Transcript Physics 207: Lecture 2 Notes

Lecture 30
To do :
• Chapter 21
Examine two wave superposition (-wt and +wt)
Examine two wave superposition (-w1t and -w2t)
Review for final (Location: CHEM 1351, 7:45 am )
Tomorrow: Review session, 2103 CH at 12:05 PM
• Last Assignment
 HW13, Due Friday, May 7th , 11:59 PM
Physics 207: Lecture 30, Pg 1
Standing waves
 Waves traveling in opposite direction “interfere” with each other.
If the conditions are right, same k (2p/l) & w, the
superposition generates a standing wave:
DRight(x,t)= a sin( kx-wt ) DLeft(x,t)= a sin( kx+wt )
Energy flow in a standing wave is stationary, it “stands” in place.
Standing waves have nodes and antinodes
D(x,t)= DL(x,t) + DR(x,t)
D(x,t)= 2 a sin(kx) cos(wt)
The outer curve is the
amplitude function
Anti-nodes
A(x) = ±2 a sin(kx)
when wt = pn n = 0,1,2,…
k = wave number = 2π/λ
Nodes
Physics 207: Lecture 30, Pg 2
Standing waves on a string
 Longest wavelength allowed:
½ of the full wave
Fundamental: l/2 = L  l = 2 L
v
2
L
lm 

m
fm
m  1,2,3,...
Recall v = f l
v
fm  m
2L
Overtones m > 1
Physics 207: Lecture 30, Pg 3
Vibrating Strings- Superposition Principle
 Violin, viola, cello, string bass
D(x,0)
 Guitars
 Ukuleles
 Mandolins
Antinode D(0,t)
 Banjos
Physics 207: Lecture 30, Pg 4
Standing waves in a pipe
Open end: Must be a displacement antinode (pressure minimum)
Closed end: Must be a displacement node (pressure maximum)
Blue curves are displacement oscillations. Red curves, pressure.
Fundamental:
l/2
l/2
l/4
Physics 207: Lecture 30, Pg 5
Standing waves in a pipe
lm  2 L
lm  2 L
lm  4 L
fm  m v
2L
m  1,2,3,...
fm  m v
2L
m  1,2,3,...
fm  m v
4L
m  1,3,5,...
m
m
m
Physics 207: Lecture 30, Pg 6
Combining Waves
Fourier Synthesis
Physics 207: Lecture 30, Pg 7
Superposition & Interference
 Consider two harmonic waves A and B meet at t=0.
 They have same amplitudes and phase, but
w2 = 1.15 x w1.
 The displacement versus time for each is shown below:
A(w1t)
B(w2t)
C(t) = A(t) + B(t)
DESTRUCTIVE
INTERFERENCE
CONSTRUCTIVE
INTERFERENCE
Physics 207: Lecture 30, Pg 8
Superposition & Interference
 Consider A + B, [Recall cos u + cos v = 2 cos((u-v)/2) cos((u+v)/2)]
yA(x,t)=A cos(k1x–2p f1t)
yB(x,t)=A cos(k2x–2p f2t)
Let x=0, y=yA+yB = 2A cos[2p (f1 – f2) t/2] cos[2p (f1 + f2) t/2]
and |f1 – f2| ≡ fbeat = = 1 / Tbeat
f average≡ (f1 + f2)/2
A(w1t)
B(w2t)
t
C(t) = A(t) + B(t)
Tbeat
Physics 207: Lecture 30, Pg 9
Exercise Superposition


The traces below show beats that occur when two different
pairs of waves are added (the time axes are the same).
For which of the two is the difference in frequency of the
original waves greater?
