Experiment 13 The sonometer

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Projects
Now for some choice!
The experiments in the Projects segment have been collected to demonstrate a variety of physical
principles and make use of a large suite of experimental measuring techniques and analytical
tools. No continuous theme connects the experiments; rather, they are self-contained. A simple,
but important physical principle or procedure forms the basis of each.
You will find that many of the projects are more open-ended than those in other segments. This
has been a conscious decision by those developing the laboratory course, with the aim of
developing your skills in experimental design and evaluation. Don’t feel as though you are
expected to meet the challenge without help - your demonstrators have been chosen for their
ability to encourage you in this process.
Organisation
In this laboratory you and your partner will make a selection from the available experiments.
The equipment is randomly arranged throughout the room so you will move from table to table
during the segment.
Upon your arrival in the laboratory, inspect all the apparatus supplied for your particular
experiment. See how you can optimally manage it to reduce errors during the data collection.
Rearrangement of the relevant equations (given as part of some experiments) invariably suggests
the most sensitive way of calculating the final quantity required.
Both demonstrators in the laboratory are available to help you improve your data collection and
analysis, but only one will be concerned with the marking of your book. Both partners should be
involved in the planning, execution and calculation phases of all experiments, but particularly so
in this laboratory. As in the other laboratories, your logbooks must be written independently.
Before you leave the laboratory each week, select a number of experiments (2 or 3) that you would
want to do the following week. Inform your demonstrator of your preferred option and s/he will
make a record of it in the Projects Booking Book. Make sure you remember your selection (write
it in your prac-manual, maybe) as this is the experiment you will be doing in the next week.
Where class sizes permit, a maximum of two students may do any one experiment in a given
week.
Preparation
The notes provided in this manual are not the pracs themselves—these will be provided to you in
the laboratory. They consist instead of a brief synopsis of the experiment and the prelab exercises
that you should complete before you arrive.
These prelab exercises are designed to assist you in understanding the experiment. In addition to
the assessment contribution, students attempting the exercises will be able to walk into the
laboratory with some pre-prepared questions for the demonstrator and will thus find themselves
with more time available to complete the prac. Remember, you will be give credit for an honest
attempt at the prelab exercises. They are designed to start your thinking.
If you require a copy of the experiments that you have completed you should inform your
demonstrator at the end of the laboratory session and s/he will make one available to you.
Physics 121/2 and 141/2 Laboratory Manual
P-1
P-2
Physics 121/2 and 141/2 Laboratory Manual
Experiment 1
Physics of the guitar
Note that you may not choose this experiment and Experiment 13, The Sonometer.
References
121/2: Section 17.2.
141/2: Sections 17.11 to 17.12; 18.6; 31.4 (Electric Guitars).
In this experiment you will investigate some of the physical and musical principles behind the
functioning and construction of the guitar. In order to do this you will need to learn some basic
music theory from the lab notes provided to you in the class.
In these notes we will pre-empt the experiment by examining the vibrations of a guitar string
from a purely geometrical viewpoint.
When a guitar string is plucked it vibrates in all of the modes available to it. Some of these modes
are prohibited by physical constraints and die out almost instantly. Other modes are strongly
favoured by the geometry etc and are sustained for a long time. These are the notes we hear when
a guitar string is plucked, and it turns out that these are the standing waves of the string.
The first (or fundamental) standing wave may be represented as:
n=1
L
 = 2L
You can see this as the amplitude envelope of the string as it vibrates: note that the ends must
have zero amplitude as they correspond to the fixed ends of the string. Also note that the whole
of the string oscillates in phase: it is not a travelling wave. We can thus identify by inspection the
wavelength of the fundamental mode as twice the length of the string, ie =2L.
Similarly, the second harmonic can be represented by:
n=2
=L
L
Prelab Exercise:
Draw the first four standing waves on a string of length L with n = 1,2,3,4. Write down their
wavelengths in terms of n and L. Hence derive a general expression for the wavelength of the nth
harmonic.
=
=
=
=
In general,=
Physics 121/2 and 141/2 Laboratory Manual
P-3
Experiment 2
The speed of sound
In this experiment you will measure the speed of sound using time of flight techniques. That is,
you will directly measure the time of propagation of a sound wave over a variety of distances to
determine the rate at which the sound wave travels. A spark generator will be used to produce a
