feb 2008c.m. bali1 physics of sound grade11 with cro and signal generator c.m. bali, bsc (hons) msc...
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Feb 2008 C.M. Bali 1
PHYSICS of SOUND Grade11
With CRO and Signal Generator
C.M. Bali, BSc (Hons) MSc MEd TeacherDept of Mathematics and Science (Physics)Miles Macdonell Collegiate, Winnipeg
Feb 2008 C.M. Bali 2
Grade 11Physics Outcomes for Sound
• S3P-1-17: How sounds are produced, transmitted and detected • S3P-1-18: Analyze an environmental issue concerning sound• S3P-1-19: Design, construct, test, and demonstrate technological devices to
produce, transmit, and or control sound waves for useful purposes• S3P-1-20:Intereference of sound waves. • for resonance• S3P-1-22: Speed of sound in air• S3P-1-23: Effect of temperature and other factors (including materials) on the
speed of sound. • S3P-1-24: Doppler Effect (Qualitative treatment) • S3P-1-25: Decibel scale (Qualitative, example of sound)• S3P-1-26: Applications of sound waves (hearing aid (for Mr. Bali), ultrasound,
stethoscope, cochlear implants (also perhaps for Mr. Bali))• S3P-1-28:Examine the octave in a diatonic scale in terms of frequency
relationships and major triads.
Feb 2008 C.M. Bali 3
Cathode-ray Oscilloscope (CRO)or oscilloscope for short
Feb 2008 C.M. Bali 4
Sound of Music(You can even play the game ‘Guess that Tune’)
Some musical instruments and their characteristic sound
Click on the link below
Choose the instrument
Click play
• http://www.dsokids.com/2001/instrumentchart.htm
• As you play a tune, you should display the sound pattern on the CRO.
S3P1-17
Feb 2008 C.M. Bali 5
Experiments with CROand signal generator
1. Frequency of a tuning fork (Slide 9)2. Experiment to explore frequency, amplitude, pitch, intensity and
loudness of Sound (Slide15)3. Comparing loudness at different distances (Slide 18)4. Beat Frequency (Slide 21)5. Speed of sound using progressive waves (Slide 30)6. Speed of sound using stationary waves (Slides 42 and 43)7. Standing waves on strings. Speed of waves on strings (Slide 45)8. Quality (timbre) (Slide 47)9. How to show the velocity of a wave depends on tension and mass
per unit length of string. (Slide 49)10. Resonance (Slides 59)11. Resonance (Slide 62)12. Demonstration of phase and measurement of speed of sound (68)13. Variation of intensity of sound with distance (Question) (Slide 70)
Feb 2008 C.M. Bali 6
The function of oscilloscope
The basic function of an oscilloscope is to plot a graph of
Voltage versus time
Feb 2008 C.M. Bali 7
Voltage versus time
Voltage (Y)
1cm*5ms/cm =0.005 s
2cm*2v/cm = 2 volts
2cm*5ms/cm=10 ms =0.010 s = period
Time (X)
Y AMPLIFIER: 2V/cm TIMEBASE, t : 5ms/cm
Timebase
Ch 1Y amplifier
Feb 2008 C.M. Bali 8
SOUND:The maximum range of human hearing
includes sound of
Frequencies15-18,000 Hz
(Microsoft Encarta Encyclopedia. CD-ROM. 2000)
Feb 2008 C.M. Bali 9
Measurement of frequency of a tuning fork
Microphone PreamplifierStorage / Analogue
switch
Time base
To CH 1
S3P1-17S3p1-19
See next slide for details
Save All switch
Feb 2008 C.M. Bali 10
Measurement of the frequency of a tuning fork
1. Connect the apparatus as in the previous slide. All switches should be in the up position.
2. Turn oscilloscope on. Adjust the intensity and focus control.3. Timebase should be calibrated (Turn ‘Var Sweep’ clockwise).4. Strike the tuning fork and hold the prongs close to the mike.5. Turn the time base switch until a good, well spaced, fairly stable
sine signal is obtained. 6. Press Storage / analogue switch in. Freeze the trace by pressing
in the Save All button.
