University of Canterbury
PHYS 101
LABORATORY MANUAL
School of Physical and Chemical Sciences
Christchurch
New Zealand
Revised June 2021
PHYSICAL CONSTANTS
Constant
Symbol
SI Unit
Speed of light in a vacuum
𝑐
2.998×10 8
ms-1
Universal gravitational constant
𝐺
6.673×10 -11
m3s-2kg-1
Avogadro’s Number
𝑁𝐴
6.022×10 23
mol-1
Universal gas constant
𝑅
8.314
Jmol-1K-1
Charge of an electron
𝑒
-1.602×10 -19
C
Electric permittivity of free space
𝜖0
8.854×10 -12
Fm-1
Magnetic permeability of free space
𝜇0
4π×10-7
NA-2
Rydberg constant
𝑅
1.097×10 7
m-1
-273.15
°C
8.99×10 9
Nm2C-2
Absolute zero of temperature
Coulomb force constant
2
Value
𝑘=
1
4𝜋𝜖0
Geophysical Constants - Christchurch, January 1991
Constant
Symbol
Value
SI Unit
Gravitational acceleration
𝑔
9.81
ms-2
Horizontal component of earth’s magnetic field
𝐵ℎ
2.10×10 -5
T
Vertical component of earth’s magnetic field
𝐵𝑣
5.42×10 -5
T
Angle of inclination
-68°35′
Angle of declination
23°33′ E +5′ per year
(2014)
240
V
50
Hz
Mains voltage
Mains frequency
SI Units
Quantity Unit
Symbol
Definition
𝑚
Defined
𝑘𝑔
Defined
second
𝑠
Defined
Temperature kelvin
𝐾
Time
Number of Atoms mole
Defined
𝐶
𝐴𝑠
Resistance Ohm
𝛺
𝑉𝐴−1
Defined
Inductance henry
𝐻
𝑊𝑏𝐴−1
mol
Defined
Frequency hertz
𝐻𝑧
𝑠 −1
𝑉
𝑊𝐴−1
𝑁
𝑘𝑔𝑚𝑠 −2
𝑃𝑎
𝑁𝑚−2
Potential difference volt
𝑊𝑏
Magnetic Flux weber
Symbol Definition
𝐴
Length metre
Mass kilogra
m
Quantity Unit
Current ampere
Charge coulomb
Force newton
𝑉𝑠
Pressure pascal
Magnetic Flux tesla
Density
𝑇
𝑊𝑏𝑚−2
Capacitance farad
𝐹
𝐶𝑉 −1
𝐽
𝑁𝑚
𝑊
𝐽𝑠 −1
Energy|Work joule
Power watt
Standard Scientific Prefixes
1018
exa
E
103
kilo
k
10-6
micro
μ
1015
peta
P
102
hecto
h
10-9
nano
n
1012
tera
T
10-1
deci
d
10-12
pico
p
109
giga
G
10-2
centi
c
10-15
femto
f
106
mega
M
10-3
milli
m
10-18
atto
a
3
CONTENTS
PHYSICAL CONSTANTS ...................................................................................................................................... 2
CONTENTS ........................................................................................................................................................... 4
PEOPLE ................................................................................................................................................................. 5
COURSE INFORMATION ..................................................................................................................................... 6
INTRODUCTION AND UNCERTAINTIES ........................................................................................................................ 8
CONSERVATION OF ENERGY .................................................................................................................................... 10
ANGULAR VELOCITY AND MOMENTUM ................................................................................................................... 14
LATENT AND SPECIFIC HEAT.................................................................................................................................... 18
ABSOLUTE ZERO ...................................................................................................................................................... 22
CAPACITANCE ......................................................................................................................................................... 24
SUPERCONDUCTIVITY .............................................................................................................................................. 29
APPENDIX A: SAMPLE EXPERIMENT AND WRITE UP ............................................... ERROR! BOOKMARK NOT DEFINED.
APPENDIX B: UNCERTAINTIES.................................................................................................................................. 46
APPENDIX C: UNCERTAINTY EXERCISES................................................................................................................... 52
APPENDIX D: MANUAL GRAPHICAL ANALYSIS ........................................................................................................ 55
APPENDIX E: COMPUTER ASSISTED GRAPHICAL ANALYSIS ...................................................................................... 59
APPENDIX F: MEASUREMENTS ................................................................................................................................. 61
APPENDIX G: WRITING LAB REPORTS ..................................................................................................................... 62
APPENDIX H: SAMPLE LAB REPORT ......................................................................................................................... 63
4
PEOPLE
Course Coordinator:
Assoc. Prof. Adrian McDonald
Laboratory Supervisor:
Cliff Franklin
Edited By: Raphael Nolden, Alex Chapman, Crissy Emeny
5
COURSE INFORMATION
Learning Outcomes
1. To learn how basic scientific equipment is used.
2. To make clear, concise, accurate records of scientific
experiments.
3. To analyse experimental data graphically.
4. To estimate the precision of an experimentally
determined result.
5. To critically discuss and analyse an experiment.
6. To present the findings in a formal report.
Assessment
The assessment for the lab component of this course
consists of two parts: The weekly lab work and write up
in the lab book and the formal lab reports. This
contributes 15% to the final mark for this course overall
which is calculated as follows:
Assessment
Lab work and lab book write up
Formal reports (best mark from two)
Weight
5%
10 %
You must pass the lab component if you wish to pass
the course as a whole.
Organisation
Throughout the semester, you will complete a different
physics experiment each week which complements
what you are learning in lectures. The experiments run
for three hours, and will often take you the full allocated
time. Please turn up a little early so you are ready to start
working when the session starts. It is important that you
prepare for the lab before you arrive by reading through
the experiment. If you fail to do so, you will struggle to
finish the session in time.
You will receive your marks via a series of checkpoints
throughout the lab. The first checkpoint will be assessed
shortly after the start of the lab, and usually checks that
you understand the theory and methods which will be
used so it is important that you read through the lab and
any relevant appendices and sections of the text book
before you arrive. Please bring this manual to every
session as you will need it to complete the lab.
Work is completed in groups of three or four and you get
to choose who you work with. It is very important that
the work is undertaken equally by every member of the
group so that everyone gets as much as possible out of
the lab. Please ask the demonstrators for assistance if
you have any trouble understanding the lab book,
equipment or results. They are there to help. Pay
attention to the feedback from your demonstrator as you
are expected to take this into account and improve the
quality of your write up as the semester progresses. Ask
for further explanation if something is unclear.
Please ensure that you have attached the sticker to the
front of your book and completed your details including
the lab class and time. It is also important that you write
your name on the lab book and this one so that they can
be found if lost. Lost property is held by the lab
supervisor in room 303.
6
Lab Work
The marks for the lab work will be based on a series of
checkpoints throughout the lab. These marks are
allocated on an individual basis and reflect your
successful completion of each checkpoint. Each lab is
marked out of 5. The first checkpoint will always require
you carry out the following write up in your lab book:
• Title of the experiment,
• Date,
• Aim of the experiment,
• Names of your lab partners,
In addition to answering some questions from your
demonstrator to demonstrate your understanding of the
experiment. The fifth checkpoint will be awarded when
you have successfully completed the experiment
including a written conclusion and answered some
questions.
Formal Reports
Up to two formal reports need to be submitted to
complement the weekly write up of experiments. These
will generally be due towards the end of each term. The
reports will be announced one week before they are
due. Further information on the lab reports including the
marking schedule can be found on Learn. A soft and
hard copy of the report must be submitted before the
deadline.
Absences
A zero mark will apply for all work which is not submitted
or labs which are not attended unless an adequate
medical or personal situation prevents this. Please
contact the lab supervisor in room 303 within one week
to make arrangements. Documentary evidence such as
a medical certificate will be required to gain an
exemption. If required, a repeat session of the missed
experiential will be organized.
Equipment
The equipment you will use in the laboratory is
specialized, and heavily-used. Ask for assistance if you
are unsure of how it should be used. Never use
excessive force on the equipment and keep the lab
bench tidy so nothing gets pushed onto the floor. Please
leave the bench as you found it by tidying away all of the
equipment at the end of the lab. Please let us know if
something is accidentally damaged, these things
happen, but you will be charged if it was willfully done.
Some of the equipment you will use to perform the
experiments has the potential to be dangerous if
incorrectly
used.
Please
always
follow
the
demonstrator’s instructions and ask if you are unsure.
The following rules apply in all first year labs and will be
strictly enforced:
• Enclosed footwear must be worn at all times: No
sandals, Jandals or bare feet. This is an OSH
requirement and you will be asked to leave if
you come without.
• No Food or drink is permitted in the labs except
water in sipper bottles
Do not use the lifts in the event of a fire, calmly walk
down the stairs and assemble on the grass on the North
West side of the building.
In an earthquake take shelter under benches and beams.
Keep away from glass and shelves. Leave the building
once the shaking has stopped, especially when there is
a risk of severe damage or fire. Be prepared for
aftershocks.
Dishonest Practice
Dishonest practice of any kind is a serious academic
offence. Never copy another person's work or cheat in
any way. Submitting lab work which is not your own is a
serious matter and official disciplinary action can be
taken.
Dishonest practice includes plagiarism, collusion,
copying and ghost writing.
• Plagiarism is the presentation of any material
(text, data, figures or drawings, on any medium
including computer files) from any other source
without clear and adequate acknowledgement
of the source.
•
•
•
Collusion is the presentation of work performed
in conjunction with another person or persons,
but submitted as if it has been completed only
by the named author(s).
Copying is the use of material (in any medium,
including computer files) produced by another
person or persons with or without their
knowledge and approval.
Ghost writing is the use of another person or
persons (with or without payment) to prepare all
or part of an item submitted for assessment.
NO EXCUSES! NO SECOND CHANCES! NO FURTHER
WARNINGS!
Students may be referred to the University Proctor when
dishonest practice is involved in tests or other work
submitted for credit. The instructor may also choose not
to mark the work.
Tips
• Read through the experiment before coming to the lab
– including the summary of the theory. Reading before
you arrive means you’ll have a good grasp of what
you’ll be doing, and can get straight into it.
• It is important to note the purpose of these
experiments is not focused on achieving the most
accurate results, i.e. getting as close possible to the
accepted value. It is to develop your abilities in
conducting experiments, and core capabilities such as
following experimental methods, problem solving, data
collection, and analysis. Sometimes, your experiment
will go awry, and your results will not agree with
accepted values. This is just a reflection of doing
scientific work in the real world. Do not fret –
inaccurate data allows you to present your deeper
understanding of the content when drawing
conclusions.
• Download the Desmos Scientific Calculator app, or
any similar scientific calculator onto your phone that
allows you to easily edit long calculations. The labs
often involve lengthy calculations, and being able to
easily go back and fix errors can save a lot of time.
7
EXPERIMENT 0
INTRODUCTION AND UNCERTAINTIES
This equation can be integrated to give Newton’s Law of
Cooling:
Introduction
Welcome to your first laboratory session! Make sure you
have had a read over the important information on the
previous pages. In this lab you will learn how to write up
an experiment and calculate uncertainties. First, you will
be using a selection of measurement devices to
measure the thickness of a paving stone, and estimate
the uncertainty associated with each measurement.
You’ll then be calculating the rate at which a cup of
coffee cools from some provided experimental data, and
its associated uncertainty. Please ask a demonstrator if
you have any questions at any point during the lab.
Checkpoint One: How thick is that paving
stone, really?
Please set up your lab book per the Lab Work subsection of the Course Information section above. Read
Appendix B: Uncertainties for an introduction on
uncertainties, accuracy vs. precision, and more. Now,
take turns to measure the thickness of the paving stone
using the three types of ruler, a metre stick, a 30cm ruler
and Vernier calipers (see Appendix F for an explanation
of these). Estimate the size of the uncertainty for each
one. Compare your results with the others in your group.
How precise are your results? How accurate are they?
See Appendix B for an explanation of the difference
between accurate and precise. To obtain your first
checkpoint, show your results to a demonstrator, and
show your understanding by answering some questions.
Checkpoint Two: Processing experimental
data
A laboratory demonstrator is enjoying a nice, hot cup of
coffee, and wants to know how quickly it is cooling
down. She reasonably posits that the coffee in her mug
will cool at a rate proportional to the difference between
the temperature of the coffee and the temperature of the
room. That is to say, if there is a large difference in
temperature, it will cool faster than if there was a small
difference. Mathematically she expresses this rate of
cooling as:
𝑑𝑇
= −𝛼𝛥𝑇
𝑑𝑡
where 𝛥𝑇 = 𝑇𝐶 − 𝑇𝑅 is the difference between 𝑇𝐶 , the
temperature of the coffee, and 𝑇𝑅 , the temperature of
the room, t is the time, and 𝛼 is the cooling constant.
8
𝛥𝑇 = 𝑇0 𝑒 −𝛼𝑡
where 𝑇0 is the initial temperature difference, at time t
= 0 s. To test the theory, the lab demonstrator brews
another cup of coffee and measured the temperature of
the liquid periodically for 35 minutes. The average room
temperature during the experiment was 20.2 ˚C. The
results she obtained are summarized in Table 1.1 below.
Time t ± 1 (s)
0
180
420
660
900
1140
1410
1620
1860
2100
Coffee Temp 𝑇𝐶 ± 0.5 C
84.2
81.8
78.3
75.1
74.2
69.5
67.4
64.8
62.7
61.1
Table 1.1 Measuring the temperature of a cup of
coffee over time
𝑇𝑅 = 20.2 ± 0.5 ˚C
Your task is to use the results of this experiment to
determine a value for the constant 𝛼 and its associated
uncertainty. To do so, follow the procedure below.
1. Copy the table above into your book, leaving room
for two extra columns on the right hand side.
2. We will determine 𝛼 by by graphically representing it
as a linear gradient (making it the slope of a line). To do
this we need a linear equation. Convert Newton’s Law of
Cooling into a linear equation by removing the
exponential component.
Hint: a linear equation is one of the form y = mx + c.
What here is “x”?
3. Complete the two empty columns in the table with the
necessary information to plot the linear equation
obtained above, and ask the demonstrator for
checkpoint two.
Checkpoint Three: Graphing the results
Plot the graph of the appropriate data to allow you to
calculate  and its uncertainty. Refer to page 55 in
Appendix D: Manual Graphical Analysis for help. More
details about how to complete uncertainty calculations
and good graphing practice are discussed in
Appendices A and B. Show the completed graph to a
demonstrator to obtain checkpoint three.
Checkpoint Four: Calculating 𝜶
Calculate a value for  Show your result to a
demonstrator to obtain checkpoint 4.
Checkpoint Five: Writing a conclusion
Write a suitable final conclusion, which includes:
• What the aim of your experiment was,
• Your results, and whether they seem reasonable or
not.
• Whether these results satisfied your aim, and if not,
why not.
