AME140
Homework 1 – oscillations, strings
(Due on Tuesday 9/16 in class)
Problem 1 – The bouncy car
It’s time to replace the shock absorbers on your car when it bounces like a ball
every time you hit a pothole. Let’s say that you count the bounces after hitting a
particularly big pothole and you note that there are 4 complete bounces in 6 seconds.
What is the spring constant for each of the four springs (one at each wheel of the car) if
the car weighs 1800 kg, fully loaded? Express the answer in units of Newtons/meter.
Problem 2 – Helmholtz resonator bass trap
A common acoustical problem in a room is the presence of
low-frequency acoustic resonances, which make a space sound
“boomy”.
One way to absorb the low frequency acoustic
oscillations in a room is to place a Helmhotz resonator bass trap,
usually several of them, in the room. A simple way to build a bass
trap is to build a ported box with a tube protruding into it. You
can then place some acoustic absorbing material in the box to
damp the Helmholtz oscillations. Let’s say that you have a piece
of PVC pipe with a 8” inside diameter that is 3 feet long that you
are going to use for the project. What volume should the box
have to make the bass trap have a resonance at 80 Hertz? (Use
340 m/sec for the speed of sound.)
Problem 3 – Solutions of the differential equation for a string
x
By differentiation and substitution prove that y (x ,t ) = sin(nw 0t )×sin(np ) is a
L
2
2
d y (x ,t )
d y (x ,t )
solution of the wave equation for a string of length L: m
, where
=T
2
2
dt
dx
c
. Note that the speed of a wave on the string is c = (T/µ)1/2 where T is the
L
tension in the string and µ is the mass per unit length of the string.
w0 = p
AME140
Problem 4 – Bass guitar string tension
Compute the tension of each string on a standard bass guitar with a string length
of 34” (0.864 meters). The table below gives the fundamental frequencies of each string
and the mass per unit length of each string.
String
Frequency
(Hz)
Mass/length
(kg/m)
G
98.0
0.0323
D
73.4
0.0691
A
55.0
0.1007
E
41.2
0.1560
Wave speed
(m/sec)
Tension
(Nts)
Tension
(lbs)
Note that 1 Nt (Newton) of force equals 0.225 Pounds of force.
Problem 5 – Nodes of modes of strings
If you play a guitar with a string length of 25.5” where would you pluck the string
to avoid exciting the string’s 4th harmonic? Where would this point be to avoid exciting
the 6th harmonic? Express your answer as the distance of the plucking point from the
bridge of the guitar.