7. Measure peak to peak horizontal distance and multiply it by time base scale.
8. Frequency = 1 / period (Hz)
9. Before repeating, press out Storage/analogue.
Feb 2008 C.M. Bali 11
Tuning Fork Observations
T Fork Timebase Pk to Pk (x)
Period/sec Frequency/Hz
C126 2 ms/cm 3.9cm 7.8ms = 0.0078s 1/0.0078=128 Hz
D288 1 ms/cm 3.5cm 3.5 ms = 0.0035s 1/0.0035 =285 Hz
E320 1ms/cm 3.1cm 3.1 ms=0.0031s 1/0.0031= 322 Hz
Feb 2008 C.M. Bali 12
% Error% uncertainty
(I
% Error *100
For Fork C126 :
(128 126)% *100 1.6%
126
uncertainty in measured value% Uncertainty = *100
Measured ValueFor Fork C126 :
0.1cm%Uncertainty = *100 2.5%
3.9cm
Measured value Accpted valueAccpted value
ErrorHz
am assuming oscillscope controls are 100% accurate)
Feb 2008 C.M. Bali 13
Sound: Longitudinal Mechanical Waves
• Sound is a longitudinal wave, but they are drawn as transverse waves, to make the information clear. Sound is a mechanical wave as it requires a material medium such as air or metal or any other material for its transmission as well as a source of energy that disturbs the material medium.
• Click on the link below to see longitudinal and transverse waves. Pay attention to the source of disturbance and the material medium that transmits the disturbance..
• Focus on any one particle (a black dot) of gas to see its to and fro movement in the direction of the flow of the wave’s energy: in this picture sound wave progresses horizontally to the right.
• http://www.kettering.edu/~drussell/Demos/waves/wavemotion.html
Feb 2008 C.M. Bali 14
Sound Characteristics
Frequency
Amplitude
Pitch
Intensity
Loudness
Quality or Timbre
Feb 2008 C.M. Bali 15
Experiment to explorefrequency, amplitude, pitch, intensity and loudness of
Sound
Microphone Preamplifier
Time base
To CH 1Speaker
Signal Generator
Constant distance
Frequency dial
Amplitude dial
Frequency display
See next slide for details
D
S3P-1-19,1-27
Feb 2008 C.M. Bali 16
Frequency and Pitch
• Set up the experiment as in the previous slide (15).
• Turn on the oscilloscope and the signal generator.
• Investigate the pitch by listening to the sound from the speaker as the frequency is increased and then decreased. Keep amplitude (peak to peak distance on the oscilloscope) and D constant.
• Students should see that the higher the frequency the higher the pitch.
• (You can do this by using tuning forks of various frequencies, but it is difficult to keep the amplitudes constant)
S3P-1-27
Feb 2008 C.M. Bali 17
Intensity and loudness of Sound
• See slide # 10 for experimental set up.• Intensity = power per m2 (W m – 2)
• Power is proportional to the square of the amplitude of the wave. (Follows from SHM)
• Amplitude of the CRO trace is proportional to the amplitude of the sound wave hitting the microphone.
• Thus intensity of sound is proportional (not equal) to the square of the amplitude of the CRO trace. [CRO measures voltage]
• Loudness:
(in be
C
D
D C
D D10 10
C C
Suppose sound C has intensity = I
and sound D has intensity = I
If I >I , then D is louder than C.
Loudness of D relative to C is given by :
I IdB = 10* log ( decibells) ; B =log
I I)lls
A
2D D
10 102CC
Since intensity is proportinal to the square of the CRO trace amplitude,
A AdB = 10* log =20* log where A is amplitude
A
S3P-1-27
Feb 2008 C.M. Bali 18
Comparing loudness at different distances
• Set up the experiment as shown below (also see slide 10).• Place microphone 50 cm from the speaker and adjust the volume on the signal
generator to get a CRO trace of 1 cm amplitude.• Move the microphone by 20 cm at a time towards and away from the speaker.
Find the dB level of the sound relative to the reference point.