A. Pair 1
B. Pair 2
C. The frequency difference was the same for both pairs of waves.
D. Need more information.
Physics 207: Lecture 30, Pg 10
Superposition & Interference
 Consider A + B, [Recall cos u + cos v = 2 cos((u-v)/2) cos((u+v)/2)]
yA(x,t)=A cos(k1x–2p f1t)
Let x=0,
yB(x,t)=A cos(k2x–2p f2t)
y = yA + yB = 2A cos[2p (f1 – f2)t/2] cos[2p (f1 + f2)t/2]
and
|f1 – f2| ≡ fbeat = = 1 / Tbeat
f average≡ (f1 + f2)/2
A(w1t)
B(w2t)
t
C(t) = A(t) + B(t)
Tbeat
Physics 207: Lecture 30, Pg 11
Review
 Final is “semi” cumulative
 Early material, more qualitative (i.e., conceptual)
 Later material, more quantitative (but will employ major
results from early on).
 25-30% will be multiple choice
 Remainder will be short answer with the focus on
thermodynamics, heat engines, wave motion and wave
superposition
Physics 207: Lecture 30, Pg 12
Exercise Superposition


The traces below show beats that occur when two different
pairs of waves are added (the time axes are the same).
For which of the two is the difference in frequency of the
original waves greater?
A. Pair 1
B. Pair 2
C. The frequency difference was the same for both pairs of waves.
D. Need more information.
Physics 207: Lecture 30, Pg 13
Organ Pipe Example
A 0.9 m organ pipe (open at both ends) is measured to have it’s
first harmonic (i.e., its fundamental) at a frequency of 382 Hz.
What is the speed of sound (refers to energy transfer) in this
pipe?
L=0.9 m
f = 382 Hz and f l = v with l = 2 L / m (m = 1)
v = 382 x 2(0.9) m  v = 687 m/s
Physics 207: Lecture 30, Pg 14
Standing Wave Question
 What happens to the fundamental frequency of a pipe, if
the air (v =300 m/s) is replaced by helium (v = 900 m/s)?
Recall: f l = v
(A) Increases
(B) Same
(C) Decreases
Physics 207: Lecture 30, Pg 15
Chapter 1
Physics 207: Lecture 30, Pg 16
Important Concepts
Physics 207: Lecture 30, Pg 17
Chapter 2
Physics 207: Lecture 30, Pg 18
Chapter 3
Physics 207: Lecture 30, Pg 19
Chapter 4
Physics 207: Lecture 30, Pg 20
Chapter 4
Physics 207: Lecture 30, Pg 21
Chapter 5
Physics 207: Lecture 30, Pg 22
Chapter 5 & 6
Physics 207: Lecture 30, Pg 23
Chapter 6
Chapter 7
Physics 207: Lecture 30, Pg 24
Chapter 7
Physics 207: Lecture 30, Pg 25
Chapter 7 (Newton’s 3rd Law) & Chapter 8
Physics 207: Lecture 30, Pg 26
Chapter 9
Chapter 8
Physics 207: Lecture 30, Pg 27
Chapter 9
Physics 207: Lecture 30, Pg 28
Chapter 10
Physics 207: Lecture 30, Pg 29
Chapter 10
Physics 207: Lecture 30, Pg 30
Chapter 10
Physics 207: Lecture 30, Pg 31
Chapter 11
Physics 207: Lecture 30, Pg 32
Chapter 11
Physics 207: Lecture 30, Pg 33
Chapter 12
and Center of Mass
Physics 207: Lecture 30, Pg 34
Chapter 12
Physics 207: Lecture 30, Pg 35
Important Concepts
Physics 207: Lecture 30, Pg 36
Chapter 12
Physics 207: Lecture 30, Pg 37
Angular Momentum
Physics 207: Lecture 30, Pg 38
Hooke’s Law Springs and a Restoring Force
 Key fact: w = (k / m)½ is general result where k reflects a
constant of the linear restoring force and m is the inertial response
(e.g., the “physical pendulum” where w = (k / I)½
Physics 207: Lecture 30, Pg 39
Simple Harmonic Motion
Maximum
potential
energy
Maximum
kinetic
energy
Physics 207: Lecture 30, Pg 40
Resonance and
damping
 Energy transfer is
optimal when the
driving force varies at
the resonant
frequency.
