Question

Suppose a guitar is tuned to drop D tuning. This means that the notes from the 6th
string to 1st string is D-A-D-G-B-E. Compute what the spring constants of each string
must be once each has achieved its tension after being tuned. This requires a differential equation.

Answer 1

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.