burst of sound which will be detected by the microphone at some position S at some time later.
The amount of time later that the sound is detected will allow a measurement of the speed of
propagation of the sound wave. Schematically, this will occur as in the diagram below:
travelling
sound wave
spark
microphone
s?
s?
As you can see from the picture, there is some difficulty in determining S accurately. Since the
time we measure will indicate when the sound is detected by the microphone, it appears that we
need to know exactly where the sound is detected. Apart from the fact that this is a tricky
technical question, requiring much knowledge about the construction of this particular
microphone, it is also quite Zen: where is the microphone? Is it at the front or the rear of the piezo
crystal which creates the voltage output?
These objections may seem to be overstated, but they highlight a fundamental measuring
problem: sometimes it is technically impossible to measure a quantity precisely. In this case we
have an (unremovable) systematic error (look up Notes on Confidence Limits in Experimental
Physics if you are unsure what a systematic error is.) Nevertheless, it may be possible to obtain
an extremely good result by being clever about the experiment.
In order to see how this might be done it is necessary to look at the analysis of the experimental
data. In the above example we were going to find the speed of sound using v = S / t. We can
write the S that we measure, SM, as a combination of the ‘true’ S, ST, (which we mortals cannot
know) plus some extra amount S which represents the amount by which we were wrong. That
is, SM = ST + S. Then the result we would get from the above calculation would be vM = vT + S /
t. Thus our result would be fundamentally wrong. (Note that this result becomes more accurate
as we let S (and thus t) become large.)
The challenge facing us is to wonder whether we might find the true v, provided we do not have
to do too much work to get it. Part of the clue to doing this may be found in the above expression,
SM = ST + S. Notice that if we move the microphone to a second position, S2M will relate to S2T
with the same S. That is, S2M = S2T +S. From this we might notice that the difference between
these two measurements, S, is independent of S. That is;
S = S2M
–
S1M
=
(S2T +S) – (S1T +S) = S2T – S1T
So, while our measurements of the distances are clearly incorrect, the difference between them
remains absolutely correct. Thus, if our analysis were to rely only on the change of distance, we
might expect it to give the correct result. This can be done by rethinking our analysis of the
dS
velocity of sound: let us now use v 
.
dt
Of course, this velocity is the gradient of an S vs t graph. Let us look at our graph, to see what is
going on…
P-4
Physics 121/2 and 141/2 Laboratory Manual
SM = ST + S
S
( t , SM )
ST
S
t
It can be seen from this view of the data that an analysis based on the gradient will be
independent of the systematic error S. In fact, there is no longer any good reason to minimise
the systematic error, S: the value of v obtained will be ‘good’ however large S.
Returning to the ‘old’ way of getting v, ie v = S / t, we can interpret this calculation as the
gradient of a line from the origin to any point on the SM line. Thus we can see that this result has
little to do with the speed of sound but that it will return a more accurate value of v if the
measured data point is at (large S, large t).
Prelab Question:
Will the result be similarly affected by a systematic error in t? (In order to answer this
conclusively you may need to go through the algebra, as above.)
Experiment 3
Microwave optics
References
121/2: Section 39.1, 40.1 and 37.4.
141/2: Section 17.11, 37.3 to 37.4 and 34.6.
The electromagnetic spectrum consists of all possible electromagnetic waves satisfying v = c = f .
Thus, visible light is an electromagnetic wave with properties that are reasonably familiar to us:
reflection, refraction, diffraction etc. The microwaves you will use have a frequency of
approximately 1010 Hz, which is about 1/100,000th that of visible light.
Prelab Question:
What is the approximate wavelength of the microwaves that you will use?
Physics 121/2 and 141/2 Laboratory Manual
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As a consequence of the macroscopic wavelength of these microwaves you will be able to directly
measure the electric field component of the microwaves, and you will be able to investigate and
interpret light-like phenomena on a macroscopic scale.
In particular, you will directly measure and interpret the polarisation of electromagnetic
radiation (please refer to your textbook if you are not familiar with polarisation). You will also set
up a standing microwave field to measure the wavelength of the microwaves. Further
experiments examining the diffraction of microwaves and the reflection, transmission and
absorption of microwaves by various materials will also be available.
Experiment 4
Equipotentials and field lines
References
121/2: Sections 23.3 and 25.4.
141/2: Sections 23.3 and 25.3.
In this experiment you will study the field which surrounds an electric charge. This electric field,
E, has very similar properties to that of the gravitational field, g, differing only in that:

electric charges, q, can be either positive or negative, whereas gravitational ‘charges’, m,
are always positive

opposite electric charges attract and like gravitational charges attract.

The ‘strength’ of the electric field is far greater than that of the gravitational field.
These differences, however small, produce the world we see. This world is generally populated by
electrically neutral objects (down to the atomic scale) due to the attraction of unlike charges. The
microscopic size of this scale is testament to the greater inherent strength of the electric force.
Gravitationally we see the existence of large clumps of matter (planets, galaxies, etc). If the
gravitational force were comparatively as strong as the electric force we would feel significant
gravitational attraction to small objects, which we do not.
The upshot of these properties, comparatively speaking, is that we can, with the aid of a voltage
supply, observe the local (near the field source) and global (far away from the field source)
properties of the electric field with a relatively small experiment. To understand the physics of
the electric field it can be useful to exploit the similarity with the gravitational field. As a
preparatory exercise, consider the gravitational field:
Prelab Question:
If U is the gravitational potential energy on a ‘flat’ earth (U = m g h) show that the field lines (g)
are at right-angles to the equipotential surfaces.
Hint:
P-6
Write out the condition for an arbitrary vector lying on an equipotential surface, and
show that this vector is always at right-angles to g. (recall that an equipotential
surface is the collection of points with U = 0)
Physics 121/2 and 141/2 Laboratory Manual
Experiment 5
The speed of light
References
121/2: Section 34.3.
141/2: Sections 33.5 to 33.7.
Using Maxwell’s equations it can be shown that light is an electromagnetic wave with a speed of
propagation given by:
1
c
 0 0
where 0 and 0 are the permittivity and permeability of free space. These quantities provide a
measure of the response of space to B and E fields. Inasmuch as this is so, it should not be
surprising to see that light, as an electromagnetic wave, has a speed which is dependent only on
these quantities.
In this experiment you will use a resonant inductor-capacitor ( L-C ) circuit to measure the
product ., from which you can obtain a measurement of the speed of light. In these notes we
hope to give you some understanding of the functioning of the L-C circuit shown below.
+
+
Signal
Generator
L
C
(Inductor)
-
C.R.O.
-
In this circuit the inductor (L) and capacitor (C) are in parallel. This means that their effect on
the circuit is similar to that of resistors in parallel, i.e.(for our purposes):
1
RTOTAL
 1
RCAPACITOR
 1
RINDUCTOR
So, if we know how the C and L act in the presence of oscillating voltages, we will know how this
circuit behaves.
Frequency dependence of the ‘resistance’ of the capacitor
In the electronics laboratory you will have seen that a capacitor is an open circuit (break in the
wire) which provides no resistance to rapidly varying voltages. Recall that this is because the
plates of the capacitor are charged up by the applied voltage, and that if this potential changes
rapidly enough the capacitor does not get to be ‘fully charged’, and thus does not become
significantly resistive. In fact, the characteristic of the capacitor is such that its effective
‘resistance’ is inversely proportional to the frequency of the applied voltage. That is:
RCAPACITOR 
1
2fC
Frequency dependence of the ‘resistance’ of the inductor
The inductor is simply a piece of wire wrapped into a large coil. When a slowly varying
(approximately DC) voltage is applied to it, it acts as a piece of wire. When the applied voltage is
rapidly varying, however, the current sets up a magnetic field in the coil which acts to oppose a
change in the applied voltage. Thus the inductor becomes ‘resistive’ at high frequencies. (Note
Physics 121/2 and 141/2 Laboratory Manual
P-7
that this occurs for the slowly varying voltage also, but that the decay time of the B field is much
quicker than the rate of change of the applied voltage, and thus the effect of the decaying B field
is minimal.) Thus the characteristic of the inductor is such that its effective ‘resistance’ is
proportional to the frequency of the applied voltage. That is:
RINDUCTOR  2fL
Graphically these become (note the logarithmic scale chosen for the frequency axis)
80
60
R capacitor (effective)
C = 3.2 microF
Effective
Resistance
()
R inductor (effective)
L = 6.4 mH
40
R total (effective)
20
0
100
Frequency
(Hz)
1000
10000
The parallel resistance formula stated earlier really just means that for parallel resistors it is the
lower of the two resistors which passes the most current and thus dominates the overall
conductivity of the circuit. Thus we can see that at high f the capacitor becomes unresistive and
for low f the inductor is unresistive. Somewhere in the middle the total R is a maximum.
Prelab Exercise:
If the effective resistance of the circuit is given by:
R EFFECTIVE