Once complete, show your conclusion
demonstrator to achieve checkpoint 5.
to
the
Exercises for Experiment 0
Work through the exercises in appendix C when the
experiment has been completed. They must be
completed at home if this is not possible within the
assigned lab time.
9
EXPERIMENT 1
CONSERVATION OF ENERGY
Purpose and Introduction
Welcome to your next laboratory session. It is important
that you have had a read over the information on Course
Information section. Make sure you have also worked
through the exercises from the end of Lab 0 (located in
Appendix C) on uncertainties before attempting this lab.
In today’s lab, you will be exploring Hooke's Law, and
the principle of energy conservation. You will use a
spring-loaded cart on a ramp to experimentally
determine the spring constant of a spring, and
determine whether energy in the system is conserved.
To do this, you will using the data collection software
Logger Pro, and a selection of software-controlled
position and force sensors.
Checkpoint One: Slidin’ on by -
they converted into? Below is a table of a few different
types of energy, and their formulae.
Type of Energy Formula
1
Kinetic
𝐸𝑘𝑖𝑛𝑒𝑡𝑖𝑐 =
2
𝑚𝑣
Gravitational
Potential
𝐸𝑔𝑟𝑎𝑣 = 𝑚𝑔ℎ
Elastic Potential
𝐸𝑒𝑙𝑎𝑠𝑡𝑖𝑐 =
1
Thermal
2
𝑘𝑥
2
2
𝐸𝑡ℎ𝑒𝑟𝑚𝑎𝑙 = 𝑚𝐶𝛥𝑇
Symbol Quantity [Units]
understanding the theory
𝑘 Spring constant [_____]
The cart you’ll be using in this experiment is equipped
with a spring which can be compressed, locked, and
then released in order to propel the cart up the track,
before it rolls back down again under the influence of
gravity. The demonstrator will show you how to set this
up correctly. Have a play with the cart on the track, and
observe its motion.
𝑔 Grav. acceleration [m/s²] (9.81m/s²)
ℎ
Height [m]
𝑚 Mass [kg]
𝑣
Velocity [m/s]
𝑥 Spring displacement [m]
The force required to compress any spring is directly
proportional to its extension or compression. This is
described by Hooke’s law:
𝐹 = −𝑘𝑥
where 𝐹 is the force in Newtons required to
compress/extend the spring, 𝑥 is the displacement in
metres of the spring from its equilibrium position, and 𝑘
is a constant called the spring constant. The negative
indicates that the force required is in the opposite
direction to the displacement.
As you may know, energy can not be created or
destroyed, only converted into different forms. As the
cart rolls up the slope and back down again, it
experiences different forces, and possess different
forms of energy, which are continuously converted from
one form to another while the cart is in motion. In a
perfectly isolated (i.e., in a vacuum), frictionless set-up,
the cart would roll up then down the slope, bounce and
repeat, oscillating forever. However, we don’t expect
this to happen in our experiment. Why is this? What
types of forces act on the cart? Are they conservative,
or non-conservative forces? What kinds of energy are
present, and what other types of energy are
10
𝐹 Force [N]
𝐸 Energy [J] or [kgm²/s²]
𝐶 Specific heat capacity [J/kg˚C]
There is the important notion of a “system” that we have
thus far been using rather candidly. For the purpose of
this lab, let us define the system as purely the track, the
cart and all its internal components. Their surroundings
are not part of the system.
The question you’ll be striving to answer over the course
of this lab is the following: is energy conserved in our
system as it moves up and down the slope? Let’s get
started.
Write up your aim, date, lab partners, and title, and ask
your supervisor for your first checkpoint. You will be
answering some questions based on this section.
Checkpoint Two: Setting up, and
𝑥 [m]
experimentally determining 𝒌
0.010
First off, choose someone in your group who has printing
credit on their UC account, or can easily top up. Get
them to log on to a lab computer at your desk and open
Logger Pro®. Now carry out the following calibration
measures:
0.015
• Check that the selector on the motion detector (at the
higher end of the track) is set to the ‘cart ’setting.
0.030
• Calibrate the force sensor by first taking the cart off
the track, then clicking Experiment —> Zero, and
zeroing the only force meter in the pop up menu.
• Calibrate the motion sensor by placing the cart with its
spring extended on the track so that it is resting against
the force meter, and again clicking Experiment —>
Zero, this time zeroing only the motion sensor in the pop
up menu. Ensure it is pointing directly at the opposite
end of the ramp when calibrating. Keep all objects out
the motion sensors path when calibrating. Cntrl+0 is the
shortcut to get back to this menu quickly.
• Check that the force sensor reads zero when no force
is being applied to it, and that the position sensor reads
zero when the cart is resting against the force sensor
with its spring out (both to within a few decimal places).
• Click the collect button, give the cart a test push so
that it moves up the track, and wait for the default 10
seconds of data collection time to elapse. You should
hear a rapid clicking as the motion sensor scans. Look
at the results that come up to check that it is tracking
well, and that the cart is dipping into the negatives on
the position scale when it hits the force sensor.
• Change the direction of the sensors if the force or
direction is shown to be negative when a force is applied
or the cart moves up the track. You can do this by
clicking Set Up Sensors —> Show All Interfaces, right
clicking on the desired sensor, and clicking Reverse
Direction.
𝐹 [N]
𝑘 ± _____ [N/m]
0.020
0.025
method you can think of, and measure the force
required to compress this to the 5 different positions
listed in the table below. Make use of the ruler scale
attached to the ramp. Use the real time force sensor
read out to collect your results. Copy the table below
into your lab book, and record your results.
Estimate the uncertainty in your force measurements by
looking at the fluctuations which occur in the live
readout. This will be large. The steadier you hold the
cart, the smaller it will be.
Calculate a final value for the spring constant 𝑘 by
taking the average of the five values calculated.
Determine its associated uncertainty using half of the
range of the values of 𝑘 you calculated.
Show your 𝑘 value and its associated uncertainty to the
demonstrator for checkpoint two.
Checkpoint Three: Collecting data
You will now be gathering the data necessary to see how
the total energy of the cart changes over time. Copy the
table on the next page into your lab book, and record all
of your measurements into it.
1. Measure the mass of the cart. Determine the angle of
the track and its uncertainty using trigonometry and a
ruler. You will need to use the brute force method (see
Appendix B: Uncertainties) to calculate the uncertainty.
2. Compress the spring on the cart, lock it, and place it
• Change the sample rate of the motion sensor. Click
Experiment —> Data Collection, and set the sample rate
to 500 samples per second. The motion sensor will no
longer give a live read out due to the increased sample
rate; this is normal behaviour.
on the track so that it is resting against the force sensor.
Quickly check your test results with a demonstrator to
make sure everything is set up correctly, then proceed
with the experiment.
4. Show your results to a demonstrator, and save them
3. Click collect, wait for the motion sensor to start, and
release the cart by hitting the button with a hard object
like a ruler. Try not to impart any momentum to the cart
as you release the spring.
(File —> Save). Now is a good time to exchange email
addresses so that everyone in the group can have
access to the graph.
To determine your spring constant 𝑘 , push the spring in
the cart against the force sensor using the steadiest
11
Quantity
Symbol
Initial compression of spring (distance into the
negatives from 0 m)
𝑥𝑖
Velocity just after leaving the spring
𝑣𝑖
Maximum distance travelled up slope
𝑑𝑚𝑎𝑥
Velocity just before first collision
𝑣𝑓
Maximum compression of spring during the first
collision
𝑥𝑓
Position of randomly selected point
𝑥𝑟
Velocity of randomly selected point
𝑣𝑟
Mass of the cart
𝑚
Spring constant
𝑘
Angle of track
˚
5. Use the examine button
and stat button
to
look at the values on the graph required to fill out the
table. The examine button allow you to see specific
values as you run the cursor along the graph, and the
stat button can be used to find the minimum, maximum
and average value within a selected area.
6. Print out a copy of these graphs and clearly label all
of the points used in the table below. This is best done
by clicking File —> Print.
Value
Uncertainty
Units
• At the maximum compression during the first collision
i.e. when the spring is compressed to its maximum.
• At your random point part way up or down the slope between the release and collision points above.
Compare these energies. Do the results correspond to
an expected trend? Where did most of the energy get
“lost”? What happened to this energy?
7. Estimate the necessary uncertainties by opening a
Show your results to a demonstrator, and tidy the bench
so that it is ready for the next group.
new instance of Logger Pro, and running the experiment
with no cart on the track, clicking autoscale
, and
using half the range of data.
Checkpoint Five: Conclusion and exercises
Show your results to a demonstrator for your next
checkpoint.
Attempt the following exercises.
1. A rubber ball that heats up if it is deformed is initially
Checkpoint Four: Energy calculations
You now have the necessary information to scientifically
determine (read: figure out with numbers) whether
energy is conserved in our system or not. To do so, carry
out the following.
1. Calculate the total energy of the system and its
uncertainty at 6 different locations:
• Before the spring is released.
• Just after the cart loses contact with the spring.
• At the top of the slope.
• Just before the first collision.
12
at rest. It is allowed to fall, bounce, and just land on
a platform halfway down from its initial drop point
where it comes to rest, as shown in Figure 1.1. If it
initially has six arbitrary units of gravitational
potential
energy,
describe
the
energy
transformations it undergoes by filling out the table
on the next page. Assume there are no energy
losses to its environment.
2. Now assume the same experiment is carried out, but
Write a suitable conclusion for the experiment,
including the aim, your results, and whether these
results matched your expectations and/or satisfied
your aim. See Checkpoint 5 in the previous
experiment and the sample experiment in the
appendix for more information.
the ball is made of lead. Instead of bouncing, the ball
converts all of its kinetic energy into thermal energy.
From how high would a 1kg ball of lead with specific
heat capacity 𝐶 of 125 J/kg˚C need to be dropped
to heat it by 1˚C? Again, assume no external energy
losses.
1.
2.
4.
3.
6.
5.
Figure 1.1 Schematic of ball being dropped from an initial maximum height, bouncing, and
coming to rest on a platform halfway between the drop point and the minimum height.
Position
Description
1.
Initially
2.
Just after being
released
3.
Just before impact
4.
During maximum
compression
5.
Just after impact
6.
At rest on the platform
𝐸𝑔𝑟𝑎𝑣
𝐸𝑘𝑖𝑛𝑒𝑡𝑖𝑐
𝐸𝑒𝑙𝑎𝑠𝑡𝑖𝑐
𝐸𝑡ℎ𝑒𝑟𝑚𝑎𝑙
XXXXXX
13
EXPERIMENT 2
ANGULAR VELOCITY AND MOMENTUM
Purpose and Introduction
can be seen across many quantities and formulae in
Welcome to your third lab. In this session you will be
exploring the principles of angular momentum and
energy conservation. You will be dropping a stationary
disk onto a spinning one, observing the results, and
attempting to determine whether angular momentum or
rotational kinetic energy is conserved during the
collision.
Checkpoint One: Flipping bottles and
What makes one object harder to accelerate in a
straight line than another? You may immediately answer
that it is due to the differences in their mass - one is
heavier than the other, and so is harder to push.
Although mostly correct, such an answer neglects a
certain subtlety. Mass is a measure of how much matter
is in an object. How hard it is accelerate in a straight line
is a property called inertia, and although it is solely
dependent on its mass, it is reserved to describe this
specific property.
This distinction may seem pointless at first, but it sets
the stage for the following. What is the property that
describes how difficult it is to get something to spin?
Again, you may say mass, but in contrast to before,
mass is not the only factor in this case. As you may
know, it is much easier to roll a drink bottle between your
hands than get it to spin end to end (as one may do for,
say, a bottle flip). How hard it is to accelerate circularly,
to spin, depends not only on the mass, but how the mass
is distributed around the chosen axis of rotation.
Different shapes of the same mass rotate more or less
slowly for the same spinning force. This property is
called rotational inertia - the resistance an object has to
being spun around a particular axis.
It is this connection between inertia and mass that
allows us to make the parallel between the formula for
kinetic energy:
1
2
(2.01)
and rotational kinetic energy:
1
𝐸𝑅𝐾 = 𝐼𝜔2
2
(2.02)
Rotational inertia 𝐼 is, in a very real sense, the rotational
analogue of mass, just as angular velocity 𝜔 is the
rotational counterpart of linear velocity. This similarity
14
Symbol/
Rotational Quantity
Formula
𝑥 Position [m]
𝜃
Angular displacement [rad]
𝑣
Velocity [m/s]
𝜔
Angular velocity [rad/s]
𝑎
Acceleration
𝛼
Angular acceleration [____]
𝑚
dropping discs - understanding the theory
𝐸𝐾 = 𝑚𝑣 2
Symbol/ Linear
Formula Quantity
Mass [kg]
𝐼
Rotational inertia [kgm²]
𝐹 = 𝑚𝑎 Force [N]
𝜏 = 𝐼𝛼 Torque [Nm]
𝑝 = 𝑚𝑣 Momentum
𝐿 = 𝐼𝜔 Angular momentum [kgm²/s]
Table 2.1 Showing a selection of linear mechanical quantities, and
their angular counterparts.
mechanics, as shown in the Table 2.1 below.
In all collisions, momentum is conserved (overall). This
is true of both types of momentum, angular and linear.
This is because momentum can only be transferred to
other objects, not converted into some other quantity. In
a sense, it has nowhere else to go. This may sound a
little non-sensical, but consider kinetic energy in a
collision. While it can be also be transferred to other
objects, it has the ability to be converted into other
quantities, namely thermal energy, sound, potential, to
name a few. This is why it may or may not be conserved.
We call collisions where kinetic energy is conserved
elastic, and those where it is not inelastic. All of those
principles remain true when we are working in the
rotational realm.
The rotational dynamics apparatus you’ll be using for
this experiment (shown in Figure 2.1) consists of a main
body which is accelerated, and a second removable
disc that is dropped onto the main body. Once the
removable disc lands on the main body, the system
rotates as one. The angular speed will be monitored
using a photosensor and a timing plate. The timing plate
has an array of 180 holes punched into it through which
the photosensor shines an infrared light. The
photosensor detects how many holes per second pass
over top of it, and converts this into an angular velocity.
This data is then collected and displayed in Logger Pro
on a lab computer.
Because of its rotation, the main body of the apparatus
possess some initial, non-zero amount of rotational
kinetic energy, and angular momentum. When the
removable disc collides with it, will the rotational kinetic
energy of the system be conserved?
The same process can be used to derive EQ 2.06 if
rotational kinetic energy is conserved.
Removable disc
𝐼1 𝜔12
𝜔𝑓 = √
Felt pad
Main body
(main disc and axle)
(axle radius: 1.5cm)
𝑖
𝐼1 +𝐼2
(2.06)
In both cases, we need to know the rotational inertia of
both the removable disc, and the main body. The first is
given by the standard formula for 𝐼 for a disc:
1
𝐼 = 𝑚𝑟 2
2
(2.07)
Hook
The main body isn’t a standard shape, and so the
derivation of its rotational inertia requires a little more
work.
