Question

It is desired to produce a musical note having a frequency of 280 Hz. (air temp 20°C so speed of sound = 343 m/s)

a.) what tension should be placed on a guitar string that has a mass per unit length of 6.80 grams/ meter? the string has a length of 18 inches between anchor points.

b.) an organ pipe open at one end and closed at the other end is used to produce the same note. what should be the length of the pipe?

Answer 1

Part A: We have:

Since the end of the string are fixed, the ends are nodes by definition.

$f_{n} = \frac{n}{2.L}.\sqrt{\frac{T}{\mu }}$ $f_{n} = \frac{n}{2.L}.\sqrt{\frac{T}{\mu }} \Rightarrow T = 445,8 [kg]$

where we made that 1 [in] = 0,0254 [m], and the value of n used: n = 1, because string musical instruments vibrates in the first harmonic, wich is called "fundamental frecuency" (in higer harmonic orders the tension will be too big)

in the ecuation:

n: bucle number

L: length of the string

u: Density

f: frecuency

Part B: Then we have an organ pipe open at one end and closed at the other, so the ecuation in this case is:

$f_{n} = n\frac{v_{sound}}{4.L}$ $f_{n} = n\frac{v_{sound}}{4.L}\Rightarrow L = 0,31 [m] = 12 [in]$

where:

n: armonic number

L: lenght of the pipe

be careful with the unit of measurement!!!

good luck!!!