 Types of motion
 Undamped
 Underdamped
 Critically damped
 Overdamped
Physics 207: Lecture 30, Pg 41
Fluid Flow
Physics 207: Lecture 30, Pg 42
Density and pressure
Physics 207: Lecture 30, Pg 43
Response to forces
States of Matter and Phase Diagrams
Physics 207: Lecture 30, Pg 44
Ideal gas equation of state
Physics 207: Lecture 30, Pg 45
pV diagrams
Thermodynamics
Physics 207: Lecture 30, Pg 46
Work, Pressure, Volume, Heat
T can change!
Physics 207: Lecture 30, Pg 47
Chapter 18
Physics 207: Lecture 30, Pg 48
Thermal Energy
Physics 207: Lecture 30, Pg 49
Relationships
Physics 207: Lecture 30, Pg 50
Chapter 19
Physics 207: Lecture 30, Pg 51
Refrigerators
Physics 207: Lecture 30, Pg 52
Carnot Cycles
Physics 207: Lecture 30, Pg 53
Work (by the system)
Physics 207: Lecture 30, Pg 54
Chapter 20
Physics 207: Lecture 30, Pg 55
Displacement versus time and position
Physics 207: Lecture 30, Pg 56
Sinusoidal Waves (Sound and Electromagnetic)
Physics 207: Lecture 30, Pg 57
Doppler effect
Physics 207: Lecture 30, Pg 58
Chapter 21
Physics 207: Lecture 30, Pg 59
Standing Waves
Physics 207: Lecture 30, Pg 60
Beats
D(0, t )  2 A cos( 2p
f1  f 2
2
| f1  f 2 | f beat  1 / Tbeat
t ) cos( 2p
|
f1  f 2
2
f1  f 2
2
t)
| f avg  1 / Tavg
Physics 207: Lecture 30, Pg 61
Lecture 30
• Assignment
 HW13, Due Friday, May 7th
I hope everyone does well on their finals!
Have a great summer!
Physics 207: Lecture 30, Pg 62
Example Interference


A speaker sits on a pedestal 2.0 m tall and emits a sine
wave at 340 Hz (the speed of sound in air is 340 m/s, so l =
1.0 m ). Only the direct sound wave and that which reflects
off the ground at a position half-way between the speaker
and the person (also 2 m tall) makes it to the persons ear.
How close to the speaker can the person stand (A to D) so
they hear a maximum sound intensity assuming there is no
“phase change” at the ground (this is a bad assumption)?
t1
t0
d
A
t0
B
D
h
C
The distances AD and BCD have equal transit times so the
sound waves will be in phase. The only need is for AB = l
Physics 207: Lecture 30, Pg 63
Example Interference

The geometry dictates everything else.
AB = l
AD = BC+CD = BC + (h2 + (d/2)2)½ = d
AC = AB+BC = l +BC = (h2 + d/22)½
Eliminating BC gives
l+d = 2 (h2 + d2/4)½
l + 2ld + d2 = 4 h2 + d2
1 + 2d = 4 h2 / l  d = 2 h2 / l – ½
= 7.5 m
t1
t0
7.5
A
t0
B
3.25 C
4.25
D
Because the ground is more dense than air there will be a phase
change of p and so we really should set AB to l/2 or 0.5 m.
Physics 207: Lecture 30, Pg 64
Sample Problem
 The figure shows a snapshot graph D(x, t = 2 s) taken at t = 2 s
of a pulse traveling to the left along a string at a speed of 2.0
m/s. Draw the history graph D(x = −2 m, t) of the wave at the
position x = −2 m.