Timing plate
Figure 2.1 The experimental set up, showing the removable disc that
is dropped onto the main body as it revolves. A felt pad ensures a
smooth impact.
What about the angular momentum? These are the
questions we are seeking to answer in this laboratory.
Let us derive the equations we will be using for this
experiment. If we assume that angular momentum is
conserved during our collision, the initial total angular
momentum of the system must be equal to the final total
angular momentum, as shown in EQ 2.03 below:
𝐿𝑖 = 𝐿𝑓
(2.03)
Looking at Table 2.1, we can see that fundamentally, the
total torque on a system is equal to its rotational inertia
multiplied by its angular acceleration (the angular
equivalent of 𝐹 = 𝑚𝑎 ). As the weight falls and
accelerates the main body, what torques act on the
main body?
There is the torque due to the force 𝐹 of the weight
pulling on the axle of radius 𝑅 = 1.5 cm, and also a
frictional torque 𝜏𝑓 . As the apparatus accelerates at rate
𝛼1 then, the total torque is equal to:
𝐹𝑅 − 𝜏𝑓 = 𝐼1 𝛼1
When the weight falls off the hook, and the main body
decelerates, we have:
−𝜏𝑓 = 𝐼1 𝛼2
As 𝐿 = 𝐼𝜔, and the system we have here is made up of
two separable bodies, EQ 2.03 can be expanded as:
I1 ω1i +I2 ω2i =I1 ω1f +I2 ω2f
(2.04)
Where 𝐼1 refers to the rotational inertia of the main body,
𝜔1𝑖 refers to its initial angular velocity, 𝐼2 to the rotational
inertia of the removable disc, and so on.
(2.08)
(2.09)
We can subtract EQ 2.9 from EQ 2.8 to give:
𝐹𝑅 = 𝐼1 (𝛼1 − 𝛼2 )
(2.10)
which can in turn be rearranged to give the rotational
inertia of the main body:
𝐼1 =
𝐹𝑅
𝛼1 −𝛼2
(2.11)
Simplify EQ 2.04 to show that
𝜔𝑓 =
𝐼1 𝜔1𝑖
𝐼1 +𝐼2
(2.05)
Demonstrate your understanding to the demonstrator
by answering some questions for Checkpoint One.
if momentum is conserved.
15
Checkpoint Two: Figuring out the weight
force
1. Attach the scale pan to the scales as shown in the
left panel of Figure 2.2 and add weights until it reads
around 0.2 kg. Record the weight and its uncertainty as
m1.
2. Add the string and hang the weights over the pulley
as shown in the right panel of Figure 2.2 and measure
the weight again, m2.
3. Why are these results different and which one should
be used as the mass to obtain the force in equation
2.11? Think about how the scales are designed to work.
Figure 2.2 The two methods for measuring the weight.
- Starts collecting data (120 seconds).
- Stops data collection.
- Auto rescales the graph.
- Creates a fit line.
- Gives the value of a particular point.
Checkpoint Three: Finding the rotational
Figure 2.3 Logger Pro controls
inertia and frictional torque
1. Level the apparatus by using the spirit levels and
Checkpoint Four: What was conserved?
ensure that it is firmly clamped to the table.
Collecting the data
2. Get someone with printing credit to log into the
1. Spin up the main body to approximately 13 rads-1 with
computer, and follow the onscreen instructions to open
find a premade configuration of Logger Pro. Now is also
a good time to exchange email addresses so that
everyone in the group can access the graph and data.
2. Line up the removable disk, about 2 cm above the
3. Double click on the file to start Logger Pro with the
correct preloaded configuration.
4. Remove the removable disk, setting it aside. Spin up
the main body and collect some data (see Figure 2.3) to
familiarise yourself with the equipment. Note how the
disk naturally slows over time because of the frictional
torque.
the felt pad on top.
first, and drop it onto the spinning disk after 20 seconds.
Be sure you don’t impart any angular velocity to the disk
when you drop it, as this will affect your final results. The
graph should look something like Figure 2.3 if this was
done correctly.
3. Label the graph and print a copy for each lab book.
4. Label the angular velocity before and after the
collision.
5. Attach the string to the small hook and carefully wind
5. Measure the radius of the removable disk and use
it around the axle. Attach the weights to the string and
hang it over the pulley.
this to calculate its inertia using the equation on table 1.
6. Click collect, and after 10 seconds release the
6. Calculate the expected angular velocity after the
weight.
collision using the assumption that angular momentum
was conserved, including uncertainties.
7. Ensure that the weight disconnects when it has
7. Repeat this calculation assuming instead that the
unwound so the disk continues to spin freely.
angular kinetic energy was conserved, again including
uncertainties.
8. Once you have satisfactory data (see Figure 2.3) add
fit lines to both gradients (𝛼1 and 𝛼2 ).
9. Label the graph and print a copy for each book.
10. Calculate 𝐼1 and 𝜏𝑓 .
Save the data in case it is needed for a formal report.
16
8. Compare these results to the observed final angular
velocity and note which of the predictions was correct.
9. Calculate the rotational kinetic energy which was lost
in the collision.
Save the data in case it is needed for a formal report.
Figure 2.3 Showing sample data for Checkpoint Three and Four
Checkpoint Five: Conclusion and tidy up
Was there any linear kinetic energy involved in the
collision? If so, estimate its magnitude and explain
where it came from and where it went.
Write a suitable conclusion for the experiment, including
the aim, your results, and whether these results
matched your expectations and/or satisfied your aim.
See Checkpoint 5 in Experiment 0 and the sample
experiment in the appendix for more examples and
information on writing a good conclusion.
17
EXPERIMENT 3
LATENT AND SPECIFIC HEAT
Purpose
Welcome to your next lab session. In this lab you will find
the specific heat of a metal and the heat of vaporisation
and heat of fusion of water by heating and cooling
different materials, and measuring the heat flow
between them.
If a sample of ice at melting temperature is placed in
warm water, we could form the following heat balance
equation:
heat gained to melt ice + heat used to heat melted ice
+ heat lost by water = 0
(3.05)
Expressed mathematically, EQ 3.05 becomes:
𝑚𝑖𝑐𝑒 𝐿𝑓 + 𝑚𝑖𝑐𝑒 𝑐𝑤 𝛥𝑇𝑖𝑐𝑒 + 𝑚𝑤 𝑐𝑤 𝛥𝑇𝑤 = 0
Checkpoint One: What’s so specific?
(3.06)
Understanding the theory
which can be used to find the heat of fusion of water.
Specific heat capacity 𝑐 is a measure of the amount of
The latent heat of vaporisation 𝐿𝑣 describes the heat
required in a phase change from liquid to gas. The heat
required to vaporise a sample is given by EQ 3.07
below:
heat 𝑄 (required to change the temperature of a 1 kg
mass of a given material by 1K. Every material has a
unique specific heat capacity. Materials with a high
specific heat require more heat for their temperature to
be raised by the same amount as a material with lower
specific heat. The amount of heat required to change
the temperature of a sample is given by EQ 3.01 below:
𝑄 = 𝑚𝑐(𝑇𝑓 − 𝑇𝑖 )
(3.01)
How could we work out the specific heat of an unknown
sample? One method of doing so is to observe what
temperature increase the unknown sample undergoes
when we put a known quantity of heat into it. So if we
dip an sample of known mass and temperature into a
beaker of water of known mass and temperature, then
measure its final temperature (assuming no heat is lost
to the surroundings), we could calculate the sample's
specific heat, because:
heat lost by sample + heat gained by water = 0 (3.02)
Mathematically, EQ 3.02 is saying:
𝑚𝑠 𝑐𝑠 ∆𝑇𝑠 + 𝑚𝑤 𝑐𝑤 ∆𝑇𝑤 = 0
(3.03)
𝑄 = 𝑚𝐿𝑣
(3.07)
If a sample of steam at vaporisation temperature is piped
in warm water the following equation must hold:
heat lost to condense steam + heat lost to cool condense
steam + heat gained by water = 0
(3.08)
Expressed mathematically, EQ 3.08 becomes:
𝑚𝑠𝑡𝑒𝑎𝑚 𝐿𝑣 + 𝑚𝑠𝑡𝑒𝑎𝑚 𝑐𝑤 𝛥𝑇𝑠𝑡𝑒𝑎𝑚𝑤𝑎𝑡𝑒𝑟 + 𝑚𝑤 𝑐𝑤 𝛥𝑇𝑤 = 0
Symbol Quantity [Units]
𝑄
Heat energy [J]
𝑐
Specific heat [J/kg˚K]
𝐿𝑓
Latent heat of fusion [J/kg]
𝐿𝑣 Latent heat of vaporisation [J/kg]
𝑚 Mass [kg]
which can be rearranged for 𝑐𝑠 :
𝑐𝑠 =
𝑚𝑤 𝑐𝑤 ∆𝑇𝑤
𝑚𝑠 ∆𝑇𝑠
𝛥𝑇 = 𝑇𝑓 − 𝑇𝑖 Change in temperature [K]
(3.04)
to find the specific heat of the sample.
The heat of fusion 𝐿𝑓 is the amount of heat required to
change the phase of a substance at its melting point
from solid to liquid. The heat required to melt a sample
is given by EQ 3.05 below:
𝑄 = 𝑚𝐿𝑓
18
(3.04)
Table 3.1 Legend of useful quantities and their units
which can be used to find the latent heat of vaporisation
of water.
Complete your usual write up, then demonstrate your
understanding to the demonstrator by answering some
questions for Checkpoint One.
Value Quantity
Thermometer
4186 J/kg˚K Specific heat of water
385 J/kg˚K Specific heat of copper
897 J/kg˚K Specific heat of aluminium
134 J/kg˚K Specific heat of tungsten
129 J/kg˚K Specific heat of lead
333.55 kJ/kg Latent heat of fusion of water
2260 kJ/kg Latent heat of vaporisation of water
Table 3.2 Reference values for the latent heats of fusion and
vaporisation, and specific heat of water.
Figure 3.1 Showing the experimental set up as required for
Checkpoint Four
Checkpoint Two: Hot metal - determining
the specific heat of the unknown metal
1. Fill the steam generator about 3/4 full of water and
turn it on.
2. Pick one metal sample, and measure its mass. Also
measure the mass of the calorimeter.
3. Put cold water into the calorimeter, checking that it
covers the sample when it is suspended by the string
Make sure the sample is not resting on the bottom of the
calorimeter. Measure the mass of the calorimeter and
water together.
Checkpoint Three: Melting ice - determining
the latent heat of fusion of water
1. Measure the room temperature, 𝑇𝑟𝑜𝑜𝑚.
2. Half fill the calorimeter with water which is
approximately 15K above room temperature using the
hot water jug provided.
3. Measure 𝑚𝑐𝑤, the mass of the calorimeter and water.
4. Measure 𝑇𝑖 , the initial temperature of the water.
5. Slowly add a small handful of ice cubes to the water,
hang the sample in water and leave it there until it
reaches equilibrium (~2 minutes).
drying the ice as much as possible with a towel before
adding it. Stir until each handful has melted before
adding the next.
5. Measure the temperature of the cold water just
6. Stop adding ice once the temperature has gone
4. When the water in the steam generator is boiling,
before the heated sample is suspended in it. Be sure to
wipe the sample dry before it is placed in the cold water.
roughly as far below room temperature as it was initially
above and keep stirring until all of the ice has melted.
6. Gently raise and lower the sample in the cold water
7. Measure 𝑇𝑓. the final temperature of the water.
trying to avoid it hitting the bottom or rising above the
surface. Simultaneously stir the water with the
thermometer and measure the maximum temperature
which is reached.
Use this data to find the specific heat of the sample.
Calculate its uncertainty and compare this final value
with the accepted value.
8. Measure
𝑚𝑐𝑤𝑖 , the mass of the calorimeter, water,
and melted ice.
Use this data to find the heat of fusion of water.
Calculate the uncertainty and compare this final value
with the accepted value.
How does the specific heat of the sample compare with
that of water? Was any heat lost and if so where and
how? Discuss how heat loss affected the result.
19
Checkpoint Four: Condensing steam determining the latent heat of vaporisation
of water
1. Set up the steam generator as shown in Figure 3.1.
Position the tubing so that only the very tip will be
immersed in the water.
2. Add ice until the water in the calorimeter at least 10
K cooler than room temperature from Checkpoint Three.
3. Minimise the amount of water which condenses in the
tube and drips in water by turning up the power level so
that the steam is exiting at a rapid rate. Proceed once
the generator has been boiling for about a minute.
4. Measure the mass of the calorimeter and water 𝑚𝑐𝑤,
and the temperature of the water 𝑇𝑖 .
5. Place the lid on the calorimeter ensuring that the tip
of the tube is submersed and stir with the thermometer
until the temperature is about 10 K above room
temperature. The steam will likely make a loud
bubbling/popping sound as it enters the water - this is
the sound of the steam bubbles imploding, and is
normal.
6. Remove the tip from the water and keep stirring,
noting the maximum temperature 𝑇𝑓 .
7. Measure the mass of the calorimeter and water and
condensed steam, 𝑚𝑐𝑤𝑠 .
Use this data to calculate the heat of vaporisation of
water, and its uncertainty. Compare this final value with
the accepted value.
20
Checkpoint Five: Conclusion and questions
Once the experimental write up is finished, tidy away the
apparatus so the bench is ready for the next group.
Please ensure that the water is drained from the
apparatus and any spills have been cleaned up. Then
answer the following questions in the lab book.
1. Why is it important to avoid dripping boiling
water into the calorimeter during the heat of
vaporisation experiment?
2.
How is the measured value of 𝐿𝑣 affected if
some of the steam escapes into the air instead
of condensing in the water?
Write a suitable conclusion for the experiment, including
the aim, your results, and whether these results
matched the accepted values.
21
EXPERIMENT 4
ABSOLUTE ZERO
Purpose
Welcome to your next lab session. In this lab you will be
experimentally determining a numerical value for the
absolute zero of temperature using the ideal gas law.
Checkpoint One: Cold and colder understanding the theory
Symbol Quantity [Units]
IMPORTANT
𝑃 Pressure [Pa] or [N/m²]
Safety glasses must be worn at all times during this lab. Liquid
nitrogen is substantially
colder[m³]
than the surrounding objects in
𝑉 Volume
the lab. When it comes into contact with a warmer object, it
rapidly boils and 𝑛
causes
burns of
if it
comes into
extended contact
Number
molecules
[mol]
with eyes or skin.
𝑅 Gas constant, 8.314 [J/mol K]
Electronics will become damaged if exposed to rapid cooling.
Temperature
Do not place the 𝑇
thermometer
into[K]
the liquid nitrogen.
