Your Comments
P
This unit is a little bit easier. Also suggestion for next year, have midterm 3 before
thanksgiving break
I'm so glad that my four years of music theory is finally paying off!
Could you please continue to explain the concept of "Standing Waves"? The whole
notion that two waves superimposed upon one another as being "standing" really just
went over my head!
Out of the blue question, but is it possible to do research in physics after 211?
What about constructive and destructive interference? When does that happen?
Sorry I took 30 seconds on this prelecture...I'm having my first ever semi-emotional
exchange with a member of the opposite sex and my brain just can't handle anything at
all
Physics makes playing the guitar sound very interesting, too bad I dont have any
musical talent whatsoever
Forget about guitars... can I bring my saxophone to class?
My friend wants to bring her saxophone to class, so can I bring my piano?
Bilbo Baggins? More like Bilbo Swaggins, amIright? In all seriousness though, it would
brighten my spirits very much to hear that the physics department appreciates the work
of Mr. Tolkien. Thief! Thief, Baggins! We hates it, we hates it forever!
These are concepts are pretty tricky. In fact, I have a feeling they are going to HERTZ
my grade. HEYOOOOO!
they found water on mercury Looks like we may also have found carbon on Mars
You may have noticed that I'm a week behind on the homework,
and that I've been blowing through the pre-lectures. Suffice to
say that I have not been executing well and am balancing playing
catch-up with not driving myself out of my mind. I -will- re-visit
the pre-lectures, and I -will- get the homework done by the 80%
deadline. Just need a little time to get back on my feet.
80% deadlines for units 21 & 22 moved to Friday morning (Dec/7)
Please don’t procrastinate on units 23-26. Everything
must be finished by December 14 at 8am.
Mechanics Lecture 24, Slide 2
Average: 82%
(no scaling)
Everyone got full credit (3 points)
for Question 15 (blame me)
Mechanics Lecture 24, Slide 3
Clicker Question
Combined exam: Friday Dec 14 at 1:30 pm.
Conflict exam: Wednesday Dec 19 at 1:30 pm.
A: I have checked my schedule already. I will take the combined final.
B: I have checked my schedule already. I will take the conflict final.
C: I have not checked my schedule but I will do so today.
“I just thought that if you had time could you please again discuss
what the final exam is going to be like. How many questions and are
the problems going to be similar to the ones on the hour exams.”
“Will there be any practice finals? I find practice problems to be the
best way to study for tests.”
“How will the final be structured? Will it be 50% Unit 4 and 50% Units
1-3, or will it be 25% for each Unit, or something else entirely?”
There is now a final practice exam online.
Mechanics Lecture 17, Slide 4
Physics 211
Lecture 24
Today’s Concepts:
A) Superposition
B) Standing Waves
Mechanics Lecture 24, Slide 5
The Wave Equation
“Can you please go in detail about f(x+vt)?”
Mechanics Lecture 24, Slide 6
CheckPoint
y
v y( x, t ) = P( x - vt)
Case A
x
y
Case B
y( x, t ) = ?
v
x
Suppose a pulse in Case A described by the function y(x,t) = P(x-vt).
Which of the following functions described the pulse in Case B?
A) y(x,t) = P(x + vt)
B) y(x,t) = -P(x + vt)
C) y(x,t) = -P(x - vt)
Mechanics Lecture 24, Slide 7
y
v y( x, t ) = P( x - vt)
Case A
A) y(x,t) = P(x + vt)
B) y(x,t) = -P(x + vt)
C) y(x,t) = -P(x - vt)
x
y
Case B
y( x, t ) = ?
v
x
A) When x and t have the same signs, the graph
moves to the left.
B) the signs inside need to be the same, but the function
is negative.
C) It is just negative of the one in case A.
Mechanics Lecture 24, Slide 8
Superposition
Q: What happens when two waves “collide?”
A: They ADD together!
Movie (super_pulse)
Movie (super_pulse2)
Mechanics Lecture 24, Slide 9
Superposition
The wave equation we derived last time is
linear. (It has no terms where the variables
x,t are squared.)
d 2x 1 d 2x
= 2 2
2
dy
v dt
For linear equations, if we have two separate solutions, A and B, then A + B
is also a solution!
A(1t)
B(2t)
C(t) = A(t) + B(t)
DESTRUCTIVE
INTERFERENCE
CONSTRUCTIVE
INTERFERENCE
Mechanics Lecture 24, Slide 10
Beats
Can we predict this pattern mathematically? Of course!
Just add two cosines and remember the identity:
A cos(1t ) + A cos(2t ) = 2 A cos(Lt )cos(H t )
where L =
1
(1 - 2 )
2
and  H =
1
(1 + 2 )
2
Demo
cos(Lt)
cos(Ht)
Mechanics Lecture 24, Slide 11
Standing Waves
What happens when two waves having the same frequency but
moving in the opposite direction meet?