1 

  2fC 
2fL 

1
Show that the resistance of the circuit will be a maximum when the frequency satisfies:
f 
1
2 LC
From this it can be seen that, if L and C can be related to  and , then a measurement of the
speed of light might be obtained by measuring the resonant frequency of the LC circuit. A similar
argument applies to the series LC circuit which you will use to determine the speed of light.
P-8
Physics 121/2 and 141/2 Laboratory Manual
Experiment 6
The saw tooth wave generator
References
121/2: None - see notes below.
141/2: Sections 28.8 and 28.7.
In this experiment you will investigate a circuit which can be used to provide the time base signal
for the CRO. In the process of doing this it is hoped that your understanding of the operation of
the CRO will be extended. In addition you will investigate the properties of an electronic device
known as a neon diode.
Prelab Exercise:
Before you attempt to build this circuit it is useful to appreciate exactly what the time base circuit
does. Therefore it is suggested that you read the introductory notes for the projects experiment
The Cathode Ray Oscilloscope (Experiment 15). Briefly summarise the role of the time-base circuit
here, and then complete the prelab exercises for Experiment 15..
Experiment 7
The pendulum
References
121/2: Sections 5.2 - 5.3, 6.2; Chapter 15 (introduction), Section 15.1; 15.3 and 15.4.
141/2: Sections 5.5, 5.6 (Weight and Tension); 16.1, 16.2; 16.4 and16.6.
In this experiment you will use a pendulum to measure g, the acceleration due to gravity.
NOTE: the following (full)
derivation is provided for the
interested student only. You
are not required to memorise
the details, but must follow its
overall structure.

m g cos 

m g sin 
mg
Physics 121/2 and 141/2 Laboratory Manual
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Derivation: the period of a pendulum for   1
From the above diagram it can be seen that the restoring force on the pendulum has the form:
F ( )  mg sin 
Let us suppose for now that the approximation sin    is valid. Then we have:
F ( )  mg
Now, we also have from first principles that:
F   ma   m  
so that:
 
g


Prelab Exercise:
Confirm by substitution that:



 ( t)  sin t
g 
 
is a possible solution to this equation.
This equation describes SHM with a period T given by T  2

.
g
Using this equation you will try to find a value of g by measuring T for a number of pendulum
lengths,  . In obtaining this result you will hope that the approximation sin    is a valid in
this case. For an approximation to be experimentally valid it must not induce an error in the
result which is larger than the experimental error associated with the apparatus.
Prelab Exercise:

P-10
Sketch a graph of y = x and y = sin x (x in radians) on the same axes to show that the
approximation x  sin x may be valid for a sufficiently small range of x.
Physics 121/2 and 141/2 Laboratory Manual
To check that the approximation sin   is valid it is necessary to look at the implications of the
approximation explicitly.

Suppose that the length of the pendulum can be measured to 2 parts in 1000 accuracy
(i.e.  2 mm in 1 m). Roughly what do you think will be the maximum  that you can swing
the pendulum at while still obtaining an accurate value of g ?
Experiment 8
The acceleration due to gravity
References
121/2: Chapter 2 (introduction); Sections 2.1 - 2.6.
141/2: Sections 2.1 - 2.8 (2.7 and 2.8 in particular).
In this experiment you will use time-of-flight techniques and some interesting analysis to
determine the acceleration due to gravity. The experiment consists of timing the flight of a bob as
it falls through a number of distances. The bob will not ‘start’ from rest, but will have the same
initial velocity for each drop. Conceptually, the ball will follow the path shown in the diagram
below:
Ball Released:
t<0
s<0
v=0
Ball passes first sensor:
t=0
s=0
v0  0
Ball passes second sensor:
t=t
s=s
Physics 121/2 and 141/2 Laboratory Manual
P-11
Prelab Exercise:

Starting from:
a
d2s
dt 2
 g  ( constant )
for a mass falling in a constant gravitational field, derive the equation of motion describing
the position of a particle at any time t.
In the course of this experiment you will measure (s, t) data (as shown above) for the mass.

Using your equation of motion, consider how you might create a linear graph whose
gradient is proportional to g. Describe your strategy here.