Why do we have an absolute zero? As matter cools, the
average kinetic energy of the molecules drops - the
molecules slide, spin, jiggle, and vibrate less and less as
it gets colder and colder. If a material made of atoms is
consistently cooled, it will solidify into a near-perfect
crystal state due to the molecules reducing to their
ground energy state - when this happens, every electron
is bound and the atoms no longer have sufficient energy
to move. This occurs -273.15 ˚C, which is the definition
of 0 K or absolute zero. To go below this would be mean
that the molecules are moving less than not at all - a bit
difficult to achieve.
Checkpoint Two: Setting up and collecting
The ideal gas law
2. Navigate
𝑃𝑉 = 𝑛𝑅𝑇
(4.01)
describes how a closed system of ideal gas responds
when pressure, number of moles and temperature is
varied. For a cooling system, the data can be
extrapolated to determine the absolute value of
temperature.
Looking at EQ 4.01, we can see that the temperature 𝑇
is linearly proportional to 𝑃 if 𝑅 , 𝑛 and 𝑉 are held
constant. So if we had a sample of gas, and measured
the resulting change in pressure due to a change in
temperature, we could extrapolate to calculate a value
for absolute zero.
Demonstrate your understanding to the demonstrator by
answering some questions for Checkpoint One.
Table 4.1 Legend of useful quantities and their units
data at atmospheric pressures
1. Log into the computer using an account which has
sufficient credit to print graphs for everyone in the
group.
to ‘Student\Class\PHYS\phys101\Ideal
Gas ’and double click on the top file. This will launch the
correct configuration of Logger Pro, which will include
the correctly formatted graphs to be used.
3. Check that the sensors are working by comparing the
output for temperature and pressure to the expected
atmospheric conditions. Normal room temperature will
be
~20˚C,
pressure
will
be
~103kPa.
4. Check that the temperature increases if you hold
your hand against the probe and the pressure drops
when the air is pumped out of the bulb.
5. Press the “Collect” button to start collecting data.
Note that data is only entered into the table when the
“Keep” button is pressed. Familiarise yourself with the
apparatus before starting the experiment as it is not
possible to remove data points once they have been
recorded.
Check your initial results with the demonstrator before
proceeding with the experiment
22
6. Set up the following so that they are full enough to
surround the bulb as much as possible:
• Beaker of hot water (~80°C)
• Beaker of warm water (30-40°C)
• Beaker of cold tap water
• Beaker of iced water
• Thermos flask of liquid nitrogen
7. Place the bulb and temperature probe into the beaker
with the ~80 ˚C water, and allow them to reach
equilibrium. Once this has occurred, press the “Keep”
button to enter the data in the table.
Checkpoint Four: Analysis to determine
absolute zero
You should now have a graph containing all three sets
of data. Add lines of best fit to each one by clicking and
dragging across each data line and pressing the fit line
button. Add a title using ‘graph options ’in the
options menu and rescale the axes to show where the
lines intersect (temperature from -300 - +100°C and
pressure from -20 – 120 kPa). Print a copy of this graph
for each person.
The fit lines will also display their corresponding
equations which are of the form:
8. Repeat Step 7 for each beaker of water, from hottest
𝑦 = 𝑚𝑥 + 𝑐 .
to coldest. Do not test the liquid nitrogen yet.
9. Place the bulb and not the temperature probe into the
liquid nitrogen flask and wait for it to reach equilibrium.
Click “Keep” to enter the data into the table.
10. Copy the four temperature values for water from
column 2 into column 3 (scroll sideways on the table)
and add the temperature of liquid nitrogen (-196°C)
manually. Check that the trendline is approximately
linear.
11. Press stop and then save the latest run by clicking
Experiment —> Store latest run in the experiment menu.
Checkpoint Three: Collecting data below
atmospheric pressures
We’ll now be repeating the above procedure twice, once
with the glass bulb partially evacuated so the pressure
is 25 kPa below atmospheric pressure, and again with
the pressure 50kPa below atmospheric pressure.
1. Place the glass bulb into the beaker of warm water
and bring it to room temperature. Open the valve, and
evacuate the bulb using the hand pump. Ensure the
valve is closed again when the desired pressure is
reached.
2. Repeat the procedure again using all of the beakers,
including the liquid nitrogen.
3. Save the run as above.
4. Repeat Checkpoint Three again with the bulb
evacuated to the pressure is 50 kPa below atmospheric
pressure.
(4.02)
Equate each of these pairs of lines and solve them to
find the temperature where they intersect. Find the
average of these three values of absolute zero and use
half the range in the data as an estimate of the
uncertainty.
Now, remove the liquid nitrogen points and recalculate
absolute zero of temperature. To do this, go back into
the data tables and highlight the liquid nitrogen data.
Then select ‘strike through ’in the edit menu. Delete the
old regression lines and insert new ones before
repeating the steps above to get another set of values
for absolute zero where the liquid nitrogen data is
ignored.
Compare the two values of the absolute zero of
temperature. Discuss with your group whether the your
result without the liquid nitrogen data is accurate.
Checkpoint
Five:
Conclusions,
and
questions.
Please empty the water and ice into the sink and ensure
any spills have been cleaned up so the bench is ready
for the next group. Then answer the following questions
in the lab book before asking the demonstrator for your
final checkpoint.
1. Why does the bulb have to be surrounded by ice
and water? Would ice give a 0°C reading?
2. Why are the three slopes so different on these
graphs?
Write a suitable conclusion for the experiment, including
the aim, your results, and whether these results
matched the accepted value.
23
EXPERIMENT 5
CAPACITANCE
capacitance, and use this plot to determine the
dielectric constant of Mylar and paper.
Purpose
Welcome to your next lab. In this lab you will build a
capacitor and use it to understand how the capacitance
of a parallel plate capacitor varies with the area of the
plate, and the plate separation. You will use this data to
determine the dielectric constants of Mylar and paper.
You will also learn about commercial capacitors and
their manufacturing tolerances.
Complete your usual write up, then demonstrate your
understanding to the demonstrator by answering some
questions for Checkpoint One.
Checkpoint One: Stored up charge understanding the theory
A capacitor is a device that can store charge by
maintaining an electric field between two plates of an
electrically conductive material. Capacitors are found in
many electronics including filters, power supplies and
cell phones. They are often the source of the buzzing
one can hear in household electronics. Capacitance is
defined as:
𝐶=
𝑄
𝑉
Note that equation 5.02 ignores real world
complications such as fringing fields which are the
portions of the electric field that extend beyond the edge
of the plates. This can become significant when the
perimeter to area ratio of the two plates becomes too
large. This is a problem as the fringing fields can lead to
an apparent increase of capacitance.
(5.01)
Checkpoint Two: Testing commercial
capacitors
Before we construct our own capacitor, we will first
measure some commercially available capacitors.
Generally, there are two naming schemes that are
employed by the industry. The first is shown in Figure
5.01, the second in Figure 5.02.
i.e., capacitance is a measure of how much charge a
capacitor can store for given applied voltage. The
capacitance of an ideal, parallel-plate capacitor is
defined as:
𝐶=𝜅
𝜖0 𝐴
𝑑
(5.02)
An important factor in capacitance is the dielectric used.
A dielectric is the material that fills the space between
the plates, and can increase or decrease the
capacitance depending on its dielectric constant 𝜅 . The
dielectric constant of vacuum is 1. We can also see that
capacitance increases as the area of the plates 𝐴
increases, but decreases as their separation 𝑑
increases. In this experiment we will be changing these
variables. We will then plot the resulting change in
Figure 5.01 Showing one common naming scheme for commercial
capacitors. For example, 204 K 300V would be read as 20 ± 2 x 104
pF at 300V. When no tolerance is given, the standard is 20%
Symbol Quantity [Units]
𝑄
Charge [Coulomb]
𝑉 Voltage [V]
𝐶 Capacitance [Farad] or [C/V]
𝜅 Dielectric constant
𝐴 Area [m²]
𝜖0 Permittivity of free space [8.854 x 10⁻¹² F/m]
Table 5.1 Legend of useful quantities and their units
24
Figure 5.02 Showing another common naming scheme for
commercial capacitors. For example, 700n M 300V would be read as
700 ± 140 nF at 300V.
Figure 5.01 The experimental set up for Checkpoints Three and
Four.
Make a table in your lab book with the following five
columns:
• Description of Capacitor
• Stated Value
• Measured Value
• Stated Tolerance
• Measured Tolerance
5. Turn on the LCR meter, and select an appropriate
range. Zero the meter using the small calibration dial at
the bottom left. Ensure that it is set to measure
capacitance.
6. Measure each of the capacitors by attaching the LCR
meter to each capacitor’s metal pins. Make sure to first
ground the capacitor by touching the pins on the copper
plates in front of you, then measure the capacitance of
each and record the results in your table.
7. Calculate the percentage difference between the
stated and measured value.
Do your measured value agree with the stated values
within tolerance?
Show your results to a demonstrator.
Checkpoint Three: Building a capacitor with
varying area to determine the dielectric
constant of mylar"
Create a capacitor using the two copper PCBs as
plates, separated by a sheet of 110 micron thick Mylar.
The dielectric constant of Mylar is 2.2. To measure this
value, complete the following
1. Copy Table 5.01 below into Excel.
25
2. Place one plate on the bench with the copper side
facing up. Align the Mylar sheet to maximise the area
while ensuring the two copper plates do not touch.
Measurement One
Height
(m)
Position
Width
(m)
Area
(m2)
Checkpoint Four: Building a capacitor with
Measurement Two
𝐶 (nF)
Height
(m)
Width
(m)
Area
(m2)
Averages and Results
𝐶 (nF)
Height
(m)
δwidth
(m)
Width
(m)
δheight
(m)
Area
(m2)
δArea
(m)
𝐶 (nF)
A4 page
¾ page
½ page
¼ page
⅛ page
Table 5.01 Required information to determine the capacitance of the parallel plate and dielectric constant of Mylar.
varying plate separation to determine the
3. Place the other plate on top, copper side facing
dielectric constant of paper
down, and place the lab book and paving stone on top
while maintaining the alignment of the plates.
Rebuild the capacitor with paper as the dielectric, and
repeat the previous experiment while varying plate
separation instead of plate area. Paper has a thickness
of 110μm, and a dielectric constant of 𝜅 = 2.0.
4. Carefully measure the height and width of the
overlapping area.
1. Set up the capacitor with two sheets of paper as the
below.
dielectric and use the lab book and paving stone to
maintain the alignment of the layers.
6. Discharge the capacitor, and then reduce its size to
2. Build each of the five different capacitors described
5. Measure the capacitance and record this in the table
approximately ¾ of a page. This is easiest to do if you
keep the height constant and simply change the width.
in Table 5.02. Measure the corresponding capacitance.
7. Repeat Step 6 for each capacitor size. Once
separations have been measured, rebuild and measure
each capacitor a second time.
capacitance of the five different areas have been
measured, rebuild and measure each capacitor a
second time. This is important to get a realistic estimate
of the uncertainty in the capacitance.
8. Calculate averages for each measured quantity. Be
sure to make use of Excel’s inbuilt functions.
9. Calculate the uncertainties for each of these
averages using the uncertainty rules. Refer to page 50.
10. Re-arrange EQ 5.02 to plot a graph with a gradient
that will allow the dielectric constant of Mylar to be
calculated.
11.Estimate the uncertainty of the value obtained for
the dielectric constant using lines of best and worst fit.
Does your final value for the dielectric constant of Mylar
agree with the accepted value?
26
3. Once capacitance of the five different plate
4. Complete the remaining calculations in the table.
±𝐶
(nF)
5. Re-arrange EQ 5.02 to plot a graph with a gradient
that will allow the dielectric constant of paper, and its
uncertainty, to be calculated.
Does your measurement of the dielectric constant of
paper agree within uncertainty with the accepted value?
Checkpoint Five: Exercises and Conclusion
Once the experimental write up is finished, tidy away the
apparatus so the bench is ready for the next group. Then
answer the following question in the lab book.
1. How could this setup be used to identify an
unknown material and what parameters should
be altered to get the best results?
2. Is it physically reasonable to have a κ of less
than 1?
3. Does the method carried out in Checkpoint
Three give an accurate measurement of the
capacitance of a parallel plate capacitor and if
not, why not?
Write a suitable conclusion for the experiment, including
the aim, your results, and whether these results
matched the accepted value.
# of pages
Plate Separation 𝑑 (m)
1Τ𝑑
𝐶 (nF)
𝐶 (nF)
(m-1)
(Measurement One)
(Measurement Two)
𝐶𝑎𝑣𝑒𝑟𝑎𝑔𝑒 (nF)
±𝐶 (nF)
2
4
8
16
24
Table 5.02 Required information to determine the capacitance of the parallel plate and dielectric constant of paper.
27
28
EXPERIMENT 6
SUPERCONDUCTIVITY
and would have very wide-reaching applications. There
is much research still to be done in this field.
Welcome to your next lab session. In this lab you will get
to experiment with superconductors and liquid nitrogen.
You will learn about the Meissner effect, and use it to
levitate a magnet above a superconductor. You will do
this in order to determine the critical temperature of a
superconductor and look at its characteristic curve. You
will then compare this to the characteristic curve of
copper.
Checkpoint One: Zero resistance understanding the theory
The resistance (measured in Ohms, or 𝛺 ) of an object
is a measure of how well current (moving charged
particles) can flow through it. A high resistance means
electricity can not flow through the object easily (it
“resists” the current) - a higher voltage would be
required to produce the same current flow compared to
an object with low resistance. This relationship is
described by Ohm’s Law:
𝑉 = 𝐼𝑅
(6.01)
Most wires that we use in electronics today have some
amount of resistance. This is a problem, as energy is
wasted as heat in the process of powering other
components.
In contrast, a superconductor is a material that exhibits
zero resistance below a certain temperature, called its
critical temperature 𝑇𝑐 . Superconductivity is a relatively
new phenomena in physics. It was first discovered in
mercury in 1911, but only appears when the mercury is
a few degrees above absolute zero. In 1986, IBM Labs
in Switzerland discovered the first High Temperature
Superconductors
(HTS),
which
become
superconducting when immersed in liquid nitrogen (77
K). This breakthrough allowed superconductivity
applications to be cheaper and more accessible, but the
critical temperature is still too low to be suitable for
everyday technology.
A fundamental property of a superconductor is that
when it transitions into a superconducting state, all
internal magnetic fields disappear and it will deflect any
external magnetic fields. This is the Meissner effect - the
expulsion of all magnetic fields from within the material.
A side effect of this is that a magnet will levitate if it is
placed above it, as the external magnetic field of the
magnet can not enter it, so are bent and deflected away,
just like they would in the presence of another magnet.
This is an easy way of identifying if a material is a superconductor.
The resistance of all material changes with changing
temperature. How it changes is called its characteristic
curve. An example is shown below.
Resistance
Purpose
𝑇𝑐
Temperature
Figure 6.1 Typical characteristic curve for a
superconductor. Notice the sharp drop in resistance
once the critical temperature is reached.