Physics 207: Lecture 30, Pg 65
Sample Problem
 History Graph:
2
-2
2
3
4
6
5
time (sec)
7
Physics 207: Lecture 30, Pg 66
Example problem
 Two loudspeakers are placed 1.8 m apart. They play tones of
equal frequency. If you stand 3.0 m in front of the speakers, and
exactly between them, you hear a maximum of intensity.
 As you walk parallel to the plane of the speakers, staying 3.0 m
away, the sound intensity decreases until reaching a minimum
when you are directly in front of one of the speakers. The speed
of sound in the room is 340 m/s.
a. What is the frequency of the sound?
b. Draw, as accurately as you can, a wave-front diagram. On your
diagram, label the positions of the two speakers, the point at
which the intensity is maximum, and the point at which the
intensity is minimum.
c. Use your wave-front diagram to explain why the intensity is a
minimum at a point 3.0 m directly in front of one of the speakers.
Physics 207: Lecture 30, Pg 67
Example problem
 Two loudspeakers are placed 1.8 m apart. They play tones of
equal frequency. If you stand 3.0 m in front of the speakers,
and exactly between them, you hear a maximum of intensity.
 As you walk parallel to the plane of the speakers, staying 3.0 m
away, the sound intensity decreases until reaching a minimum
when you are directly in front of one of the speakers. The
speed of sound in the room is 340 m/s.
Constructive
 What is the frequency of the sound?
Interference
v = f l but we don’t know f or l
in-phase
DRAW A PICTURE
Destructive
Interference
out-of-phase
Physics 207: Lecture 30, Pg 68
Example problem
 Two loudspeakers are placed 1.8 m apart. They play tones of
1.8 m
equal frequency. If you stand 3.0 m in front of the speakers,
and exactly between them, you hear a maximum of intensity.
 v = 340 m/s.
PUT IN GEOMETRY
v = f l but we don’t know f or l
Constructive
AC - BC = 0 (0 phase difference)
Interference
3.0
m
AD - BD = l/2 (p phase shift)
in-phase
A
AD = (3.02+1.82)1/2
BD = 3.0
C
l = 2(AD-BD) =1.0 m
Destructive B
Interference
out-of-phase
D
Physics 207: Lecture 30, Pg 69
Example problem
 Two loudspeakers are placed 1.8 m apart. They play tones of
equal frequency. If you stand 3.0 m in front of the speakers,
and exactly between them, you hear a maximum of intensity.
b. Draw, as accurately as you can, a wave-front diagram. On your
diagram, label the positions of the two speakers, the point at
which the intensity is maximum, and the point at which the
intensity is minimum.
c. Use your wave-front diagram to
explain why the intensity is a
minimum at a point 3.0 m
directly in front of one of the speakers.
Physics 207: Lecture 30, Pg 70
Sample problem
 A tube, open at both ends, is filled with an unknown gas. The
tube is 190 cm in length and 3.0 cm in diameter. By using
different tuning forks, it is found that resonances can be excited
at frequencies of 315 Hz, 420 Hz, and 525 Hz, and at no
frequencies in between these.
a. What is the speed of sound in this gas?
b. Can you determine the amplitude of the wave? If so, what is it?
If not, why not?
Physics 207: Lecture 30, Pg 71
Sample problem
 A tube, open at both ends, is filled with an unknown gas. The
tube is 190 cm in length and 3.0 cm in diameter. By using
different tuning forks, it is found that resonances can be excited
at frequencies of 315 Hz, 420 Hz, and 525 Hz, and at no
frequencies in between these.
What is the speed of sound in this gas?
L=1.9 meters and fm = vm/2L
315 = v m/2L
420 = v (m+1)/2L
525 = v (m+2)/2L
v/2L = 105
v = 3.8 x 105 m/s =400 m/s
b) Can you determine the amplitude of the wave? If so, what is it?
If not, why not?
Answer: No, the sound intensity is required and this is not known.
Physics 207: Lecture 30, Pg 72
Sample Problem
 The picture below shows two pulses approaching each other on
a stretched string at time t = 0 s. Both pulses have a speed of
1.0 m/s. Using the empty graph axes below the picture, draw a
picture of the string at t = 4 s.