MicrophoneSpeaker
50 cm 100cm0 cm
Signal generator
Distance = 50 cm A (50) = 1. 0 cm Relative Loudness = 0
dist = 30 A (30) = dB = 20 * log {A(50) / A(30)} =
dist = 10 cm A(10) =
dist = 70 A(70)- =
dist = 90 cm A(90) =
S3P-1-27
Feb 2008 C.M. Bali 19
Double Beam OscilloscopeIt has two inputs CH1 CH2 and a common timebase
Therefore, you can display and compare
two signals (graphs) simultaneously.
Feb 2008 C.M. Bali 20
Beat Phenomenon
When two sound waves
of different frequencies
interfere
we hear sound
whose amplitude
varies
periodically.
Feb 2008 C.M. Bali 21
How to show beat phenomenon and measure beat frequency
Signal Gen 1
Signal Gen 2
To Red and Black terminals
To red and black terminals
To CH1To Ch 2
Speaker 1 Speaker 2
499Hz
510Hz
Timebase
Measure Beat period
Beat periodStore All switch
Storage / analogue switch
Bt freq = 1 / Bt period Ch1, Ch2, Dual, Add SWITCHCh1,Ch2, Dual, Add SWITCH
OutcomesSP3-1-17Sp3-1-19S3P-1-20
See slide 26 for details
Feb 2008 C.M. Bali 22
Measurement of beat frequency
When two sinusoidal functions of different frequencies are added
we get:
Time
Displacement
TbBeatB
1f =
T
Feb 2008 C.M. Bali 23
Beat Frequency FormulaIf two sinusoidal disturbances, y1 and y2, of frequencies, f1 and f2, are added,
the principle of superposition gives:
1 1 1 1
2 2 2 2
1 2 1 2
1 2 1 2
1 2
sin where 2
sin where 2
sin sin
2sin *sin2 2
2sin represents2
the slow varying amplitude of
α - β α+ βsin
2 2
y t f
y t f
Y y y t t
Y t t
Using : sinα+sinβ = 2sin
t
1 2
the enveloping curve (amplitude modulation) in the previous slide.
The envelope amplitude is zero when = n2
Therefore, sound has zero amplitude with an angular period of .
Therefore, time
t
beat1 2
1 2 1 2
,period, T for zero sound is given by 2
2 2,
21 1 2,
2 2 2
beat
beat
beatbeat
T
Therfore T
fTherefore f fT
Not for grade 11students but suitable for grade 12
Feb 2008 C.M. Bali 24
Beat Frequency Formula: A simple approach using relative speed
• Imagine two athletes, A and B, running to and fro on a straight track, a total distance of 2L.• Suppose A takes x seconds and B takes y seconds to complete one to and fro distance on the track .• Therefore, x and y are the periods of A and B• Frequency of A =1/x, and of B= 1/y• Let us assume they will be running for some time!• We want to know how often they meet or cross each other. To be more precise, how many times per
second they will cross each other; that is, what is their beat frequency?
m
m
2LSpeed of A = /sec.
x2L
Speed of B = /sec.y
Consider B to be at rest, at the start line, and A running relative to B.
2L 2L y - xSpeed of A relative to B = - =2L
x y xy
Time for A to go round once, and meet
BA
B again2L xy
=Beat period = =y - xy - x2L
xy
1 1 y - x y xFrequency with which this occurs = = = = -
xyBeat Period xy xy xyy - x
1 1⇒ - = f - f
x y
Students should be able to do this proof
SP3-1-20
Feb 2008 C.M. Bali 25
Beat Frequency Formula: A simple approach using relative angular speed (in degrees)
• Imagine two athletes, A and B, running around a circular track.• Suppose A takes x seconds and B takes y seconds to go round the track once.• Therefore, x and y are the periods of A and B• Frequency of A =1/x, and of B= 1/y• Let us assume they will be running for some time!• We want to know how often they cross each other. To be more precise, how many times per second they will cross
each other; that is, what is their beat frequency?• You can go through this argument using radians, and will get the same result.
0
0
0 0
degrees
degrees
360The Angular speed of A = /sec.x
360The angular speed of B = /sec.
yConsider B to be at rest, at the start line, and A runs relative to B.