The superconductor used in this lab is a ceramic called
Perovskite which has the chemical formula:
Y Ba2 Cu3 O7
The problem with most HTS is that they are brittle or
inflexible which makes them impractical for use in
cables. In this lab you will be experimenting with a
flexible superconducting wire, developed in New
Zealand as a possible solution to this problem.
In 2020, researchers at the University of Rochester
became the first in the world to successfully develop a
room
temperature
super-conductor
called
carbonaceous sulphur hydride, with a critical
temperature of 15 ˚C. Unfortunately, the material only
exhibits superconductivity at pressures of around 267
Gigapascal, or about three quarters the pressure found
in the centre of the Earth. An easy-to-use and
manufacture room-temperature super-conductor is
considered the holy grail in renewable energy storage,
29
Temp. (K)
60
70
80
90
100
110
120
130
140
150
160
170
180
190
200
210
220
230
240
250
260
270
280
290
0
1
2
3
4
5
6
7
8
9
7.60
7.53
7.46
7.40
7.33
7.26
7.19
7.12
7.05
6.99
6.92
6.85
6.78
6.71
6.64
6.56
6.49
6.42
6.37
6.33
6.29
6.25
6.21
6.17
6.13
6.09
6.05
6.01
5.97
5.93
5.90
5.86
5.83
5.79
5.75
5.72
5.68
5.64
5.60
5.56
5.52
5.48
5.44
5.41
5.37
5.34
5.30
5.27
5.23
5.20
5.16
5.13
5.09
5.06
5.02
4.99
4.95
4.91
4.88
4.84
4.81
4.77
4.74
4.70
4.67
4.63
4.60
4.56
4.53
4.49
4.46
4.42
4.39
4.35
4.32
4.28
4.25
4.21
4.18
4.14
4.11
4.07
4.04
4.00
3.97
3.93
3.90
3.86
3.83
3.79
3.76
3.73
3.69
3.66
3.63
3.60
3.56
3.53
3.50
3.47
3.43
3.40
3.37
3.34
3.30
3.27
3.24
3.21
3.18
3.15
3.12
3.09
3.06
3.03
3.00
2.97
2.94
2.91
2.88
2.85
2.82
2.79
2.76
2.73
2.70
2.67
2.64
2.61
2.58
2.53
2.52
2.49
2.46
2.43
2.40
2.37
2.34
2.31
2.29
2.26
2.23
2.20
2.17
2.14
2.11
2.08
2.05
2.02
1.99
1.96
1.93
1.90
1.87
1.84
1.81
1.78
1.75
1.72
1.69
1.66
1.64
1.61
1.59
1.56
1.54
1.51
1.49
1.46
1.44
1.41
1.39
1.36
1.34
1.31
1.29
1.26
1.24
1.21
1.19
1.16
1.14
1.11
1.09
1.07
1.04
1.02
0.99
0.97
0.94
0.92
0.89
0.87
0.84
0.82
0.79
0.77
0.74
0.72
0.69
0.67
0.65
0.62
0.60
0.58
0.55
0.53
0.50
0.48
0.45
0.42
0.40
0.38
0.36
0.34
0.32
0.30
0.28
0.26
0.24
0.22
0.20
0.18
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
-0.02
-0.04
-0.06
-0.08
-0.10
-0.12
-0.14
-0.16
-0.18
Table 6.1 To find the critical temperature of the superconductor, locate the closest voltage value (in mV) inside the grey
area of the table to your observed value. The row corresponds to a range of temperatures within 10 K, the rows to the
first digit within that range. Example: a voltage of 1.72mV corresponds to a temperature of 217 K.
Complete your usual write up, then demonstrate your
understanding to the demonstrator by answering some
questions for Checkpoint One.
IMPORTANT
Safety glasses must be worn at all times during this lab. Liquid
nitrogen is substantially colder than the surrounding objects in
the lab. When it comes into contact with a warmer object, it
rapidly boils and causes burns if it comes into extended contact
with eyes or skin.
Checkpoint Two: Determining the critical
temperature of Perovskite
1. Attach the two leads of the superconductor in the
aluminium holder to one of the blue Minipa multimeters,
one to Com (ground), and the other to the voltage
terminal, and set the meter to measure mV.
3. Wait for it to come into thermal equilibrium with the
liquid nitrogen (approximately two minutes, or when the
boiling reduces), then lift the superconductor holder by
the pins out of the liquid nitrogen with the tweezers, and
place it flat on the bench.
4. Carefully position the magnet to levitate above the
superconductor with the tweezers. This can be tricky to
achieve, but be persistent.
5. Immediately when the superconductor ceases
levitating, note the voltage on the multimeter and use
this to find the critical temperature of the
superconductor using Table 6.1 above.
6. Repeat this measurement five times, discard any
outlying values or results, and calculate the average
critical temperature of the superconductor. Calculate its
uncertainty from the half the range in the values you
have acquired. Show this result to a demonstrator.
2. Fill the polystyrene tray with liquid nitrogen and
Checkpoint Three: Determining the
submerge the superconductor entirely.
characteristic curve of a superconductor
Although we are trying to obtain the characteristic
curves of a superconductor and copper, it is not
30
Figure 6.1 The experimental setup for determining the characteristic curve of a sample. Note this diagram is
not to scale.
possible to measure the very small resistances involved
in this experiment - instead, you’ll be recording both
current and voltage as the temperature changes, then
calculating the resistance from EQ 6.01.
9. Plot the characteristic curve of the superconductor
to determine the critical temperature, and estimate its
uncertainty.
You can either start with Checkpoint Three, or
Checkpoint Four, and do the other once complete with
the first.
How does the critical temperature found compare to
that found in Checkpoint Two?
1. Set up the experiment as shown in Figure 6.01 below.
characteristic curve of copper
Use the blue Minipa multimeters for the voltage
readings, and the yellow meter for the current readings.
Set up the three multimeters side by side, as you will be
recording their screens using your cellphone camera.
2. Submerge the wire into the liquid nitrogen and wait
for it to reach thermal equilibrium (approximately three
minutes, or when the boiling reduces).
3. Turn on the power supply and check that you have
some measure of current, approximately 0.6A.
4. Check that when the superconducting wire is
removed that the voltages start to change smoothly. If
this does not occur, check with a demonstrator that the
experiment is correctly set up.
5. Start recording a video of the three multimeters, wait
five seconds, then take the sample out of the liquid
nitrogen. Record the multimeters for ~3 minutes, or until
the figures on the multimeters stop changing rapidly.
6. In an Excel spreadsheet, set up five columns for your
three recorded values (temperature probe voltage,
voltage probe voltage, current), and two resulting values
(the temperature, from Table 6.1, and the resistance).
7. Review the first minute of the video and extract the
voltage values at roughly five second intervals. For the
remaining four minutes, extract the voltage values at
roughly fifteen second intervals.
8.Convert the temperature probe values from mV to K
Checkpoint Four: Determining the
1. Repeat the procedure from Checkpoint Three using
the copper sample instead of the superconductor.
Why does the characteristic curve of the copper differ
from the superconductor?
Checkpoint
Five:
Conclusion
and
Exercises
Tidy up the bench so that it is ready for the next group.
Then attempt the following exercises.
1. A toroid of superconducting material can be used to
store electrical power, because the current will continue
to flow. This persistent current induces a magnetic field
around the device which is described by Ampere’s law
for a current inside a toroid:
3
𝐼𝑡 =
2𝐵𝑡𝑜𝑟𝑜𝑖𝑑 (𝑥 2 +𝑟 2 )2
𝜇0 𝑟 2
(6.02)
The field strength can be measured using a compass,
this describes the combined force from the toroid and
the horizontal component of earth’s magnetic field which
can be related as:
𝐵𝑡𝑜𝑟𝑜𝑖𝑑
𝐵𝑒𝑎𝑟𝑡ℎ
= 𝑡𝑎𝑛 𝜃.
(6.03)
Combine these equations, and calculate the current in the
toroid. Relevant constants can be found on the opposite
page, and at the front of the book.
using Table 6.1. Use an average of the current to
convert the voltage probe values to resistance.
31
2. Some of the current will decay within the
superconducting toroid. This happens because of flux
use this technology to store power to serve the public
electricity grid?
Symbol Quantity [Units]
𝐵𝑡𝑜𝑟𝑜𝑖𝑑 Magnetic field due to the current in the toroid [Tesla]
𝐵𝑒𝑎𝑟𝑡ℎ The horizontal component of the Earth’s magnetic field [Tesla]
𝑥 Distance from toroid [m]
𝑟 Radius of toroid [m]
𝜇0 Magnetic permeability of free space [N/A²]
𝐼𝑡 Current in toroid [A]
Table 6.2 Legend of useful quantities and their units for Checkpoint Five.
creep and flux flow. This decay rate is described by:
Write a suitable conclusion for the experiment, including
Quantity Quantity [Unit]
𝑟 9.5 x 10-3 m
𝑥 13 x 10-3 m
𝜃 10˚
𝑅 1 x 10-15 Ω
𝐿 5 nH
Table 6.3 Legend of useful quantities and their units for Checkpoint Five.
𝑅
𝑓 = 𝑒 −( 𝐿
)𝑡
(6.04)
where 𝑓 is the fraction of power that remains.
Calculate how long it the power to decay by half. Is this
decay time sufficient for a utility company that wants to
32
the aim, your results, and whether these results
matched within uncertainty, and matched the accepted
value.
APPENDICES
33
APPENDIX A
SAMPLE EXPERIMENT: THE LASER
The size of the airy disk produced by a single small
hole in an opaque screen is given by
Purpose
In this lab, you will be determining the wavelength of
light emitted by a HeNe laser and find the diameter of
a Lycopodium spore.
Checkpoint One: Introduction to lasers and
their applications
𝑟 = 1.22𝜆
𝐿
𝑑
(1.2)
This pattern can be intensified by using a large number
of small holes instead of one. Babinet’s theorem states
Variable Meaning
Lasers produce monochromatic (single wavelength),
coherent (in phase) light which has a lot of applications
in industry and research. The wavelength of a laser
can be found using the diffraction pattern which is
produced when the beam is reflected off a ruler onto a
wall.
𝑟 Radius of first dark ring
𝜆 Wavelength [m]
L Distance from screen to hole [m]
𝑑 Diameter of the hole [m]
Table 1.1 Legend of variables for Equation 1.2
that the holes can be replaced by small opaque dots
on a transparent screen. This makes it possible to
measure the diameter of the particles in a powder.
Checkpoint Two: Experimental procedure
to determine laser wavelength
Figure 1.1 Showing a schematic of the diffraction pattern produced
by a laser reflecting off a ruler.
1. Set up the laser so it is facing a wall before turning
it on.
The relation between the angles at which the peaks in
intensity of light appear, their order, and the
wavelength of the light is described by the equation:
𝑚𝜆 = 𝑑(𝑐𝑜𝑠𝜓 − 𝑐𝑜𝑠𝜙𝑚 )
Variable Meaning
𝑚 Order number
𝜆 Wavelength [m]
𝜙𝑚 Angle between order 0 and order m
𝜓
½ angle between order 0 and direct
beam
Table 1.1 Legend of variables for Equation 1.1
34
(1.1)
2. Attach a piece of paper to the wall to collect data
and mark the point where the laser hits without
deflection.
3. Set up the ruler so it has a very small angle from the
horizontal. Find m0 the point where the laser is
reflected if no interference occurred. This can be
done by reflecting the beam off the un-ruled surface
of the ruler.
4. Align the ruler so the laser light is reflected from the
millimetre ruled area of the surface and use this
alignment for the remainder of the experiment.
5. Measure L, the distance between the reflection
point and the wall. Carefully consider the size of the
uncertainty for this measurement.
6. Mark each of the bright spots on the paper.
7. Turn off the laser and remove the paper from the
wall. Measure the values of h and sm and calculate
ψ.
𝑠𝑚 ±
Order
[m]
𝜙+𝜓 [˚]
𝛿(𝜙 + 𝜓) [ ˚ ]
𝜙 [˚]
𝛿𝜙 [ ˚ ]
𝑐𝑜𝑠(𝜙)
𝑐𝑜𝑠(𝜙)
1
2
3
4
5
Table 5.02 Required information to determine the wavelength of light produced by a HeNe laser.
Checkpoint Three: Analysis to determine
laser wavelength
1. Make a copy of Table 5.02 and calculate all of the
quantities and uncertainties needed to complete
the analysis. Use tan-1 (
𝑠
) to find (Φ + ψ).
𝐿
2. Rearrange Equation 1.1 to the standard linear form
and plot a graph to find the wavelength and its
uncertainty.
Checkpoint Five: Conclusion and tidy up
Once the experimental write up is finished, tidy away
the apparatus so the bench is ready for the next group.
Then answer the following questions in the lab book.
1. Laser beams are often visible in movies, why does
this not happen here? Suggest something which
could be used to make the beam visible.
3. Compare this result with the accepted wavelength
of a HeNe laser of 632.8 nm.
Checkpoint Four: Determining the
diameter of a Lycopodium spore
1. Place a new piece of paper on the wall and replace
the ruler with a slide containing Lycopodium
powder. This slide should be placed perpendicular
to the beam from the laser. Trace the fringe of the
first dark band and turn off the laser. Measure the
diameter of the ring, being sure to note a
reasonable uncertainty
2. Calculate the diameter of the Lycopodium powder
and its uncertainty and compare it to the accepted
value of 33 μm.
35
Sample Write Up
36
37
38
39
40
41
42
43
44
45
APPENDIX B: UNCERTAINTIES
What are uncertainties and why are they so important?
The uncertainty of a measured value expresses how accurately it was measured. For example a brick’s length may be
stated as 14.3 ±0.2 cm. This means that the brick is 14.3 cm long but could be 0.2 cm longer or shorter.
FIGURE B.1 THE LONGEST AND SHORTEST LENGTH OF THE BRICK.
Thus the brick could be somewhere between 14.1 cm and 14.5 cm long.
All measurements which are taken in the lab have some form of inaccuracy, this results in uncertainty in the data. When
this data is used to make calculations, the uncertainties need to be propagated through the analysis to ensure that the
final value is meaningful and expresses its accuracy. Scientists need to verify their results by comparing them with
other, independent measurements, which is impossible if the uncertainties are not included in the analysis.
Consider the following example.
One group of scientists has spent years developing a new system to measure distances using a pulse of sound and
another group has achieved the same measurement using a laser. They want to validate their new technology and
both measure the distance between two buildings which have a well-established separation of 57.32 m. They each
take a measurement of this distance, the sound group claims the distance to be 57.35 m and the laser group 57.31
m. As it stands, these results do not agree and so at least two of the results must be wrong. However, an important
element is missing: The uncertainties. When these are considered, the accepted separation is defined as 57.32 ±
0.02m. The laser group measured 57.31 ± 0.01m and the sound group measured a value of 57.35 ± 0.04 m, so all
three results actually agree within the limits of their respective experimental uncertainties.
