Physics - Standing waves on guitar

1. Pitch Perfect 2
By: Teresa Chang

2. What is a capo?
A capo is a clamp used on neck of stringed
instruments, such as guitars, banjos and
mandolins
Purpose:
Shorten playable length of strings to
increase the pitch (aka the frequency)

3. Why use a capo?
The fingerings (orientation of fingers on each
string) remains unchanged when a capo is used
compared to when it is open (without a capo).
In other words, capos make playing higher notes
easier!

4. Standing waves on guitars
Since the strings on a guitar
are fixed on both ends (via
tuning pegs and bridge base
of guitar), when a string is
plucked, standing waves are
created as the wave
oscillates back and forth,
and reflected from fixed
ends

5. Problem
Part A: If I pluck an open string (without a capo) of linear mass
density 7.28g/m on a guitar, the tension of the guitar string I’m
playing is 96.882 N and I measure the length of string as 70cm,
what is the frequency of the string without a capo on? Assume
all frequencies produced on the guitar are fundamental
frequencies.
Part B: If I want to use a capo to play a higher note that is double
the frequency of the open string, at what length from the bridge
of the guitar should I place my capo in order to produce this new
pitch?

6. Part A- Step 1: Write down known
values

7. Part A- Step 2: Plug into equation
Use the fundamental frequency equation:

8. Part B- Step 1: Write down known
values
Tension and linear mass density are properties that
remain unchanged from part A because you are using the
same string.

9. Part B- Step 2: Manipulate
equation and plug values in
1) Use fundamental
frequency equation.
2) Multiply both
sides by 2L2.
3) Divide both
sides by f2.
4) Divide both
sides by 2.
5) Plug values in.
Result: The new
frequency is
proportional to ½ the
length of the string of
the old frequency.

10. Making sense of your
answer
The frequency calculated from part A corresponds to the string E2
(you can Google this). From part B, to play a pitch of 164.8Hz, the
capo must be 35cm away from the bridge which corresponds to
the 12th fret of the guitar. Your new note is now E3. You can
actually measure this!
Length of E2 string measured from tuning
peg to bridge base of guitar
Length of E2 string measured is
70cm

11. Making sense of your
answer
Unfortunately, on classical guitars, the 12th fret is where the sound
board is, so you can’t actually put a capo there. But it works on
acoustic guitars!
Length of string from 12th fret to bridge base
of guitar
Length of E3 string measured is
35cm