360 360The angular speed of A relative to B = -
x y
A B
y - x=360
xy
Time for A to go roumd once, and meet B again360 xy=Beat period = =
y - xy - x360xy
1 1 y - x y xFrequency with which this occurs = = = = -
xyBeat Period xy xy xyy - x
1 1⇒ - = f - f
x y
Students should be able to do this proof
Feb 2008 C.M. Bali 26
Measurement of Beat frequencyExperimental Procedure
1. Set up the experiment as in slide 21.
2. Set Signal 1 to 490 Hz, and Signal 2 to 509 Hz. You should hear beating sound.
3. Set Oscilloscope to: Dual, time base to 20ms/Div, Storage/ analogue switch in, and Y1 and Y2 to an appropriate scale. You should see two traces somewhat out of phase.
4. Now press the Add switch.
5. With the storage / analogue switch in, Press Store All switch.
6. This should freeze the trace.
7. Measure Beat period.
8. Calculate Beat frequency.
Feb 2008 C.M. Bali 27
Beat Frequency Observations
Speaker 1
Freq / Hz
Speaker 2
Freq / Hz
Time base Beat period
Beat period /seconds
Beat frequency/ Hz
510 514 50 ms/ cm
5.6 cm 0.050*5.6=0.280 s
1/0.280=
3.57= 4Hz
201This was a good graph
210 20 ms/cm
5.4cm 0.108 s 9.25 Hz
1010This was a good graph
900.5 2 ms/ Div
4.5cm 0.090 s 11 Hz
Feb 2008 C.M. Bali 28
Sound as Progressive wave
A simple argument for students
• Progressive waves carry energy from one point to another.• As the wave particle oscillates once, the wave moves forward by
one wavelength.• If the particles oscillates f times per second, the wave will move
forward f wavelengths.• Thus:
*f
*f
Feb 2008 C.M. Bali 29
Speed = frequency * wavelengthA more rigorous derivation
y= a sin(ωt - kx) where y = displacement of wave, a= amplitude(ωt - kx) is called the phase of the wave.ω= 2πf
2πk =λ
Suppose in time Δt, the wave progresses by a a horizontal distance of Δx.This means that the of the wave is the same at t and at t+ Δt.This means the is the same at these two times.Thus, (ωt - kx)= (ω(t+ Δt)- k(x+ Δx)
(ωt - kx)=(ωt+kx+ωΔt - kΔx) ωΔt - kΔx= 0 kΔx
disp
= ωΔt
lacement
velocity
ase
=
ph
Δx ω 2πf= = =2πΔt k
v
λ
= fλ
Not for students
Feb 2008 C.M. Bali 30
Measurement of speed of soundusing progressive waves
A B C
E
F
0cm100 cm
A = Signal generatorB = SpeakerC = Mini MicrophoneD = Block of wood for supportE = PreamplifierF = Mitre rule
D
Note this reading when both
signals are in Phase
Move D until both signals are in phase again.
Note this point
This distance, d = One wavelength
Timebase
Frequency can be readDirectly from Signal generatorOr from the oscilloscope trace
To Ch1 input To Ch2
Double beam
S3P-1-17, 1-19,1-22
See slide 32 for details
Dual mode
Feb 2008 C.M. Bali 31
Measurement of speed of soundUsing progressive waves
1. Set up the experiment as shown on the previous slide
2. We know from basic experiments that speed of sound is about 340 m / s.
3. We want to measure a wavelength of say 20 cm. This informs us that the frequency should be around 340 / 0.20 = 1700 Hz.