FIGURE B.2 MEASUREMENTS OBTAINED USING DIFFERENT TECHNIQUES AND THEIR AREA OF
AGREEMENT.
Accurate vs. Precise
These terms are often used interchangeably - however, in science they have very specific meanings. If a measurement
is accurate, it is close to the “true” value, what that measurement, be it a length, weight mass, etc actually is in reality.
If measurement is precise, it means the spread of the results is small. This translates to the size of the uncertainty; high
precision means low uncertainty. Low precision, high uncertainty.
46
Sources of Uncertainty
There are two main sources of uncertainty: Systematic Uncertainties and Random Uncertainties.
Systematic Uncertainties
Systematic uncertainties occur due to systematic errors in the equipment or methodology and tend to be biased, i.e.
they add a consistent error/variation to the measurement. This could be due to:
• Poorly calibrated machines – e.g. the zero mark does not correspond to zero.
• Inaccurate equipment – such as a stop watch running slow, or ruler which is longer or shorter than it claims.
• Consistent improper reading or output – such as misreading a dial or screen either through parallax or
misunderstanding of the scale etc.
• Assumptions that are made – such as ignoring friction or heat loss during an experiment.
Taking multiple measurements will not reduce these errors and they will not be detected until the analysis is done. If
systematic uncertainties are discovered, they should be discussed in the conclusion.
Random Uncertainties
These uncertainties occur because of inherent variability when the measurement is recorded, such as measuring the
period of a pendulum. Both the reaction time of the person and inability to determine exactly where the end of the swing
is will create some variability in the measured result. These results tend to be unbiased, i.e. they add a randomly varying
error/variation to the measured values, and are often caused by:
• Inability to reproduce measurements - due to unclear scales or edges when measuring position or distance,
human reaction time in time measurements or limitations in the accuracy of the equipment.
• Uncontrolled changes in the lab environment such as changes in temperature, light level, or humidity.
Random uncertainties can be reduced if multiple measurements are taken and they can be estimated as the data is
collected.
Estimating Uncertainties on Measurements
At high school students may learn to use half of the smallest division of the instrument as the uncertainty. This is a vast
oversimplification. At the university level students are expected to gain a better understanding of how uncertainties
should be estimated.
There are three main factors which need to be considered when estimating uncertainties:
• The inability to accurately determine what is being measured.
• Human factors such as reaction time and inability to use devices to their full potential.
• The limitations of the measuring device i.e. its accuracy.
Sometimes it can be difficult to determine exactly what is being measured. A common example is measuring the length
of an object, in this case it can be hard to define the edge because of an intrinsic roughness or because it is askew.
Measuring time is another good example; in a swinging pendulum it can be hard to determine exactly when a period
has ended.
The measuring device, and most importantly the user, produce significant uncertainties. Most devices have
manufacturer stated accuracies but this is often only a small part of the overall uncertainty. Consider the following
examples:
• A stop watch will have an intrinsic uncertainty which is much smaller than human reaction time and so the
measured time will usually have a much larger uncertainty than the limitations of the stop watch.
• A measurement with a ruler will usually have a greater uncertainty than half of the smallest division because
the end of the ruler is worn, the markings on the ruler are uneven and relatively thick or because of parallax.
• Another consideration to make is that many devices will fluctuate their measurements with changes in
temperature, such as rulers expanding as the material warms. These random changes should also be reflected
in the estimated uncertainty.
Note that most digital devices which are used in the first year lab have an uncertainty of ±1 of the last decimal place
which is displayed, but user induced errors will make them much larger.
For most measurements the following can be used to determine the uncertainty of the measured value:
• When a single measurement is made, the uncertainty is estimated by adding the contributions from each of
the factors mentioned above.
47
•
When a measurement is repeated the uncertainty of each measurement can be estimated by determining half
of the range of the data.
Working with uncertainties
Uncertainties can be written in two forms: Absolute Uncertainties (δ) and Relative Uncertainties (%δ). An absolute
uncertainty states how large the error is in the same unit as the measurement, for example a building is measured to
be 12.37 ± 0.05 m which means the measurement has an uncertainty of 0.05 m. A relative uncertainty expresses the
error as a percentage of the value, for example: 23.74 m ± 5.7 % which is equivalent to an uncertainty of 1.4 m.
These are used for uncertainty analysis only, all final results should be presented with absolute uncertainties.
The Greek symbol δ (or “del”) can be read as “uncertainty of”, so “δx” can be read as “the uncertainty in the quantity
x”, or “the uncertainty in x” in abbreviated form. %δx is then the percentage uncertainty in x.
Converting Uncertainties
To convert an absolute uncertainty to a percentage uncertainty:
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑈𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑡𝑦 =
𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑈𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑡𝑦
𝑉𝑎𝑙𝑢𝑒
× 100
Or in symbolic form:
%𝛿 =
𝛿𝑥
𝑥
× 100
To convert a percentage uncertainty to an absolute uncertainty:
𝛿𝑥 =
𝑥
100
× %𝛿
Example: what is the percentage uncertainty on the building which is 12.37 ± 0.05 m long?
0.05
× 100 = 0.4%
12.37
Converting this back to absolute uncertainties gives
12.37
× 0.4 = 0.05𝑚
100
Writing Uncertainties
All measurements should be recorded with their Absolute Uncertainties (as discussed above). It is important to note
that the uncertainty have to have the same number of significant figures as the number and should be written as
follows.
𝑥 = 135.42 ± 0.01 𝑚
𝑦 = 2.335 ± 0.004 𝑘𝑔
𝑟 = (2.3 ± 0.2) × 103𝑁
The value can never be more accurate than the uncertainty. For example, it makes no sense to write x = 17.58 ± 5.5
m. This is like stating that its takes 17 minutes and 35 seconds to walk to university give or take about 5 minutes.
The correct form is x = 17.5 ± 5.5 m.
Calculating uncertainties
Once the uncertainties of the measurements have been determined they need to be propagated through the analysis
so that an uncertainty can be determined for the final value being calculated.
48
Addition and Subtraction
When numbers are added or subtracted their Absolute Uncertainties are added together. This is because both parts
contribute to the final uncertainty. Even in the case of subtraction, it makes no sense if the result became more accurate
after a second uncertainty had made a contribution. For example using x = 13.2 ± 0.5 m, y = 34.6 ± 0.1
𝑧= 𝑥 + 𝑦
= (𝑥 + 𝑦) ± (𝛿𝑥 + 𝛿𝑦)
= (13.2 + 34.6) ± (0.5 + 0.1)
= 47.8 ± 0.6 𝑚
δ = “uncertainty of”
If we had:
so
δx
is the uncertainty of x
𝑧= 𝑥– 𝑦
= (𝑥 − 𝑦) ± (𝛿𝑥 + 𝛿𝑦)
= (34.6 − 13.2) ± (0.1 + 0.5)
= 21.4 ± 0.6 𝑚
The uncertainty on both of these is the same even though the result varies.
FIGURE B.3 ADDITION OF UNCERTAINTIES.
In the case of addition it is clear why the uncertainties are added together in order to find the total uncertainty on z.
FIGURE B.4 SUBTRACTION OF UNCERTAINTIES.
In the case of subtraction, it is clear that the uncertainties still need to be added to ensure the full possible range of
lengths for z is calculated.
It should be noted from the diagrams above that the value of δz is the same for both situations which agrees with the
equations above.
49
Division and Multiplication
When multiplying or dividing quantities, the Percentage Uncertainties (%𝜹) are added. As an example, consider finding
the velocity of an object moving along across a known distance 𝑑 for a known amount of time 𝑡:
𝑑 = 1.34 ± 0.03 𝑚
𝑡 = 2.7 ± 0.2 𝑠
We calculate a velocity of:
𝑣=
𝑑 1.34
=
= 0.496𝑚𝑠 −1
𝑡
2.7
The uncertainty in velocity, 𝛿𝑣 , is calculated by adding the uncertainties of the distance and velocity as percentages:
𝛿𝑣 = %𝛿𝑣 × 𝑣
𝑑
= (%𝛿𝑑 + %𝛿𝑡) ×
𝑡
When a value is squared the
Working out percentage uncertainties:
percentage uncertainty has to be
0.03
doubled because it is the same as
𝛿𝑑 =
× 100% = 2.2%
1.34
multiplying the number in twice.
0.2
𝛿𝑡 =
× 100% = 7.4%
2.7
Add percentage uncertainties to find the total:
%𝛿𝑣 = 2.2% + 7.4% = 9.6%
Convert back to absolute uncertainties for the final result:
𝛿𝑣 = %𝛿𝑣 × 𝑣 =
Thus the velocity with its associated uncertainty is:
9.6
100
× 0.496 = 0.05𝑚𝑠 −1
𝑣 = 0.496 ± 0.05 𝑚s −1
Multiplying by a Constant
When multiplying by a value with no uncertainty the percentage uncertainty remains constant.
Example 1:
𝐹 = 𝑚𝑎
𝑚 = 2.3 ± 0.2 𝑘𝑔
= 2.3𝑘𝑔 ± 8.7%
𝑎 = 9.81𝑚𝑠 −2 (𝑛𝑜 𝑢𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑡𝑦 𝑔𝑖𝑣𝑒𝑛)
thus
𝐹 = 2.3 × 9.81
= 22.563 𝑁 ± 8.7 %
= 23 ± 2 𝑁
Example 2:
∆𝐸 = 𝑚𝑔∆ℎ
𝑚 = 53 ± 3 𝑘𝑔 or 𝑚 = 53 𝑘𝑔 ± 5.7%
𝛥ℎ = 150 ± 10 𝑚 or ∆ℎ = 150 𝑚 ± 6.7%
𝑎 = 9.81 𝑚𝑠 −2 (no uncertainty 𝑔𝑖𝑣𝑒𝑛)
thus
∆𝐸 = 53 × 9.81 × 150 ± (5.7 + 6.7)%
= 77,989 ± 12.4%
= 80000 ± 10000𝐽
= (80 ± 10) × 103 𝐽
Average of a Repeated Measurement – Uncertainty is half the range
The uncertainty of a measurement can be reduced by taking multiple measurements and averaging the value. In this
case, the average uncertainty is calculated as half the range of the data. For example, consider the time it takes for a
ball to fall from a bench to the floor which has been measured four times. The recorded times are 1.1 ± 0.1 s, 1.0 ±
0.1s, 1.2 ± 0.1s, and 1.0 ± 0.1s. The mean time is 1.075 seconds. Taking half the range of the data, we obtain:
50
1.1 − 1.0
= 0.1𝑠
2
Giving a final result of 1.1 ± 0.1s.
The Brute Force Method
When uncertainties occur within functions, such as logs and trigonometric functions, the brute force method is used.
This estimates the effect of the error on the function by evaluating how the range of values with vary the result.
When 𝑥 = 𝑥 ± 𝛿𝑥, the uncertainty in 𝑓(𝑥) can be found using the following:
𝛿𝑓 = 𝑓(𝑥 + 𝛿𝑥) − 𝑓(𝑥)
Example: Find the uncertainty of 𝑠𝑖𝑛−1 (𝑂/𝐻) when 𝑂 = 5 ± 1, 𝐻 = 30 ± 1. Here, our “ 𝑥 “ is the quantity 𝑂Τ𝐻. Refer to page
49 for an explanation on working out uncertainties when dividing. So we have:
sin-1(O/H)
= sin-1(5/30)
= sin -1(0.1666)
= 9.594
δsin-1(O/H)
= sin -1(O/H + δ(O/H)) – sin-1(O/H)
= sin -1(0.1666 + 0.0388) – sin(0.1666)
= 2.268
So sin-1(O/H)
= 9.59 ± 2.27
There is an extra step here. This is the uncertainty of the
quantity O/H, not just the uncertainty of O, or H. See
page 49. Uncertainty calculations often involve multiple
steps such as this.
Summary
Estimating Uncertainties
Measurement Type
Method
Single
Estimate the error of the tool's scale divisions and then add additional error as according
to how the tool was used.
Multiple
Use half of the spread in the data to estimate the uncertainty in each measurement. Then
use the repeated measurement method.
Converting Uncertainties
Conversion
Method
Absolute to percentage
%𝑈𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑡𝑦 =
𝑈𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑡𝑦
× 100
𝑉𝑎𝑙𝑢𝑒
Percentage to Absolute
𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒𝑈𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑡𝑦 =
𝑉𝑎𝑙𝑢𝑒
× %𝑈𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑡𝑦
100
Combining uncertainties
Type
Method
Addition and Subtraction
Add absolute uncertainties.
Division and Multiplication
Add percentage uncertainties.
51
Exponents
Add the percentage uncertainty once for every order of the exponent i.e. if squared add
it twice.
Multiplication by a Constant
Keep percentage uncertainty the same.
Average of many Measurements
δxaverage = Range / 2
Brute Force Method
δf = f(x + δx) – f(x)
APPENDIX C: UNCERTAINTY EXERCISES
Calculate the uncertainties on the following quantities using the methods described in appendix B. Be sure to complete
some from each section so you are confident with all of the techniques which will be used.
Addition and Subtraction
Task 1
A train is being assembled from the following selection of carriages and engines. How long it the train?
Engine1 = 33.4 ± 0.2 m
Engine2 = 25.3 ± 0.3 m
Wagon1 = 35.7 ± 0.9 m
Wagon2 = 13.14 ± 0.07 m
Wagon3 = 33.4 ± 0.1 m
Wagon4 = 46.0 ± 0.4 m
Task 2
A Builder has a long beam x = 14.27 ±0.07 m and has a cutting list of shorter beams which need to be cut. How long
will the remaining beam be once all of the following pieces have been cut off?
Beam1 = 1.34 ± 0.02 m
Beam2 = 4.23 ± 0.02 m
Beam3 = 3.10 ± 0.02 m
Beam4 = 0.80 ± 0.02 m
Beam5 = 2.427 ± 0.005 m
Multiplication, Division and Exponents (using percentage uncertainties)
Task 3
A 220.5 ±0.1 kg rover on another planet with negligible atmosphere has 11000 ±60 J of energy as it approaches the
bottom of a hill. It turns off its motor and coasts up the hill reaching a vertical height of 13.3 ±0.1 m. Is it possible to
determine which planet this is happening on if
1
𝐸𝑘 = 𝑚𝑣 2
2
𝐸𝑝 = 𝑚𝑔ℎ
Constants
Task 4
The weight of a person on the moon is 134 ±7 N. What is their mass given that
gmoon = 1.67 ms-2
weight = mg
52
g on mars
= 3.72 ms-2
g on mercury
= 3.78 ms-2
Repeated Measurements
Task 5
The diameter of a hair is measured 5 times using a travelling microscope.
x = 70.4 ± 0.03 µm
x = 70.42 ± 0.03 µm
x = 70.41 ± 0.03 µm
x = 70.47 ± 0.03 µm
x = 70.42 ± 0.03 µm
What is the diameter of the hair and its uncertainty?
