## Tuesday, December 3, 2013

### Spring Constants for Guitar and Piano Strings

Over the break, my mom played piano and my brother played guitar for my family. This made me think of the simple harmonic motion of the guitar and piano strings (assuming no friction is involved and the strings would keep oscillating after being played). I decided to calculate the spring constant for the G and A strings on a classical acoustic guitar and a grand piano.

Each note vibrates at a specific frequency. The note G vibrates at 392.00 Hz. I used the equation f = (1/2π)√(k/m) to find the spring constant. The mass of a G string on a classical acoustic guitar is about 3.5 g or 0.0035 kg. By rearranging the equation, I found k=m(2πf)2. For the G string on the guitar, k= (0.0035 kg)(2π * 392.00 Hz)2. The spring constant equals 21,232 N/m. As the pitch of the note becomes higher, the string becomes thinner and lighter and the frequency of the note increases. I estimated the mass of the A string would be about 3.0 g or 0.0030 kg.  The frequency of an A note is 440.00 Hz. Therefore, the spring constant for a guitar A string is: k = (0.0030 kg)(2π * 440.00 Hz)2. The spring constant for the guitar A string = 22,929 N/m.

For the piano strings, I first needed to find the mass of the G and A strings. I found the mass of a piano string is about 5.0 g per meter. The length of the G string is about 0.42 m. The length of piano strings increase and become thinner as the notes become higher. I calculated mass = 0.005 kg/0.42 m = 0.0119 kg for the G string. I estimated that the A string was about 0.50 m and it’s mass was slightly lower, about 4.5 g per meter. I calculated the mass of the A string = 0.0045 kg/0.50m = 0.0090 kg (slightly less than the G string’s mass).

Using the same equation as listed previously, I found the spring constant for the piano G string is: k = (0.0119 kg)(2π * 392.00 Hz)2. For the piano, the G note spring constant = 72,190 N/m. The spring constant for the A string is: k = (0.0090 kg)(2π * 440.00 Hz)2. The piano A string spring constant = 68,787 N/m.

In summary the spring constants for the guitar and for the piano are:
Guitar:
·      G string: k= 21,232 N/m
·      A string: k= 22,929 N/m.

Piano:
·      G string: k= 72,190 N/m
·      A string: k= 68,787 N/m

The spring constants are much greater for the piano strings than the guitar strings because piano strings are stiffer and stronger than guitar strings.

#### 1 comment:

1. f = (1/2π)√(k/m) applies to a mass vibrating in the *longitudinal* direction of the spring. Strings vibrate in the transverse direction. So your calculation is not correct.

You can also understand that 21 Newton (only 2,1 kg) could never make a guitar string 1 meter longer!

:-)