Consider the effect of a systematic error in s on this graph, and think about how you might
analyse your data so that your result is independent of this error. (Look up Notes on
Confidence Limits in Experimental Physics if you are unsure what a systematic error is.)
Experiment 9
Mechanical resonance
References
121/2: Chapter 15 (introduction); Sections 15.1 - 15.2, 15.6.
141/2: Sections 16.1, 16.8, 16.9.
All objects have a natural frequency of (mechanical) resonance which depends largely on their
shape, composition and configuration. A swing, for example, has a natural frequency dependent
only on its length. If the swing is pushed at a frequency which is not (an integer fraction of) its
natural frequency it will not swing very highly. If, however, the swing is pushed at the ‘right’ rate
it will attain a large amplitude of oscillation (and ‘fun’ will be had).
In this experiment you will investigate the resonant behaviour of a spring - mass system. You
will relate this behaviour to properties of the spring (k) and the mass (m). This will be done by
measuring k as the gradient of an F vs x graph and by using a mass bar driver system to drive (=
push) the mass at resonant and off-resonant frequencies.
P-12
Physics 121/2 and 141/2 Laboratory Manual
Prelab Exercise:
Show that:
 k

x(t )  A sin t
 m


F  mx  m
d2x
dt 2
is a solution to the spring equation:
 kx
and thus that the period of the oscillation is given by:
T  2
m
.
k
Experiment 10
Measuring the wavelength of light with a ruler
References
Diffraction gratings are not discussed in detail in either text book, however, the following
references should help with the principles of interference and diffraction.
121/2: Chapter 39 (introduction); Sections 39.4 - 39.5; Chapter 40 (introduction);
Section 40.1.
141/2: Sections 36.1, 36.4, 37.1 - 37.4, 37.7.
In this experiment you will measure the
wavelength of red laser light
(approximately 6  10-7m.) with a ½ mm
graduated ruler to an accuracy as high as
0.1 % !
This can be achieved by using the ruler as
a reflection diffraction grating (shown to the
right) and observing the resultant
diffraction pattern.
Xn
n
n
Xc
Metal
ruler
i
laser
L
Physics 121/2 and 141/2 Laboratory Manual
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The light incident on the steel ruler is scattered by the engravings, each of which becomes a
source of spherical (Huygen) wavefronts. This scattering occurs at many sites along the ruler and
the light is allowed to recombine at the screen. As the light from neighbouring scratches on the
ruler travels different distances there will be a phase difference between the scattered wavefronts
and there will thus be interference.
Let us consider the light incident on the ruler in more detail:
We can use an approximation here that the outgoing (n) waves are parallel because the screen is
far away, i.e. L>>d.)
Prelab Questions:

If the incident light is coherent (in phase), what must be the path difference - in terms of
the wavelength - for the light arriving at the screen to be in phase also?

Is this the condition for constructive or destructive interference?

Using the geometry of the experiment, write an expression for the path difference in terms
of i, n and d.
When you do this experiment you will find that the central maximum (n = 0) is readily
distinguished from the other maxima. Given that you will be measuring only n and n, can you
work out how you will eventually determine n ? (Note that i is fixed but that you cannot directly
measure it with great accuracy. Can you calculate i, from your knowledge of 0 ?
P-14
Physics 121/2 and 141/2 Laboratory Manual
Experiment 11
X-ray diffraction: (lattice spacing of NaCl)
References
Prelab exercise:
Read appendices F and G for background on x-ray production and mounting crystals in the
580 Tel-X-ometer.
121/2: Page I3 (not very useful however - use the reference below).
141/2: Section 37.9.
In this experiment you will use x-rays – very short wavelength, high energy electromagnetic
radiation – to determine the separation of neighbouring atoms in a crystal of NaCl.
In order to do this you will need to understand the processes of Bragg diffraction and x-ray
production. The following notes will help you to teach yourself about Bragg diffraction…
Bragg diffraction
A NaCl crystal has a regular (face centred) cubic structure, as
shown here. For the sake of simplicity it is useful to confine our
treatment of the physics involved so that it lies entirely in one
plane. This is achieved by orienting the NaCl crystal so that its
crystal planes coincide with the plane of the x-ray source and
detector. If we do this then the crystal looks like (from ‘above’):
d
Each of the Na and Cl atoms will scatter the x-rays
in all directions, but because of the regular spacing
of the atoms in the lattice, there will be
interference of these scattered x-rays. This occurs
because the x-rays interact with all of the atoms in
the crystal, and thus there are many scattered
beams. This process involves exactly the same
physics as other wave interference phenomena you
are familiar with: when the path difference
between the scattered beams is an integer number
of wavelengths, constructive interference will
result and (if the radiation is in the visible region)
a bright spot will appear.
In practice, there will be many combinations of  and  having a maximum intensity of x-rays.
Measurement of the lattice spacing is achieved by further restricting the geometry of the
experiment. In order to find out how we may do this, let us think through the process of Bragg
diffraction.
Physics 121/2 and 141/2 Laboratory Manual
P-15
Consider the crystal to be a collection of rows of atoms. By doing this we can break our analysis
into two parts: we can look at the diffraction from one row of atoms and then we can consider the
diffraction from a collection of these rows.
Considering the x-rays incident on one of these rows:
2
1
B

D
A
Note that   
in general
because the
x-rays are
scattered.

