AME140 Homework 1 – oscillations, strings (Due on Tuesday 9/17 in class) Problem 1 – The bouncy car You know that it’s time to replace the shock absorbers on your car when every time that you hit a pothole your car bounces up and down like you’re on a trampoline. Let’s say that you count the bounces after hitting one particularly big pothole and you note that there are 4 complete bounces in 5 seconds. What is the spring constant for each of the four springs (one at each wheel of the car)? Express the answer in units of Newtons/meter. Assume that the car has a mass of 1500 kg, fully loaded with you and your friends. Problem 2 – Helmholtz resonator bass trap A common acoustical problem in a room is the presence of low frequency acoustic resonances, which tend to 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, or 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 absorb the sound. Let’s say that you have a piece of PVC pipe with a 4” inside diameter that is 2 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 125 Hertz? (Use 340 m/sec for the speed of sound.) Problem 3 – Solution of the string differential equation By taking the appropriate derivatives and substitution prove that x
y (x ,t ) = sin(nω 0t )⋅sin(nπ ) is a solution of the differential equation for a string of L
2
d y (x ,t )
d 2 y (x ,t )
c
=
T
length L, µ
, where ω 0 = π Note that the speed of a wave on 2
2
L
dt
dx
1/2
the string is c = (T/µ) . AME140 Problem 4 – Bass guitar string tension Compute the tension of each string in 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) Wave speed (m/sec) Tension (Nts) Tension (lbs) G 98.0 0.0323 D 73.4 0.0691 A 55.0 0.1007 E 41.2 0.1560 Note that 1 Nt (Newton) of force equals 0.225 Pounds of force. Problem 5 – Nodes of modes of strings If you are playing 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 from the bridge.