Physics 207: Lecture 30, Pg 73
An example
 A heat engine uses 0.030 moles of helium as its working
substance. The gas follows the thermodynamic cycle shown.
a. Fill in the missing table entries
b. What is the thermal efficiency of this engine?
c. What is the maximum possible thermal efficiency of an engine
that operates between Tmax and Tmin?
B
AB
BC
CA
NET
QH
600 J
0J
0J
600 J
QL
0J
0J
Wby
0J
DETh
600 J
-161 J
-243 J
A
Physics 207: Lecture 30, Pg 74
C
An example
 A heat engine uses 0.030 moles of helium as its working
substance. The gas follows the thermodynamic cycle shown.
a. Fill in the missing table entries
b. What is the thermal efficiency of this engine?
c. What is the maximum possible thermal efficiency of an engine
that operates between Tmax and Tmin?
A: T= 400 K
B: T= 2000 K
B
C: T= 1050 K
AB
BC
CA
NET
QH
600 J
0J
0J
600 J
QL
0J
0J
Wby
0J
DETh
600 J
-161 J
-243 J
A
Physics 207: Lecture 30, Pg 75
C
The Full Cyclic Process
 A heat engine uses 0.030 moles of helium as its working
substance. The gas follows the thermodynamic cycle shown.
a. What is the thermal efficiency of this engine?
b. What is the maximum possible thermal efficiency of an engine
that operates between Tmax and Tmin?
(T = pV/nR, pVg =const.)
A: T= 400 K=1.01x105 10-3/ 0.030/8.3
B
B: T=2000 K
pBVBg = pC VCg
(pBVBg / pC )1/g = VC
(5 (10-3)5/3 / 1)3/5 = VC = 2.6 x 10-3 m3
C: T= 1050 K
A
C
Physics 207: Lecture 30, Pg 76
The Full Cyclic Process
A: T= 400 K=1.01x105 10-3/ 0.030/8.3
B: T=2000 K
Q = n Cv DT = 0.030 x 1.5 x 8.3 DT
VC = 2.6 x 10-3 m3
Q = 0.3735 DT
C: T= 1050 K
WCA (by) = p DV = 1.01x105 1.6 x 10-3 J
AB
BC
QH
600 J
0J
QL
0J
0J
Wby
0J
355 J
DETh
600 J
-355 J
CA
NET
0J
600 J
404 J
404 J
-161 J
194 J
-243 J
0J
B
h = Wby / QH = 194/ 600 = 0.32
hCarnot = 1- TL / TH= 1- 400/2000 =0.80
A
Physics 207: Lecture 30, Pg 77
C
An example
 A monatomic gas is compressed isothermally to 1/8 of its
original volume.
 Do each of the following quantities change? If so, does the
quantity increase or decrease, and by what factor? If not, why
not?
a. The rms speed vrms
b. The temperature
c. The mean free path
d. The molar heat capacity CV
Physics 207: Lecture 30, Pg 78
An example
 A small speaker is placed in front of a block of mass 4 kg. The
mass is attached to a Hooke’s Law spring with spring constant
100 N/m. The mass and speaker have a mechanical energy of
200 J and are undergoing one dimensional simple harmonic
motion. The speaker emits a 200 Hz tone.
 For a person is standing directly in front of the speaker, what
range of frequencies does he/she hear?
 The closest the speaker gets to the person is 1.0 m. By how
much does the sound intensity vary in terms of the ratio of the
loudest to the softest sounds?
Physics 207: Lecture 30, Pg 81
An example
 A small speaker is placed in front of a block of mass 4 kg. The
mass is attached to a Hooke’s Law spring with spring constant
100 N/m. The mass and speaker have a mechanical energy of
200 J and are undergoing one dimensional simple harmonic
motion. The speaker emits a 200 Hz tone. (vsound = 340 m/s)
 For a person is standing directly in front of the speaker, what
range of frequencies does he/she hear?
