Chapter 12, Problem 15RQ

1. 15 A standard guitar, whether acoustic or electric, has six strings, all with essentially the same total length between the bridge and the nut at the tuning head. Each string vibrates at a different frequency determined by the tension on the string and the mass per unit length of the string. In order to create pitches (notes) other than these six, the guitarist presses the strings down against the fretboard, thus shortening the length of the strings and changing their frequencies. In other words, the vibrating frequency of a string depends on tension , length, and mass per unit length of the string. The equation for the fundamental frequency of a vibrating string is given by

   $\text{f=}\frac{\sqrt{T/\mu }}{2L}$

   Where

   f = frequency [Hz]

   T = string tension [N]

   µ·=  mass per unit length [kg/ m]

   L = string length [m]

   Many electric guitars have a device often called a "whammy" bar or a "tremolo" bar that allows the guitarist to change the tension on the strings quickly and easily, thus changing the frequency of the strings. (Think of Jimi Hendrix simulating "the rockets red glare, the bombs bursting in air" in his rendition of The Star Spangled Banner-a true tour de force.) In designing a new whammy bar, we test our design by collecting data on a single string of the observed frequency at different string lengths (using the fretboard) with a specific setting of the whammy bar.

Length (L) [m]	0.25	0.28	0.32	0.36	0.40	0.45	0.51	0.57	0.64
Frequency (f) [Hz]	292	241	231	205	171	165	136	129	112

1. a. Create a graph of the observed frequency data, including the power trendline and equation generated by Excel.
2. b. If the tension was reduced to half of its original value, would the frequency increase or decrease and by what percentage of the original values?
3. c. c. If the tension on the string is 125 newtons, what is the mass per unit length in grams per meter?
4. d. d. If the mass per length of the string is 3 grams per meter, what is the tension in newtons?