano frequency be?

b) After tightening the string, you now hear 3 beats per second. What was the original piano string frequency?

c) How much tension must you apply to tune the piano now that you hear 3 beats per second?

d) How many beats per second will you hear at this frequency?

Based on the beat frequency, what could yourpiano frequency be?

We know that the beat frequency we hear is the difference between the piano frequency and the tuning fork frequency. However, it may be greater or less than the tuning fork frequency.

Therefore the two possible frequencies of the piano are: f1=440+4 = 444Hz

f2=440-4 = 436Hz

After tightening the string, you now hear 3 beats per second. What was the original piano string frequency?

The piano is constrained at both ends. Therefore the fundamental frequency is: We also know that:

We can now rewrite the equation as:

As seen in the equation, when we increase tension, the frequency also increases. Since the piano frequency increased as the beat frequency decreased, the piano frequency must be approaching the tuning fork frequency of 440Hz.

Thus the original frequency of the string was 436Hz.

How much tension must you apply to tune the piano now that you hear 3 beats per second?

Since we now hear 3 beats per second, we can apply the same reasoning as before to determine the current piano frequency to be 437Hz. We can use the equation derived from the previous question to solve this problem by setting up two equations to solve for the tension in either case and then determine their difference.

How many beats per second will you hear at this frequency?

Since the frequency of the tuning fork now matches that of the piano, you will hear no beat and simply your now correctly tuned 440Hz A-note frequency.