Emech=200 J = ½ mvmax2 = ½ kA2  vmax=10 m/s
Now use expression for Doppler shift where
vsource= + 10 m/s and – 10 m/s
f
f observer 
source
vs
1 v
Physics 207: Lecture 30, Pg 82
An example
 A small speaker is placed in front of a block of mass 4 kg. The
mass is attached to a Hooke’s Law spring with spring constant
100 N/m. The mass and speaker have a mechanical energy of
200 J and are undergoing one dimensional simple harmonic
motion. The speaker emits a 200 Hz tone. (vsound = 340 m/s)
 For a person is standing directly in front of the speaker, what
range of frequencies does he/she hear?
Emech=200 J = ½ mvmax2 = ½ kA2  vmax=10 m/s
Now use expression for Doppler shift where
vsource= + 10 m/s and – 10 m/s
f observer
f source

vs
1 v
Physics 207: Lecture 30, Pg 83
An example
 A small speaker is placed in front of a block of mass 4 kg. The
mass is attached to a Hooke’s Law spring with spring constant
100 N/m. The mass and speaker have a mechanical energy of
200 J and are undergoing one dimensional simple harmonic
motion. The speaker emits a 200 Hz tone. (vsound = 340 m/s)
 The closest the speaker gets to the person is 1.0 m. By how
much does the sound intensity vary in terms of the ratio of the
loudest to the softest sounds?
Emech=200 J = ½ mvmax2 = ½ kA2  A = 2.0 m
Distance varies from 1.0 m to 1.0+2A or 5 meters where
P is the power emitted
Ratio  Iloud/Isoft= (P/4p rloud2)/(P/4p rsoft2) = 52/12 = 25
Physics 207: Lecture 30, Pg 84
An example problem

In musical instruments the sound is based
on the number and relative strengths of the
harmonics including the fundamental
frequency of the note. Figure 1a depicts the
first three harmonics of a note. The sum of
the first two harmonics is shown in Fig. 1b,
and the sum of the first 3 harmonics is
shown in Fig. 1c.
Which of the waves shown has the shortest
period?
a. 1st Harmonic
b. 2nd Harmonic
c. 3rd Harmonic
d. Figure 1c
At the 2nd position(1st is at t=0) where the three
curves intersect in Fig. 1a, the curves are
all:
a. in phase
b. out of phase
c. at zero displacement
d. at maximum displacement
Physics 207: Lecture 30, Pg 85
An example problem
The frequency of the waveform shown in Fig.
1c is
a. the same as that of the fundamental
b. the same as that of the 2nd harmonic
c. the same as that of the 3rd harmonic
d. sum of the periods of the 1st, 2nd & 3rd
harmonics
Which of the following graphs most accurately
reflects the relative amplitudes of the
harmonics shown in Fig. 1?
Physics 207: Lecture 30, Pg 86
Get the beat…
 You are studying an unusual pair of sea creatures that use low
frequency sound to interact. Each animal produces a perfect
sine wave. Your oscilloscope output appears below. What are
the respective frequencies?
displacement (meters)
2
1
0
-1
-2
0
5
10
time (seconds)
15
20
Physics 207: Lecture 30, Pg 87
Get the beat…
 You are studying an unusual pair of sea creatures that use low
frequency sound to interact. Each animal produces a perfect
sine wave. Your oscilloscope output appears below. What are
the respective frequencies?
 | f1 – f2 |= 1/ Tbeat = 1/ 5
(f1 + f2 ) /2 =1/ Tavg = 1/1
 2 f1 = 2.2 Hz  f1 = 1.1 Hz so f2 = 0.9 Hz
displacement (meters)
2
1
0
-1
-2
0
5
10
time (seconds)
15
20
Physics 207: Lecture 30, Pg 88