4. We will use the frequency of sound at 1.5 KHz to 3.0 kHz.
Sig. Gen freq Freq from CRO Wavelength Speed using CRO freq
% uncertainty
1470 Hz (34.0 -10.0) =2 4. 0 cm 0.24*1470 = 3 50 m / s About 5%
1960 Hz (36-0)/2 = 18cm 0.18*1960 = 352 m / s See next two slides
2439 Hz 29.0-14.5 = 14.5 cm 0.145 * 2439 = 353 m / s
2940 Hz 23-11.4 = 11.6 cm 340 m/sec
Feb 2008 C.M. Bali 32
Errors and uncertainties
Students can calculate
errors and uncertainties in the beat frequencies
from data in previous slide
Feb 2008 C.M. Bali 33
How to calculate uncertainties
v =fλ1 1 1f = = , ∴v = d
Period τx τxx = horizontal peak to peak distance on CRO screenτ= time base setting d = distance( peak to peak) moved by microphone=wavelength
δv δx δd∴ = +v x d
Feb 2008 C.M. Bali 34
Calculating Uncertainties• Sig. Gen freq Frequency from CRO Wavelength Speed using CRO freq• 1500 Hz 1470 Hz (34.0 -10.0) =2 4. 0 cm 0.24*1470 = 3 50 m / s
δv δx δd= +
v x dx = 3.4 ± 0.1cm (CRO Pk - Pk)
d = 24.0 ± 0.5cm (Microphone Pk - Pk)
δv 0.1 0.5= + = 0.05 = ±5%
v 3.4 24.0
Feb 2008 C.M. Bali 35
Theory of UncertaintiesErrors always add, hence absolute signs.
a b
c
a-1 b b-1 a a b
c c c+1
where x, y and z are indepenedent variables,
and f is the dependent variable, i.e. it is a function of x, y and z.
x yLet f =z
f f dfδf = δx + δy + δzx y z
ax y by x -cx yδf = + +z z z
δf δyδx δz= a +b± +cx yf
∂ ∂⇒∂ ∂ ∂
⇒
⇒ z
Thus, fractional uncertainty in f = fractional uncertainty in x* exponent value of x +...
δx is called the fractional uncertainty in x.
x
Not for students
Feb 2008 C.M. Bali 36
x ±yUncertainty in f =
p ±q
( )
( )
m mx+ y = x+ y; And e
x ym rrors always add
Let m x y
n p q
m x y
Similarly n p q
mf
nf m n
f m n
m nf f
m n
Feb 2008 C.M. Bali 37
Uncertainties
A measurement can never be perfect.
The minimum error in a measurement is equal to the instrumental error.
If the variation in the measurement is much bigger than the instrumental error, then ignore the instrumental error, but use the deviation or standard deviation from the mean as a measure of uncertainty in the mean.
Feb 2008 C.M. Bali 38
Example
1/ 2
1/ 2
1 12 2
12
Tv
Tv
v Tv T
Tv v
T
Feb 2008 C.M. Bali 39
Conditions for setting up
Stationary Waves
When two progressive wavesof the same frequency and wavelength
And traveling in opposite directions And
interfere then, at certain wavelengths,
that satisfy the boundary conditions,
Stationary waves are formed.
Feb 2008 C.M. Bali 40
Mathematics of Stationary Waves
1
2
1 2( where Y =displacement of resultant wave)
y =asin( ωt-kx): Wave moving to the right.
y =asin( ωt+kx): Wave moving to the left.LetY =y +yY =asinωtcoskx-acosωtsinkx+asinωtcoskx+acosωtsinkxY =2asinωt.coskxY =[2acoskx] sinωt
This equation shows that the maximum amplitude is 2a.And , net dispalcement, Y, is a function of distance, x. When x is such that cos kx is zero, t
( Where 2a coskx represents amplitude of Y)
hen the amplitude is zero, i.e. .When x is such that cos kx is 1, then amplitude is maximum( 2a) ,
it is a node pointit is an antinode p i.e. oint.
Not for grade 11 students
Feb 2008 C.M. Bali 41
Standing wavesFrom the University of Toronto
http://www.upscale.utoronto.ca/GeneralInterest/Harrison/Flash/#sound_waves
• http://faraday.physics.utoronto.ca/IYearLab/Intros/StandingWaves/Flash/reflect.html
• http://faraday.physics.utoronto.ca/IYearLab/Intros/StandingWaves/Flash/standwave.html
• http://faraday.physics.utoronto.ca/IYearLab/Intros/StandingWaves/Flash/sta2fix.html
• http://faraday.physics.utoronto.ca/IYearLab/Intros/StandingWaves/Flash/sta1fix.html
Feb 2008 C.M. Bali 42
Speed of soundusing stationary waves
Speaker
Signal Generator Amplitude dial
See next slide for details
S3P-1-19,1-27
Shiny reflector
MicrophoneDual Mode
Speaker signal
Microphone signal (White dotted line): almost flat showing a node position at microphone
CH 1
CH 2
Feb 2008 C.M. Bali 43
Speed of sound using Stationary Waves
1. Set up the experiment as in the previous slide (slide 42)2. Move the screen until the microphone signal is very small (see slide 45 for analogy).