C
d
Note that path 1 travels a distance AD further and a distance BC shorter than path 2. We can see
that there will be a constructive interference for any ,  where this path difference (AD-BC) is an
integer number of wavelengths.
Prelab exercise
Draw an expanded version of the diagram above and use it to show that the path difference is
given by:
P.D. =  AD–BC  = d (cos  – cos ) = n 
(nI)
In principle, this argument may be used to determine the structure of the crystal. As we have
discussed earlier, however, successive layers of the crystal will also diffract and may cause a
destructive interference at the same , . If this were to happen, the interference peak would be
missed and the results incorrect. In order to avoid this, we can force  =  so that we always have
a constructive interference from this process: by doing this we will always be observing the zeroth
order diffraction, of reflection. If this is done we will be able to observe the interference from
neighbouring layers of the crystal without having to worry about how these two effects compete.
So, let us consider diffraction from the collection of rows that make up the crystal:


d
A

C
reflecting ‘layers’
of crystal
B
P-16
Physics 121/2 and 141/2 Laboratory Manual
As before, the x-rays scattered from adjacent rows will be in phase only if the path difference is an
integer number of wavelengths. i.e.:
P.D. =  ABC  = 2  AB  = 2 d sin  = n 
(nI)
This is the Bragg condition we will use in this experiment: by holding  =  we can look at the
interference of x-rays scattered from neighbouring planes within the crystal and determine the
lattice spacing, d.
It is possibly instructive to note that, by treating the crystal in this way we have transformed it
from a single three dimensional diffracting volume into a diffraction-grating like part (a row of
scattering objects) and a thin-film like part (where the path difference is due to the separation of
the crystal planes).
If you do not understand how this analysis works, it is recommended that you read these notes
again. Ask your demonstrator a little while after the practical begins if you have any questions.
Experiment 12
Single-slit diffraction
References
121/2: Chapter 40 (introduction); Section 40.1.
141/2: Sections 37.1 - 37.4.
In this experiment you will use a laser and a photo-detector to determine the width of a narrow
slit by examining the diffraction pattern produced by the slit. After doing this you will be able to
measure the width of your hair by considering it to be an inverse slit, which (interestingly)
produces the same diffraction pattern as a slit.
To detect the maxima and minima in the diffraction pattern, you will use a photosensitive resistor
which has a resistance that is approximately inversely proportional to the light intensity
illuminating it. The photoresistor is enclosed in a lightproof container, with a small circular
aperture, as shown in fig. 1. Only light entering from (nearly) perpendicular to the detector face
will reach the photo-resistor: thus, stray light from the room should not affect your results.
Figure 1 : the construction of the photo-sensitive resistor

ohm
meter
LASER
slide
photo-sensitive resistor
Physics 121/2 and 141/2 Laboratory Manual
P-17
Prelab Question:
It may be apparent that the
diameter of the aperture  will
affect your ability to locate the
exact positions of the maxima and
minima of the diffraction pattern.
With reference to the diagram,
discuss the relationship between
the relative sizes of the
aperture/diffraction pattern and
your ability to accurately measure
the position of the features of the
diffraction pattern. (It may be
useful to consider the limiting
physics by increasing / decreasing
 and examining the problems
first encountered in each of these
limits.)
I
DETECTOR