Brute Force Method
Task 6
An angle has been measured as θ = 42.0 ± 0.3°.
What is sin(θ) and its uncertainty?
Combinations
Most real equations require a combination of these techniques. Use the same rules for dealing with uncertainties as
you use when solving equations. I.e. do any addition and subtraction first then convert to percentage uncertainties to
do the multiplication or division. Finally convert back to absolute uncertainties to use the brute force method when this
is required. Do another % uncertainty conversion if the result of the brute force method is multiplied by something else.
Task 7
Find the thickness of a wire x and its uncertainties, given
λ = (589.3 ± 0.3) nm
L = (70.12 ± .02) mm
S ’= 5.86, 5.67, 5.77, 5.75 mm
Use the formulae
Saverage = S’ average/50
x = ½ λ (L/Saverage)
Task 8
Find the wavelength of a certain spectrum λ and its uncertainties, given
θ1 = 117.47 ± 0.02°
θ2 = 79.30 ± 0.02°
d = (3.353 ± 0.005) × 10-6 m
m=2
Use the formulae
θ = ½ (θ1 - θ2)
mλ = d sin θ
Task 9
Find the focal length of a lens given that
p = 0.10 ± 0.01 m
q = 0.50 ±0.03 m
1 1 1
= +
𝑓 𝑝 𝑞
Task 10
Find the rotational inertia of disk I1 and its uncertainties, given
F = (1.74 ±0.02) N
r = 1.50 cm
1 = (0.255 ±0.001) rad/s2
2 = (-0.016 ±0.001) rad/s2.
Use the formula F r = I1 (1 – 2)
Find the final angular velocity of a combined disk f and its uncertainties, given
I1 = (9.7 ±0.2) kg m2
I2 = (7.90 ±0.05) kg m2
i = (13.201 ±0.001) rad/s.
Use the formula f = I1 i / (I1 + I2)
53
Task 11
Find the x and y component and their uncertainties of the force F1 if
F1 = 1.323 N
Angle θ1 = 30 ± °1°.
Use the formula Fx = F cosθ and Fy = F sinθ
Find the x component, y component and their uncertainties of a torque if
F = 0.539 N
direction angle θ1 = 63 ±2°
perpendicular distance r = (8 ±2) cm.
Use the formulas τx = r F cosθ and τy = r F sinθ
Task 12
Find the modulus of rigidity (G) of a rod and its uncertainties if
2𝐼𝐿
𝑇 = 2𝜋√
L = (15.0 ±0.1) cm
𝜋𝐺𝑟 4
t = (21.16 ±0.08) s, (21.19 ±0.08) s, (21.25 ±0.08) s
m = (356.51 ±0.02) g
R = (2.225 ±0.001) cm
r = (0.41 ±0.01) mm, (0.40 ±0.01) mm, (0.39 ±0.01) mm, (0.41 ±0.01) mm.
Use the formulas
T = t/20
I = (2/5) mR2
Task 13
Find the heat of fusion Lfusion and its uncertainties for water if
mcalorimeter = (14.72 ±0.05) g
mcal + water = (246.81 ±0.05) g
mcal+w+ice = (341.27 ±0.05) g
Ti = (35.4 ±0.1)o C
Tf = (1.8 ±0.1)o C
cwater = 4.19 J/g o C
Use the formulas
mwater = mcal + water − mcalorimeter
mice = mcal+w+ice − mcal + water
mice cwater (Tf – 0) + mice Lfusion = - mwater cwater (Ti – Tf)
Task 14
Find the thermal conductivity for a certain material k and its uncertainties if
mbeaker + water1 = (171.35 ±0.05) g
mbeaker + water2 = (186.59 ±0.05) g
mbeaker = (162.18 ±0.05) g
𝑚
𝑚
( 𝑎+𝑐 − 𝑎𝑚𝑏𝑖𝑒𝑛𝑡 ) 𝐿
d = (7.87 ±0.05) cm
∆𝑡𝑎+𝑐 ∆𝑡𝑎𝑚𝑏𝑖𝑒𝑛𝑡
𝑘=
∆𝑇
ta+c = tambient = (10.0 ±0.5) min
𝐴
𝑥
x = (0.460 ±0.005) cm
L = 333 J/g
ΔT = 100 oC
Use the formulas
mambient = m beaker +water1 – mbeaker
ma+c = mbeaker +water2 – mbeaker
A = ¼ d2
54
APPENDIX D: MANUAL GRAPHICAL ANALYSIS
Graphs are an important tool as they allow scientist and engineers to extract useful information from data and present
it more visually. Drawing graphs by hand enables students to gain a better understanding of this process and the
important elements of a graph.
What to Consider when Drawing a Graph
Each graph should fill as much of the page as possible, a larger graph will increase the accuracy of the results. Each
graph should be drawn on a new page of graph paper in the lab book.
Choosing the axes
The variable you are trying to calculate will be represented within the gradient of the graph. To identify which variables
are to be plotted on each axis, compare the equation you are examining to the linear form:
𝑦 = 𝑚𝑥 + 𝑐 .
Rearrange the equation you are using so that the measured values represent the x and y values, and all other variables
are represented by the gradient m or the constant c. The independent variable is usually plotted on the x axis and the
dependent variable is plotted on the y axis.
Example: An experiment is undertaken to find the wavelength of a laser (see appendix A). The distance and thus angle
to the bright spot is measured for each order. The equation
𝑚𝜆 = 𝑑(𝑐𝑜𝑠𝜓 − 𝑐𝑜𝑠𝜑𝑚 )
can be rearrange into the linear form to give (see appendix A for details)
𝑐𝑜𝑠𝜑𝑚 =
−𝑚𝜆
+ 𝑐𝑜𝑠𝜓
𝑑
Since the diffraction order, m, is the independent variable it will be plotted on the x axis, and the cosine of the angle
cos(φm) on the y. This plot will give a gradient of
−𝜆
𝑑
.
The next important step is to select appropriate scales for the axes. This can be estimated by using the equation:
𝑚𝑎𝑥𝑖𝑚𝑢𝑚𝑣𝑎𝑙𝑢𝑒 − 𝑚𝑖𝑛𝑖𝑚𝑢𝑚𝑣𝑎𝑙𝑢𝑒
𝑛𝑢𝑚𝑏𝑒𝑟𝑜𝑓𝑙𝑎𝑟𝑔𝑒𝑠𝑞𝑢𝑎𝑟𝑒𝑠𝑎𝑣𝑎𝑖𝑙𝑎𝑏𝑙𝑒
And rounding to the next practical scale size i.e. one which makes a good
readable scale. Practical scale divisions include 1, 2, 5 and 10 or some multiple
or division of 10 of these e.g. 0.02 or 20.
Example: The data collected for the x axis ranges from 1-6 and there are a
maximum of 9 squares available. Using the equation above gives 0.55 which
should be rounded up to 1 so each square will represent 1 on the graph.
Graphs should take up as much of the page
as is practical.
The spread of the data should take up most
of the graph
Axes do not have to go through zero
Label all axes, including units
Label the graph
Label each axis using the scales calculated above and add an Axis Label. This should include the variable and its unit.
Add a Title to the graph. This should express either the gradient of the graph or the value you intend to calculate.
E.g. determining the wavelength of a HeNe laser.
It should not start with ‘Graph to... ’or be of the form ‘y vs. x ’as this information is already being shown.
55
Figure D.1 Sample graph drawn using the steps outlines in this appendix. Look at appendix A to see this graph in
context and look at the data which was used to plot it.
56
Plot the data
Uncertainty bars can have different sizes for each
data point.
If the uncertainties are too small to see they can
be ignored but this must be clearly noted on the
graph.
Plot each data point using the values of x and y which have been
measured. If the dataset contains outliers these should be plotted
but clearly marked as such and ignored in the remaining gradient
analysis.
Add the Uncertainty Bars. The length of each bar corresponds to the size of the uncertainty.
E.g. If a measurement is 23 ± 2 m then this will be plotted as a point at 23 with a bar of length 2 on either side of the
data point (total length 4). These should be drawn as per sample graph. This has to be done for the uncertainties in
both the x and y direction.
Draw the Line of Best and Worst Fit
The Line of Best Fit (LOBF) should be drawn to take into account all of the data points which have not been classified
as outliers. It should attempt to find the average of these points and pass through each point considering its uncertainty.
This line should not be forced to go through zero even if the equation suggests that it will. It should represent the data
which was collected, not where it is predicted to go.
The Line of Worst Fit (LOWF) should be drawn within the range of the data point uncertainty bars. To maximise the
range of the error, start the line from the error bar edge of an upper data point, and end the line on the opposite error
edge of a lower data point.
Finding the Gradient of the line
The gradient of the best fit line has been defined to represent the relationship or the value we are investigating. Choose
two points near the ends of the line of best fit, and calculate the gradient as follows:
𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡 = 𝑚 =
𝑟𝑖𝑠𝑒
𝑟𝑢𝑛
=
𝑦2 −𝑦1
𝑥2 −𝑥1
It is important that points chosen include a large range of the data as this will reduce the size of the uncertainty in the
measurement. Repeat this procedure for the line of worst fit.
The uncertainty of the gradient can be calculated by subtracting the gradient of the line of worst fit from the line of best
fit:
𝛿𝑚 = |𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡 𝐿𝑂𝐵𝐹 − 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡 𝐿𝑂𝑊𝐹|
Find the Value of the Intercept C
Sometimes it is important to find the value of the Intercept c. In this case the axes have to be chosen so the x axis
includes a zero. To find its value simply read off the point where the line of best fit crosses the y axis. The uncertainty
of this value can be found by reading off the value where the line of worst fit crosses the y axis.
57
Summary
Ensure that the following are considered and included in each graph which is plotted.
• Rearrange the equation which is being examined/used and select the correct variables to plot on each axis.
• Choose scales which optimise the data on the page (use up as much of the page as possible) while using
workable/useful scale divisions.
• Label the axes with the variable and units used.
• Put a title on the graph which clearly explains why it was drawn. This should not include ‘graph of... ’or be of
the form ‘x vs. y’.
• Plot all data points and their corresponding uncertainty bars – these can vary for each point.
• Draw a line of best and worst fit.
• Calculate the gradient or intercept and the associated uncertainty.
58
APPENDIX E: COMPUTER ASSISTED GRAPHICAL ANALYSIS
Some graphs in the 100 level laboratories can be drawn using a computer.
The following builds on the previous appendix and explains how this can
be done using excel.
Drawing a graph using Excel
•
•
•
•
•
•
Place the two data sets to be plotted on separate columns. Add
further columns for the uncertainties for each dataset.
• Ensure that each column of data has an equal number of
points.
Select the data columns and, in the insert tab, select the scatter
plot in the charts area. Then choose the Scatter with only Markers
option as shown.
Reformat the Axes – Right click on the numbers next to the axis that needs to be changed and choose format
axis.
• Choose the maximum and minimum values for this axis so that the data covers most of the chart and
then select a suitable division size or use auto.
Repeat for the other axis if necessary.
Click on the chart and then click on the Layout tab.
Add a Title by clicking on the chart title button.
Add Axes Labels by clicking on the axes titles button and selecting the horizontal and then the vertical option.
• Ensure both the variable and its unit are shown.
Finding the velocity of a car
110
y = 5.5189x + 25.561
100
Distance d (m)
•
90
80
70
60
50
4.
6.
8.
10.
12.
14.
16.
Time t (s)
Figure E.1 Sample graph drawn using Excel.
59
•
•
•
Uncertainty Bars
• Click on the error bars button in the layout tab and select more error bar options.
• Choose custom and then click specify value. Use the dialogue box to select the positive and negative
values for the uncertainty bars. Note that most 100 level labs will have the same value for both, so
the same dataset can just be selected twice.
• Horizontal uncertainty bars should also appear, click on these to select the horizontal error bars and
repeat as above.
The Line of Best Fit (LOBF) is added by clicking on trendline and selecting more trendline options. Select the
type of line required (usually linear) and check display equation on graph. This will provide the gradient of the
LOBF.
The Line of Worst Fit (LOWF) is harder to do as excel does not provide an inbuilt function for this. Two
additional points need to be plotted and a trendline fitted to them to add this line.
• Add two new columns and add the x and y coordinates for the two points required.
• Right click on one of the data points on the plot and click on select data.
• In the dialog box click add.
• Select the x and y values and click ok.
• Add a trendline as above, again displaying the equation on the graph.
• Format the data points so they are invisible:
▪ Right click on the data point and select format data series.
▪ In Marker options select none.
All other calculations should be completed using the methods outlined for the hand drawn graphs.
60
APPENDIX F: MEASUREMENTS
When taking measurements it is important that the operation of the apparatus and read out of the data are correct.
This will keep the uncertainties to a minimum and ensure that the results are valid.
Verniers
Vernier scales increase the accuracy with which measurements can be read off a piece of equipment. They do this by
further subdividing the smallest division of the scale and allowing the position to be determined more accurately.
Verniers work by adding an additional scale to the device which slides along the main scale. This has divisions which
are slightly shorter than the main scale which allows the exact position to be read from the point where the two scales
align.
The calliper below is a common example. It shows that the specimen is between 4 and 5 units long. Using the Vernier
it is clear that it is actually 4.3 units long because the 3 on the Vernier scale is aligned with the main scale.
FIGURE F.1 VERNIER SCALE SHOWING A SPECIMEN OF SIZE 4.3 BEING MEASURED.
Not all Verniers have divisions of 10 and it is important to check what each division corresponds to.
The following show three examples. Note that the first two digits are read off the main scale and the final one comes
from the Vernier – where it aligns with the main scale.
measurement = 32.0
measurement = 32.3
measurement = 32.7
FIGURE F.2 THREE SAMPLE MEASUREMENTS USING VERNIERS.
61
APPENDIX G: WRITING LAB REPORTS
Have a careful read of the sample report which has been provided as this will give you a lot of information on what
should be contained and most importantly the style in which it should be written.
Keep the report as concise as possible. There is no minimum or maximum number of pages or words but generally
shorter is better, as long as everything has been covered.
Follow the structure as given in the marking schedule, do not try to merge sections or put information under the wrong
headings. Do not leave things out of one section because they were already mentioned.
Always use the correct format for stating the final results i.e. (result
E.g. (5.23 ±0.04) x 103 m. Never state the final result with a percentage uncertainty.
±uncertainty)
x 10x Unit.
Always state whether or not the results agreed with the expected value within the limits of uncertainty.
E.g. We found X to be 5.53 ±0.06 m which agrees with the stated value of 5.55 m within the limits of uncertainty
[Source X].
Headings – This should be self-explanatory. Note this is separate from the details on the cover sheet as this is not
considered when marking.
Abstract – This should be a single, short paragraph which can be read on its own and still provides a good overview
of the experiment and results. This should be readable on its own as should the rest of the report.