3. Now keep the screen still, and move the microphone gently.
4. Measure the distance between two nodal points: this is equal to the distance moved by the microphone between two successive points where the CRO signal is very small, while the reflector is kept at the same position. This distance is equal to half the wavelength. Thus, the wavelength is twice this distance.
5. Find speed of sound by multiplying the wavelength by the frequency, which can be seen on the signal generator.
6. Find uncertainty in the speed.
7. Find error in the speed.
Distance Wavelength Frequency Speed uncertainty Error
Feb 2008 C.M. Bali 44
Natural Frequencies and Resonance
All objects have certain natural frequencies of oscillations. For example, a swing can oscillate back and forth at a certain frequency that depends on its dimensions. If you push this swing gently but periodically, such that the frequency of your push is exactly the same as the natural frequency of the swing, then energy can flow from the your body into the swing. The energy in the swing can build up to the extent that the swing may become very unstable (very high amplitude of vibration) and even break !! Three dimensional objects have many natural frequencies of vibrations.
Show: Tacoma Narrows Bridge Falls
http://www.boreme.com/boreme/funny-2007/tacoma-bridge-p1.php
The phenomenon in which energy can pass easily from one object (A) into an other (B) if it is fed at the natural frequency of the other (B)
object is called resonance.
SEE NEXT SLIDE:
Feb 2008 C.M. Bali 45
How to set up a stationary (resonating) wave on a string
B
AD
E
A: Signal GeneratorB: VibratorC; StringD PulleyE: Weight to provide tension in string
Table
C
1st Harmonic
2nd Harmonic
3rd Harmonic
4th Harmonic
1:
2
2
n L for nth harmonic
L
n
Vibrates with small amplitude
Large amplitude of vibration
S3P-1-21
AN ANN N N
AN
Feb 2008 C.M. Bali 46
Harmonics and Overtones• See Previous Slide 45 for experimental set up.
1. Look at the frequency on the signal generator as you set up stationary waves on the string.
2. The frequency is lowest when first harmonic is set up. This frequency is called the fundamental frequency.
3. As the number of harmonics increases, frequency increases. (note: the nth harmonic has n half waves.)
4. OVERTONES: When an instrument (e.g. a taut wire as in the experiment) is plucked, it produces a sound which is a combination of many harmonics.
5. Overtones can be displayed on the CRO and analyzed and
shown to be composed of harmonics of that taut wire.
6. The first overtone is the first harmonic, the second overtone is the second harmonics, and so on.
S3P-1-27
Feb 2008 C.M. Bali 47
Quality or Timbre of Sound
• When an instrument is played it does not produce a note of just one frequency. It consists of the fundamental frequency and higher harmonics with different amplitudes. The resultant wave is the superposed sum of various harmonics. Thus the resultant wave is different from that of the fundamental frequency. The ear responds to the fundamental frequency, but it also senses the presence of higher harmonics. This sensation due to the presence of many harmonics is called the quality of sound. Thus different instruments, and even the same instrument, generate sounds of different quality, even though they may be playing the same fundamental frequency.
Play a tune from slide 4
http://www.dsokids.com/2001/instrumentchart.htm and observe the trace on the CRO
Feb 2008 C.M. Bali 48
Analysis of Quality or Timbre of SoundWith a graphing calculator
Use TI83 or the link below for a graphing calculator:
• Without Fourier analyzing, we can see that a periodic non-sinusoidal function is the sum
of many sinusoidal functions. Graphing Calculator:
http://my.hrw.com/math06_07/nsmedia/tools/Graph_Calculator/graphCalc.html
• Set window to : x min (-5), x max (5), y min (-2), y max (2); Xscl and Yscl =0.1• Mode Radians.
• Example: draw a graph of: Y = 1sin (x) + 0.75 sin (2x) + 0 sin (3x) +0.20 sin (4x): You should see that Y is periodic but non-sinusoidal, the period being different from the periods of the component harmonics,(also draw individual graphs)
• Thus quality is due the presence of many harmonics.
