Physics 2210 Fall Semester 2014 - Astronomybelz/phys2210/lecture23.pdf · Physics 2210 Fall...
Transcript of Physics 2210 Fall Semester 2014 - Astronomybelz/phys2210/lecture23.pdf · Physics 2210 Fall...
Announcements
● Unit 21 “Simple and Physical Pendula” (Nov 24th)● HW Due 11/25th as usual
● No new material Wednesday November 26th. In-class discussion of problems for last midterm, final exam. ● Unit 22 “Harmonic Waves” Monday Dec 1st
● HW due Sunday December 7th at midnight● Exam 3 (covering thru unit 19) December 3rd
● Unit 23 “Standing Waves and Superposition”, Monday December 8th
● HW Due Tuesday 9th
● Course Review December 10th
● Final Exam (December 16th)
Announcements● Final exam time is Tues Dec 16th, 3:30-5:30 PM. ● Room JFB 101
● Source - http://registrar.utah.edu/academic-calendars/final-exams-fall2013.php
US
Prelecture Feedback
● This seems too abstract compared to previous sections.
● What happens when waves collide with different wavelengths or velocities? Is a standing wave still created?
● You should play the guitar in class.
Example: A uniform string has mass 0.300 kg and length of 6.00 m. The string passes over a pulley and supports a 2 kg object. Find the speed of a pulse traveling on the string.
Q: What happens when two waves “collide?”
A: They ADD together!
Superposition
Mechanics Lecture 23, Slide 9
For linear equations, if we have two separate solutions, A and B, then A + B is also a solution!
Ct At Bt
2 1
Superposition
CONSTRUCTIVEINTERFERENCE
DESTRUCTIVEINTERFERENCE
Mechanics Lecture 23, Slide 10
B2t
A1t
The wave equation we derived last time is linear. (It has no terms where
the variables x,t are squared.)2
2
22
2 1
dt
xd
vdy
xd
Can we predict this pattern mathematically?
Just add two cosines and remember the identity:
where and
cosLt cosHt
Beats
212
1 L 212
1 H
ttAtAtA HL coscos2)cos()cos( 21
Mechanics Lecture 23, Slide 11
Standing Waves
What happens when two waves having the same frequency but moving in the opposite direction meet?
Mechanics Lecture 23, Slide 12
Wave 1 Wave 2
Wave 1Wave 2
Wave 1Wave 2
zero (Nodes)
Superposition
)cos()cos(2)cos()cos( tkxAtkxAtkxA
Changingamplitude
Stationary wave
How it Works
How to make it:
Mechanics Lecture 23, Slide 13
www.youtube.com/watch?v=3mclp9QmCGshttp://
Mechanics Lecture 23, Slide 14
We just found that the frequency of a string scales
with its diameter if the tension is the same.
This is really what you find if you examine a
guitar:
dlow
dhigh
.050”.050”.012”.012”
Mechanics Lecture 23, Slide 18
Their diameters are about half of the adjacent strings.
.028 .052 .022 .042 .014 .030
The lighter string in these pairs is one octave higher (2x
frequency).
Different Example – 12 String Guitar
Mechanics Lecture 23, Slide 19
HW Example
● A guitar string with mass density m = 2.3 x 10-4 kg/m is L = 1.04 m long on the guitar. The string is tuned by adjusting the tension to T = 108.4 N.● With what speed do waves on the string travel?● What is the fundamental frequency for this string?● Someone places a finger a distance 0.164 m from
the top end of the guitar. What is the fundamental frequency in this case?
● To “down tune” the guitar (so everything plays at a lower frequency) how should the tension be adjusted?