Figure 2: Single slit diffraction pattern
and the detector with collecting
aperture .
Experiment 13
Note:
The sonometer
You may not chose this experiment and Experiment 1, Physics of the Guitar.
References
121/2: Sections 16.3 and 16.6.
141/2: Sections 17.6 and 17.11 to 17.12.
In this experiment you will investigate the vibrations of a wire and you will try to determine how
the physical variables constraining the wire, such as length, mass density and tension affect the
resonant frequency of the wire.
As an introduction to these concepts it is useful to examine the vibrations of a string as
determined by the geometry of the situation.
When a string is plucked it vibrates in all of the modes available to it. Some of these modes are
prohibited by physical constraints and die out relatively quickly. Other modes are strongly
favoured by the geometry etc and are sustained. These are the notes we hear when a guitar
string is plucked, and it turns out that these are the standing waves of the string.
The first, or fundamental, standing wave may be represented as:
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Physics 121/2 and 141/2 Laboratory Manual
n=1
L
 = 2L
You can see this as the amplitude envelope of the string as it vibrates: note that the ends are
stationary, corresponding to the fixed ends of the string, and also that the string oscillates in
phase: it is not a travelling wave. We can thus identify the wavelength of the fundamental mode
as twice the length of the string, ie  = 2L.
Similarly, the second harmonic can be represented by:
n=2
=L
L
Prelab Exercise:
Draw the first four standing waves on a string of length L with n = 1,2,3,4. Write down their
wavelengths in terms of n and L. Hence derive a general expression for the wavelength of the nth
harmonic.
=
=
=
=
In general,=
Experiment 14
Mechanical measurement of the velocity of light
It is not immediately obvious from observation that the speed of light is finite. In 1675 Olaf
Röemer measured the speed of light by observing that the period of Jupiter’s moons varied. As
the moon orbits Jupiter, it is eclipsed by the planet for a time. Röemer noticed that the duration of
these eclipses was shorter when the Earth was moving towards Jupiter than when the Earth was
moving away. This time difference was attributed to the distance that the earth had travelled
while Jupiter’s moon was eclipsed, and to the finite velocity of light. Thus the velocity of light
was first measured to be 2  108 m/s. This value is too slow by a factor of 1/3 primarily because of
inaccurate estimates of the planetary distances involved.
Physics 121/2 and 141/2 Laboratory Manual
P-19
Earth orbiting
the sun with
(relatively)
constant speed
SOL
moon of
Jupiter having
constant period
position of
Earth when
moon
reemerges
position of
Earth when
moon
eclipsed
Jupiter
time measured for
moons occlusion
is shortened by a
time t = x / c.
In 1862 Foucault devised an experiment which employed a rapidly rotating mirror and some
accurate distances to measure the velocity of light, which Michelson then modified to measure the
velocity of light in two orthogonal directions simultaneously. In this experiment you will measure
the speed of light (wholly within the laboratory!) using apparatus similar to that used by Foucault
in his experiment. In this introduction we will consider a simplified version of the Foucault
apparatus which demonstrates the physics of the measurement of the speed of light.
The apparatus is configured as in the diagram below.
Rotating
Mirror
(RM)
LASER

Fixed
Mirror
(FM)
In this arrangement the laser is directed at the rotating mirror (RM) which, as it rotates, sends
the beam spinning around the room. In one position the beam will be reflected towards the fixed
mirror (FM) which has been adjusted so that it will reflect the light directly back to the RM. By
the time the beam has travelled back to the RM it will have rotated a bit, and thus the light
reflected from it will not go directly back to the laser. Rather, it will make an angle with the
incident beam which depends directly on the speed of light.
In order to measure the speed of light we need to relate the displacement of the return beam to
the geometry of the experiment.
screen
R
LASER
(RM)

x
D
(FM)
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Physics 121/2 and 141/2 Laboratory Manual
Prelab Exercise:
Using the above diagram to define your variables, calculate the following quantities:

How long will the light take to travel from the RM to the FM and back?

Through what angle will the mirror
rotate in this time?

Note that (2  t) is
independent of 

In order to work out the angle that the
return beam makes with the incident beam
one must note that the beam is deflected by
twice the angle that the mirror has rotated
through, as in the diagram to the right.

2t

incoming beam
–t
–t
return beam
Prelab Exercise:
Continue your analysis of the apparatus by finding:

the angle that the light is reflected from the mirror (relative to the incident light beam);

Where will the beam be found on a screen placed at some distance R from the RM?
After this calculation you should have the following expression:
x