Feb 2008 C.M. Bali 49
Experiment The speed of a wave on a string
• Set up the experiment as in slide 45. Record resonant frequency for the first, second and third harmonic. Enter the observations in the table below. Find average speed.
Different students can try same material of different thicknesses and tensions.
Precautions: Keep tension constant. Check L has not changed.
Length of string,
L / m
N = Number of loops, or harmonics
wavelength =
2L/(N) / m
Freq, f / Hz (from Sig. Gen)
v = freq * wavelength
(m / s)
1
2
3
4
Feb 2008 C.M. Bali 50
An Experiment for IB or AP Students
Problem: • The speed of a wave on a string is thought to depend on the tension
in the string. Investigate the relationship between speed and tension.
Idea:
Let us suppose that the speed depends on the tension.
We don’t know how it depends.
[Theory must always precede experiment, even if we don’t realize it.]
: ;T
v T tension in string mass per unit length of string
nv kTSo assume – requires some thought!
Feb 2008 C.M. Bali 51
Theory and Analysis of data
n
1/2
ln(v) vs. ln(T
v = kT : k is a constant.Take logs to base e :
ln(v)= ln(k)+n ln(T)Therefore a graph of should be a straight line, with slope = n.
1If n is , then the relationship is T = kT
2
)
= k T
Feb 2008 C.M. Bali 52
Experimental details
• Set up the experiment as shown in slide 45.• Conduct the experiment in second harmonic.• Keep the length of the string constant.• Apply different weights to vary the tension.• For each weight, note f on signal generator. The wavelength is L.• Therefore the speed of the wave is f * L.
• Record the observations in the table below:
Feb 2008 C.M. Bali 53
DataMass / kg
Weight (mg) / N
Wavelength / m
Frequency / Hz
Speed
m /s
1.300
1.400
1.500
1.400
1.500
Feb 2008 C.M. Bali 54
Graph of ln(V) vs. ln(T)
ln (T)
ln (v)Gradient = 0.5
O
ln (K)
Feb 2008 C.M. Bali 55
• Discussion
• Conclusion
At a more simple level,
students can plot Velocity versus Sqrt of T.
A straight line of slope 1will uphold the relationship between velocity and tension.
Feb 2008 C.M. Bali 56
Velocity and mass per unit length
: We need to discover the value of n.
If n turns out to be be , then v can be said to be equal to .
Take logs to base e:
ln ln ln
Therefore a graph of vs. should be a straight linlnv
1 k-
2
ln e o
nv k
v k n
slope n
y intercept
f
and = ln k. ln v
ln
Slope should be – 1/2
Feb 2008 C.M. Bali 57
Experimental details
• Set up the experiment as in slide 45.• You should use wires of the same material but different thicknesses. • Use the same length of wire, keep tension the same. Do experiment in 2nd harmonic
to keep calculations easy – this makes wavelength = L• Plot ln . lnv vs
L / m Mass/m Tension
Constant
f / Hz2nd Harmonic
V = f * L ln V ln (mass /m)
Feb 2008 C.M. Bali 58
Analysis
• Graph
• Calculations of slope etc
• Discussion
• Conclusion.