Introduction – This should give some background to the experiment and motivation for doing it. It should also contain
(the derivation of) all of the relevant equations which are used in the report and define any variables used. Be sure to
arrange them into the form which will be used in the analysis. It should also give a brief overview of how the experiment
was done. Be sure that this is not directly copied from the lab manual. Do some reading beyond this and site sources.
Method – This section is frequently written incorrectly. It should be in past tense and not be a list of bullet points. This
should also contain a diagram of the experimental setup.
Results – This section should contain no calculated data. It should contain the raw data and its associated
uncertainties. It should list the sources of the uncertainties and how they were estimated.
Analysis – This is where all of the data processing is done.
•
•
Be sure to explain what the calculations etc. are.
Be sure to state what the results are and what they mean at the end of a calculation e.g. The length of the
pendulum was found to be 0.475 ±0.003 m.
• Include some sample calculation and sample calculations for the uncertainties.
• The quality of the graph in the report should be better than the lab book because there is time to redraw it if
something goes wrong.
Discussion – This is where the results which were obtained are finally discussed and compared to the accepted
values. It should contain the numerical results which were calculated and their uncertainties and these should be
compared to the accepted results and any discrepancy discussed. Do not use “human error” to explain discrepancies
or anything else for that matter. Do not try to suggest that arbitrarily choosing smaller uncertainty values will improve
your results. Be sure to discuss why the results did not agree with the accepted values if applicable and how the results
could be improved if they did agree. Explain how this result is relevant to the aim i.e. does it prove what was being
tested?
Conclusion – This should summarise the results. It should start by restating the aim and include the results and their
uncertainties which were found and if this experiment has confirmed or refuted the theory. Do not add any new material
here.
References – This should contain a list of all of the sources which were used while writing the experiment. Please
format them in a consistent way and ensure that the listed sources have been referenced within the text itself.
“Lab manual” is not a sufficient reference and neither is a web address on its own. For the latter you also need to
include the date it was referenced.
62
APPENDIX H: SAMPLE LAB REPORT
Measurement of the Wavelength of a Laser using an
Unconventional Diffraction Grating and
Measurement of the Diameter of Lycopodium Spores
Alex Salkeld (11235813)
Partners:
Raphael Nolden
Chrissy Emeny
Alex Neiman
Date: 14/02/2014
Abstract:
An experiment was conducted to measure the wavelength of a helium-neon (HeNe) laser, using a standard metal ruler
as a diffraction grating, and determine the diameter of lycopodium spores using a circular diffraction pattern. The laser
wavelength was determined by shining the laser onto the metal ruler at a very low angle, so that the markings on the
ruler acted as a diffraction grating. The spacing of the dots in the resulting diffraction pattern gave the wavelength to
be (700 ± 200) nm. This agrees with the reference value of 632.8 nm [1,2]. The lycopodium spore diameter was
calculated by simply shining the HeNe laser onto a prepared slide containing the spores. The spacing of dark fringes
on the resultant diffraction pattern gave a diameter of (35 ± 1) m. This does not agree with the reference value of 33
m.
63
Introduction
Lasers are a very common tool in physics, and modern society in general. Their unique optical characteristics make
them important parts of many industries including medicine, surveying, precision cutting of many materials and
communications [3]. One of their useful properties is that they produce an intense beam of light of a single wavelength.
As nearly all optical properties depend upon the wavelength of the light involved, this wavelength is a useful quantity
to know.
The wavelength can be determined by passing the laser beam through a diffraction grating and observing the diffraction
pattern it produces. Bragg’s law
𝑚𝜆 = 𝑑 𝑠𝑖𝑛 𝜃
relates the spacing of the bright fringes of a diffraction pattern to the physical configuration of the laser and diffraction
grating. Where
𝑚 is the spectral order of the bright fringe,  is the wavelength of the laser, d is the spacing between
lines on the diffraction grating, and 𝜃 is the angle from the normal of the laser to the fringe. Using this, the wavelength
can be determined from the diffraction pattern size.
A second use for lasers is to measure very small objects. For example, the average size of dust particles or spores can
be determined using the diffraction pattern which is formed when a sample is hit by a laser of known wavelength. The
size of a diffraction pattern caused by light passing through a single small hole or a cluster of small holes is given by
𝑅 = 1.22𝜆
𝐿
𝑑
where R is the radius of the first dark fringe of the pattern,  is the wavelength of light, L is the distance between the
holes and the projected pattern and d is the diameter of the holes. This law also holds when the setup is reversed so
that the sample contains solid dots on a transparent background. Thus, the size of the particles can be determined by
measuring the size of the diffraction pattern.
This experiment was carried out to test these two functions of a helium-neon (HeNe) laser, first to measure the
wavelength of the laser, and then to measure the diameter of lycopodium spores from a prepared sample plate.
However, to measure the wavelength of the laser, a conventional diffraction grating was not used. Instead, a diffraction
pattern was obtained by reflecting the laser off of the millimetre markings on a metal ruler. This used a modified form
of the Bragg equation
𝑚𝜆 = 𝑑(𝑐𝑜𝑠(𝜓) − 𝑐𝑜𝑠(𝜙𝑚 ))
where 𝜓 is the angle that the ruler is tilted at from horizontal, and 𝜙𝑚 is the angle between the ruler and the diffraction
order of interest, as shown in figure 1 below.
64
Method
The equipment was set up as indicated in figure 1. The HeNe laser was attached to the holder so that it was parallel to
the bench. The metal ruler was held by a clamp stand in the path of the laser beam. The ruler was set at a small angle
from parallel to the bench, until a diffraction pattern was observed on the wall. The pattern appeared as a vertical line
of red dots, brightest near the bottom and fading towards the top.
Figure 1: Experimental setup. The laser was aligned parallel to the bench. The ruler was angled slightly off parallel
until the diffraction pattern was observed on the wall.
The zero order diffraction (reflection) dot was identified by sliding the ruler so that the beam was incident upon a nonmarked part of the ruler. The point where the laser was directly incident on the wall was also marked by moving the
ruler aside. The distance to several other diffraction orders from this point was measured. Thus, using trigonometry,
the angles to each fringe could be determined, and from this, the laser wavelength could be determined.
The ruler was then replaced by the lycopodium spore sample plate in the clamp stand. The plate was arranged so that
it was approximately at right angles to the laser beam. This caused a diffraction pattern to form on the wall. Moving the
sample plate from side to side allowed the first dark fringe to easily be identified. The diameter of this fringe was
measured at a number of different points. This provided the information required to determine the size of the spores.
Results
Measurements for finding the wavelength of the laser
Length from ruler to wall, L = (0.965 ± 0.008) m
Height of reflected laser beam, h = (0.068 ± 0.003) m
The uncertainties are chosen due to the sizes of the laser dots on the ruler and wall respectively.
Measured diffraction pattern heights:
m
sm (± 0.003) m
1
0.081
2
0.095
3
0.104
4
0.113
5
0.12
Table 1: Measured heights (sm) of the first five diffraction pattern fringes.
Measurements for finding the diameter of the spore
Length from the sample to the wall, L = (0.83 ± 0.01) m
The uncertainty is chosen to account for slight movement of the apparatus during measurement.
Measured diameter of the dark fringe:
65
D (± 0.001) m
0.038
0.036
0.037
0.037
Table 2: Repeated diameter measurements of the first dark fringe of the diffraction pattern. The uncertainty is
taken from the scale of the ruler.
Analysis
Finding the wavelength of the laser
The given expression for finding the laser wavelength for the ruler setup is
𝑚𝜆 = 𝑑(𝑐𝑜𝑠(𝜓) − 𝑐𝑜𝑠(𝜙𝑚 )),
Or, in a linear form,
𝜆
𝑐𝑜𝑠(𝜙𝑚 ) = − 𝑚 + 𝑐𝑜𝑠(𝜓).
𝑑
To determine
𝑐𝑜𝑠(𝜙𝑚 ) from the heights measured, trigonometry is used:
𝑡𝑎𝑛(𝜙𝑚 + 𝜓) =
The constant
𝑠𝑚
𝐿
𝜓 can be determined when 𝜙𝑚 = 𝜓, which is the directly reflected laser dot, at 𝑠𝑚 = ℎ.
𝑡𝑎𝑛(2𝜓) =
ℎ
𝐿
Therefore,
1
ℎ
𝜓 = 𝑡𝑎𝑛−1 ( )
2
𝐿
1
0.068𝑚
)
= 𝑡𝑎𝑛−1 (
2
0.965𝑚
= 2.01538°
Uncertainties
%𝛿ℎ =
%𝛿𝐿 =
0.003𝑚
= 4.4%
0.068𝑚
0.008𝑚
= 0.83%
0.965𝑚
ℎ
%𝛿 ( ) = 4.4% + 0.83% = 5.23%
𝐿
1
1
2
2
𝜓 = 𝑡𝑎𝑛−1(0.07 ± 5.23%) = 𝑡𝑎𝑛−1(0.07 ± 0.0037)
66
Using brute force method:
𝑡𝑎𝑛−1(0.07) = 4.03°
𝛿 𝑡𝑎𝑛−1 = 𝑡𝑎𝑛−1 (0.07 + 0.0037) − 𝑡𝑎𝑛−1 (0.07)
= 4.24° − 4.03°
= 0.21°
Therefore,
1
𝜓 = (4.03° ± 0.21°)
2
= (2.0 ± 0.1)°
Using this value, the remaining 𝜙𝑚 can be calculated using the same method. The results are shown in Table 3.
𝑠𝑚
)
𝐿
m
sm
(± 0.003) m
𝑠𝑚
𝑡𝑎𝑛−1 ( )
𝐿
1
0.081
4.8°
0.2°
2.8°
0.3°
0.9988
0.0003
2
0.095
5.6°
0.2°
3.6°
0.3°
0.9980
0.0004
3
0.104
6.2°
0.2°
4.2°
0.3°
0.9974
0.0004
4
0.113
6.7°
0.2°
4.7°
0.3°
0.9967
0.0005
5
0.120
7.1°
0.2°
5.1°
0.3°
0.9961
0.0005
𝛿 𝑡𝑎𝑛−1 (
𝜙𝑚
𝛿𝜙𝑚 𝑐𝑜𝑠(𝜙𝑚 ) 𝛿 𝑐𝑜𝑠(𝜙𝑚 )
Table 3: Calculations of 𝒄𝒐𝒔(𝝓𝒎 ) and its uncertainties.
Determination of the wavelength of a HeNe laser using an unconventional
1.0005
diffraction grating
cos(φm)
0.9990
0.9975
0.9960
y = -0.0007x + 0.9994
0.9945
1
2
3
4
5
Diffraction fringe order, m
Figure 2: Graph for finding the wavelength using gradient which is equivalent to
−𝝀Τ𝒅.
67
From figure 2, the line of best fit (LOBF) gradient is −𝜆Τ𝑑 = −0.00066. The line of worst fit (LOWF) gradient is
−𝜆Τ𝑑 = −0.00087. Therefore the final gradient and uncertainty is:
−𝜆Τ𝑑 = −0.00066 ± (−0.00066— 0.00087)
= −0.00066 ± 0.00021
= (−0.7 ± 0.2) × 10−3
Hence, assuming that the spacing on the ruler, d, is exactly 1 mm, the wavelength can be determined:
𝜆 = 0.0001𝑚 × (0.7 ± 0.2) × 10−3
(0.7 ± 0.2) × 10−6
= (700 ± 200)𝑛𝑚
So the wavelength is 700 ±200 nm which agrees with the accepted value of 632.8 nm [1,2] within the limits of
uncertainty.
To find the size of the lycopodium spores, the formula
𝑅 = 1.22𝜆
𝐿
𝑑
is used. Given the large uncertainty in the measured value of , the reference value of 632.8 nm will be used. The
average value of diameter of the dark fringe measured is
𝐷𝑎𝑣𝑔 =
0.038 + 0.036 + 0.037 + 0.037
𝑚
4
= (0.037 ± 0.001)𝑚
so
𝑅 = (0.0185 ± 0.0005)𝑚
Hence:
𝑑 = 1.22𝜆
𝑑 = 1.22 × 632.8𝑛𝑚
𝐿
𝑅
(0.83 ± 0.01)𝑚
(0.0185 ± 0.0005)𝑚
= 0.000035𝑚
Uncertainties
%𝛿𝐿 =
%𝛿𝑅 =
0.01𝑚
= 1.2%
0.83𝑚
0.0005𝑚
= 2.7%
0.0185𝑚
%𝛿𝑑 = 1.2% + 2.7% = 3.9%
Thus
𝑑 = 0.000035𝑚 ± 3.9%
= (0.000035 ± 0.000001)𝑚
= (35 ± 1)𝜇𝑚
So the diameter of the spore is 35 ±1 µm which does not agree with the accepted value of 33 µm [4] within the limits
of uncertainty.
Discussion
Using the ruler as a diffraction grating, the wavelength of the HeNe laser was determined to be (700 ± 200) nm. This
agrees with the reference value of 632.8 nm [1,2]. However, 200 nm is a very large uncertainty. This technique is not
a very accurate method to determine the wavelength. A big limitation is the size of the laser dots. The size of the laser
dot compared to the size of the distance between dots is very large, and so this introduces a lot of uncertainty to the
measurement. To improve upon this, the aperture of the laser could be narrowed, and an actual diffraction grating be
used, to take advantage of the greater resolution of diffraction pattern it can give. This would result in a greater spacing
between the dots in the diffraction pattern, and hence a smaller uncertainty.
The average diameter of the lycopodium spores was determined to be (35 ± 1) m. This does not agree with the
reference value of 33 m [4]. The most likely source of this disagreement is the measurement of the diameter of the
dark fringe. The fringes of the diffraction pattern were not clearly defined, which lead to a large inaccuracy in this
measurement. It is also quite likely that the sample had been moved slightly during measurements.
68
Conclusion
This experiment was conducted to determine the wavelength of a HeNe laser, and then to use this laser to measure
the diameter of lycopodium spores. The wavelength of the laser was determined to be (700 ± 200) nm, which agrees
with the accepted value of 632.8 nm [1,2]. The large uncertainty of the measurement stems from the low resolution
provided by the ruler as a diffraction grating. The lycopodium spores were determined to have a diameter of (35 ± 1)
m, which does not agree with the reference value of 33 m. A number of factors may have influenced this value,
including potential movement of the apparatus during measurement.
References
[1]
Kaye and Laby, Tables of physical and chemical constants, 14th ed. 1982 Longman Group
Ltd.
[2]
Helium-Neon Lasers
accessed: 14/02/2014
[3]
Serway, Jewett, Wilson and Wilson, Physics Vol. 2, 1st ed. 2013 Cengage Learning Australia
Pty Ltd.
[4]
Raphael Nolden, PHYS101S1 Lab Manual 2014, Dept. Physics and Astronomy, University
of Canterbury.
http://hyperphysics.phy-astr.gsu.edu/hbase/optmod/lasgas.html,
69