4DR
c
Evaluate the expected size of the displacement x for the (representative) values D = 1 m,
L = 1 m,  = 2  f = 6000 rad/sec, c = 3108 m/sec.
While this apparatus forms the basis of the equipment on your bench, it turns out that some extra
components need to be added in order to use it to make real measurements. This is because the
displacement x that you have calculated above is very small: so small that the return beam does
not fully emerge from the incident beam! When you arrive in the laboratory you will find a full
explanation of the modified apparatus in the lab notes.
Physics 121/2 and 141/2 Laboratory Manual
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Experiment 15
The Cathode Ray Oscilloscope
The CRO is a complicated-looking instrument that does a very simple job. Fundamentally, the
CRO performs the same job as the needle-pointer voltmeters that you have already used. What
makes the CRO a powerful instrument is the extra functions that it makes available to the user.
However, in order to understand these features it is useful to understand how the CRO functions
as a voltmeter first.
In order to explain the operation of the CRO we will attempt to construct it bit by bit.
Inside the CRO there is an electron gun which aims a thin beam of electrons at a phosphorescent
screen. When this beam hits the screen, the screen glows. In between the electron gun and the
screen there is a pair of deflection plates. These are two metal plates, one on each side of the
beam, which can have a potential difference applied to them. If we recall that electrons carry a
negative charge, we can see that these deflection plates can be used to deflect the electron beam
so that it hits the screen in a different place.
electron
beam
deflection of
the beam is
proportional to
the voltage on
the plates
+V
electron
gun
–V
deflection
plates
path of the
undeflected electron
beam (when there is
zero voltage on
deflection plates)
phosphorescent
CRO screen
Fortunately, the deflection of the electron beam is proportional to the voltage on the plates. Thus
if we can put the voltage we wish to measure across the deflection plates then our CRO will
represent this voltage as a displacement of the spot on the screen.
Thus the CRO can be used as a voltmeter, although as a voltmeter our CRO suffers from one
problem: it does not have an adjustment to allow for the measurement of other scales of voltages,
as a standard voltmeter does. This problem is overcome in the CRO by inserting an amplifier into
the circuit.
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Physics 121/2 and 141/2 Laboratory Manual
deflection of
the beam is
proportional to
the gain on the
amplifier
electron
gun
AMPLIFIER
with gain control
input signal
(voltage to be measured)
It can be seen from this picture that if the gain on the amplifier is turned up, the spot will be
deflected further up the screen, and thus the input signal can be magnified. This is the same as
choosing a different scale on the voltmeter. There is a small difference here that is often a source
of confusion when using the CRO: when you choose a different scale on a voltmeter you also
commit yourself to reading one of the scales printed on the voltmeter, and do so accordingly. The
CRO, however, does not have scales printed on the screen; instead, the amplifier gain control has
the scale printed on it, where the scale is quoted in volts per division (often 1division = 1
centimetre). This means that , for a given gain chosen, one division on the screen represents a
certain (stated) number of volts. Obviously (think about it!) the further you turn up the gain the
lesser the voltage input per division of deflection on the screen.
Our CRO now functions as a perfectly serviceable voltmeter! Now lets try to jazz it up a bit…
Firstly, let us add to the CRO another, independent set of deflection plates. These plates will
control the X-position of the spot independently from the other deflection plates which influence
only the Y-position of the spot. Our CRO then becomes:
Vertical (Y)
deflection
electron
gun
Y
AMP
X
AMP
Y inputs
X inputs
Horizontal
(X) deflection
It may be useful to think of this CRO as being two voltmeters whose outputs are represented at
right angles.
As stated at the start of this discussion, the CRO is just a voltmeter, albeit quite a jazzy one. You
may be aware that a standard voltmeter is often capable of measuring two types of voltages: DC
(ie, constant) and RMS (for where AC signals are concerned). These two cases are a bit limiting:
what to do if we wish to look at the structure of the voltage in detail, rather than just the average
DC or RMS level? It is possible to do this using our CRO!
In order to represent a time varying signal on our CRO, using the X axis as the time axis, we will
need to input a signal to the X axis which can pull the spot across the screen at a constant rate.
The Y-input would then alter only the vertical position of the spot as the X-input drew the spot
across the screen.
Physics 121/2 and 141/2 Laboratory Manual
P-23
Prelab Exercise:
Sketch the X-input required to draw the dot across the screen at a constant rate.
Time Base
Fortunately the CRO has an in-built circuit to perform this function. The sweep function is called
a time base circuit. The time base controls on the CRO are labelled SWEEP TIME / DIV (read as
“sweep time per division”). The rapidity with which the spot is drawn across the screen can be
adjusted by altering the gradient (but not the amplitude!) of the signal you worked out for the
above exercise.
Note that, once the spot has traced across the screen at a constant rate, representing the Y-input
voltage as a Y-deflection, you ideally want it to return to the ‘beginning’ of the screen to start
again and to re-trace the signal. (If it did not do this you might miss your signal: for example, the
AC mains run at 50 Hz. If you wanted to view the AC mains voltage in detail, you would need to
look at a signal which lasted for only 1/50th of a second. Can you see something lasting for only
1/50th of a second?)
Prelab Exercise:
Sketch the X-input required to draw the dot across the screen and back again three times at twice
the rate of your previous answer.
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Physics 121/2 and 141/2 Laboratory Manual

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