Movie (super)
Wave 1
Wave 2
zero (Nodes)
Superposition
Wave 2
Wave 2
Wave 1
Wave 1
Mechanics Lecture 24, Slide 12
How it Works
A cos(kx - t ) + A cos(kx + t ) = 2 A cos(kx) cos(t )
Stationary
wave
Changing
amplitude
How to make it:
Demo
Mechanics Lecture 24, Slide 13
Standing Waves
(N = 1)
Can you talk about how harmonics work on an stringed instrument?
Mechanics Lecture 24, Slide 14
(N = 1)
(N = 1)
Mechanics Lecture 24, Slide 15
Standing Waves
http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html
Mechanics Lecture 24, Slide 16
CheckPoint
Suppose the strings on your guitar are 24” long as shown. The
frets are the places along the neck where you can put your
finger to make the wavelength shorter, and appear as
horizontal white lines on the picture.
When no frets are being pushed the frequency of the highest
string is 4 times higher than the frequency of the lowest
string. Is it possible to play the lowest string with your finger
on any of the frets shown and hear the same frequency as
the highest string?
24
20
frets
16
12
8
A) Yes
B) No
4
Mechanics Lecture 24, Slide 17
Clicker Question
If you want to increase the frequency of a 24”
string by a factor of 4 while keeping the tension the
same, how long should the string be?
A) 4”
B) 6”
C) 8”
24
D) 12”
20
frets
16
12
8
4
Mechanics Lecture 24, Slide 18
Clicker Question
Is there a fret that you can push on this guitar to
make the string 6” long?
A) Yes
24
B) No
20
frets
16
12
8
6
4
Mechanics Lecture 24, Slide 19
Clicker Question
The highest string vibrates with a frequency that is
4 times that of the lowest string.
Compare the speed of a wave on the high string
and the low string:
A) vhigh = vlow*2
B) vhigh = vlow*4
C) vhigh = vlow*16
v=f
vlow
vhigh
Mechanics Lecture 24, Slide 20
Clicker Question
The speed of a wave on the high string is
4 times the speed on the lowest string.
If the tensions are the same, how does the mass
per unit length of the strings compare:
A) mlow = mhigh*2
B) mlow = mhigh
C) mlow = mhigh
*4
Recall
v =
2
*16
T
m
T
m= 2
v
vlow
vhigh
Mechanics Lecture 24, Slide 21
Clicker Question
Two cylinders have the same length and are
made from the same material. If the mass of
the bigger one is 16 times the mass of the
smaller one, how do their diameters
compare?
L
A) dlow = dhigh*2
B) dlow = dhigh
*4
M = mV = m L r 2
C) dlow = dhigh*16
dbig
dsmall
Mechanics Lecture 24, Slide 22
CheckPoint
The 6 strings on a guitar all have about the same
length and are stretched with about the same
tension. The highest string vibrates with a frequency
that is 4 times that of the lowest string.
If the strings are made of the same material, how
would you expect the diameters of the lowest and
highest strings to compare?
A) dlow = dhigh*2
B) dlow = dhigh*4
C) dlow = dhigh*16
dlow
dhigh
Mechanics
Lecture24,
24,Slide
Slide 23
Mechanics
Lecture
The highest string vibrates with a frequency that is 4
times that of the lowest string. If the strings are made
of the same material, how would you expect the
diameters of the lowest and highest strings to
compare?
A) dlow = dhigh*2
B) dlow = dhigh*4
C) dlow = dhigh*16
B) Since frequency is related to the 1/sqrt(mass/length),
the mass would have to be 16 times as much per unit length.
To do this, you would have to have 4 times the diameter.
dlow
dhigh
Mechanics Lecture 24, Slide 24
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:
.050”
.012”
dhigh
dlow
Mechanics Lecture 24, Slide 25
Different Example – 12 String Guitar
The lighter string in these pairs is one
octave higher (2x frequency).
Their diameters are about half of the
adjacent strings.
.028 .052
.022 .042
.014 .030
Mechanics Lecture 24, Slide 26
Homework Problem
v = L/t
T
v =
m
2
T =v m
2
m = m/L
Can we go over how changing the length
affects things. Specifically the slinky
problem on the homework last night.
http://danielwalsh.tumblr.com/post/1156601
6253/explaining-an-astonishing-slinky
Mechanics Lecture 24, Slide 27
Homework Problem
Average speed = distance / time
v= f
= 4A/P = 4Af
v
=
f
Mechanics Lecture 24, Slide 28
Homework Problem
Tnew = 2Told
mnew
Lnew
t=
vnew
1
= mold
2
Lnew = 2Lold
vnew
Tnew
=
mnew
2Told
Told
=
=2
1
mold
mold
2
= 2vold
Mechanics Lecture 24, Slide 29