Feb 2008 C.M. Bali 59
ResonanceMeasurement of speed of sound using a resonance tube
Demonstration of nodes and antinodes with sound
RB
FA
Speaker
OscilloscopePreamp
Mike
Tube
Signal generator
Freq dial
Amplitude dial
L = wavelength / 4
Metre ruleTo Ch 1 or 2
Shows Large signalAt Resonance
To show node at the bottom gently lower the microphone to the bottom of the cylinder
SP3-1-20, 1-21, 1-22
Feb 2008 C.M. Bali 60
Data
1
44.5
( ) 186
144.5
444.5*4 178 1.78
( ) 331
L cm
f R Hz
cm m
v f R ms
Feb 2008 C.M. Bali 61
Discussion of results:Note: Accepted value=332+0.6T/degC
• So at room temp, v =332+.6*20=344 m / s• End Correction: depends on the tube diameter• Measurement techniques, including parallax
Include % error
Conclusion:
Feb 2008 C.M. Bali 62
ResonanceMeasurement of speed of sound using a resonance tube
Demonstration of nodes and antinodes with sound
RB
FA
Speaker
OscilloscopePreamp
Mike
Tube
Signal generator
Freq dial
Amplitude dial
Metre ruleTo Ch 1 or 2
Shows Large signalAt Resonance
2L n
n = number of loops or harmonics, here n = 1
Tube open at both endsSP3-1-20, 1-21, 1-22
Feb 2008 C.M. Bali 63
Resonance Tube (both ends open) Observations
First resonant frequency =359 Hz
Length of tube = 45.5 cm1
45.52
91 0.91
359
327 /
2 0.52%
359 45.5
L cm
cm m
f Hz
v f m s
v fv f
LL
v f Lv f L
Feb 2008 C.M. Bali 64
Phase relationships and oscilloscope
Consider two functions (or voltage signals) y (t) and x (t) where:
( ) sin( )
( ) sin( ) sin cos cos sin
x t a t
y t a t a t a t
phase difference
Feb 2008 C.M. Bali 65
The principle of superposition gives:
Case 1: Phase Difference0
sin sin
2 sin
R x y a t a t
a t
,
( sin )
Alsoy xboth t
(1)
(2)
R=Resultant displacement as a function of time
A relation independent of time, or true at all times Y-axis
X-axis
Feb 2008 C.M. Bali 66
Case 2 : Phase Difference0 )( 180or
( ) sin( )
( ) sin( ) sin cos cos sin
x t a t
y t a t a t a t
cos180 = -1, and sin180 = 0
Therefore, ( ) sin( ) siny t a t a t
( sin ) ( sin ) 0x y a t a t
( )Andy x at all times Lissajous
relationX-axis /cm
Y-axis
Timebase off, X-Y in
Time
Y cm
Feb 2008 C.M. Bali 67
Case 3 : Phase Difference0 )
2( 90or
( ) sin( )
( ) sin( ) sin cos cos sin
x t a t
y t a t a t a t
cos 90 = 0, and sin 90 = 1
Therefore, ( ) sin( ) cosy t a t a t ( sin ) ( cos ) (sin cos )x y a t a t a t t
2 2 2 2 2 2 2
2 2 2
cos
:
sin( )x y t a t a
y a circle of radius a
Anda
at all timesx
Lissajous relation X-axis /cm
Y-axis
Timebase off, X-Y in
Feb 2008 C.M. Bali 68
How to demonstrate change in phaseSpeed of sound using this method.
A B C
E
F0cm
A = Signal generatorB = SpeakerC = MicrophoneD = Block of wood for supportE = PreamplifierF = Mitre rule
D
Timebase
To Ch1 input To Ch2
Double beam
Move the microphone , with the X-Y switch pushed inThe trace should change: you should see straight line (+ & - slope) and circleWavelength = distance moved by the microphone for the phase to change by 2pi radians.
X-Y switch Pressed IN
Feb 2008 C.M. Bali 69
0 X-axis
Y-axis
With time base off,And X -Y switch in
Sine function with double amplitude
X-Y switch out, timebase on Lissajous Figure
Add Switch on
Sine function with Zero amplitude
2
Feb 2008 C.M. Bali 70
QuestionHow would you investigate the variation of sound intensity
with distance?
Signal Gen
Spe
aker
1
Timebase
Store All switch
Storage / analogue switch
Ch1
Pre AmplifierMicrophoneDistance = x
Feb 2008 C.M. Bali 71
Doppler Effect
Use internet sources to demonstrate this effect
sound observerobserved '
sound source of sound( car)'
original fromsource of sound( car)
v ±vf = :
λuse positive if observer approaching the source of sound,
negative if receeding
v ±vλ =
f
Use negative if source of sound moving towards the observer,
use positive if moving away from the observer.
S3P-1-24
Feb 2008 C.M. Bali 72
End
C.M. Bali, BSc (Hons) MSc MEd CPhys MInstPTeacher
Dept of Mathematics and Science (Physics)Miles Macdonell Collegiate,
Winnipeg, Canada
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