1st Science
Physics Laboratory Manual
PHYC 10020
Biophysics of the Cell
2010/11
Name.................................................................................
Partner’s Name ................................................................
Demonstrator ...................................................................
Group ...............................................................................
Laboratory Time ...............................................................
Contents
Introduction
Laboratory Schedule
1
Experimental Measurements: the Bedrock of Science
2
2
Plotting Scientific Data
22
3
Newton’s Second Law
48
4
Waves and Resonance
56
5
6
Investigation into the Behaviour of Gases
and a Determination of Absolute Zero
Electrons in Atoms: The Spectrum of Atomic Hydrogen
64
74
Introduction
Physics is an experimental science. The theory that is presented in lectures has its
origins in, and is validated by, experimental measurement.
The practical aspect of 1st Science Physics is an integral part of the subject. The
laboratory practicals take place throughout the semester in parallel to the lectures.
They serve a number of purposes:
•
•
•
an opportunity, as a scientist, to test the theories presented in lectures;
a means to enrich and deepen understanding of physical concepts presented
in lectures;
the development of experimental techniques, in particular skills of data
analysis, the understanding of experimental uncertainty, and the development
of graphical visualisation of data.
Some of the experiments in the manual may appear similar to those at school, but the
emphasis and expectations are likely to be different. Do not treat this manual as a
‘cooking recipe’ where you follow a prescription. Instead, understand what it is you are
doing, why you are asked to plot certain quantities, and how experimental uncertainties
affect your results. It is more important to understand and show your understanding
in the write-ups than it is to rush through each experiment ticking the boxes.
This manual includes blanks for entering most of your observations. Additional space is
included at the end of each experiment for other relevant information. All data,
observations and conclusions should be entered in this manual. Graphs may be
produced by hand or electronically (details of a simple computer package are provided)
and should be secured to this manual.
There will be six 3-hour practical laboratories in this module evaluated by continual
assessment. Note that each laboratory is worth 5% so each laboratory session makes
a significant contribution to your final mark for the module. Consequently, attendance
and application during the laboratories are of the utmost importance. At the end of each
laboratory session, your demonstrator will collect your work and mark it. Remember,
If you do not turn up, you will get zero for that laboratory.
If you miss a laboratory through illness, talk to Thomas O’Reilly, the laboratory manager
(Room 1.30), on your return and he will attempt to reschedule your missed practical.
Name:___________________________
Date:___________
Student No: ________________
Demonstrator:_________________________
Experimental Measurements:
the Bedrock of Science
Introduction
All the technology we take for granted today, from electricity to motor cars, from
television to X-rays, would not have been possible without a fundamental change,
around the time of the Renaissance, to the way people questioned and reflected upon
their world. Before this time, great theories existed about what made up our universe
and the forces at play there. However, these theories were potentially flawed since they
were never tested. As an example, it was accepted that heavy objects fall faster than
light objects – a reasonable theory. However it wasn’t until Galileo1 performed an
experiment and dropped two rocks from the top of the Leaning Tower of Pisa that the
theory was shown to be false. Scientific knowledge has advanced since then precisely
because of the cycle of theory and experiment. It is essential that every theory or
hypothesis be tested in order to determine its veracity.
Briefly describe another theory, which when experimentally tested, was shown to be
incomplete or false.
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1
Actually, the story is probably apocryphal. However 11 years before Galileo was born, a similar experiment was
published by Benedetti Giambattista in 1553.
2
Physics is an experimental science. The theory that you study in lectures is derived
from, and tested by experiment. Therefore in order to prove (or disprove!)2 the theories
you have studied, you will perform various experiments in the practical laboratories.
First though, we have to think a little about what it means to say that your experiment
confirms or rejects the theoretical hypothesis. Let’s suppose you are measuring the
acceleration due to gravity and you know that at sea level theory and previous
experiments have measured a constant value of g=9.81m/s2. Say your experiment
gives a value of g=10 m/s2. Would you claim the theory is wrong? Would you assume
you had done the experiment incorrectly? Or might the two differing values be
compatible?
What do you think?
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The ‘margin of error’
In order to compare theory and experiment you have to know how ‘good’ your
experiment is, or to evaluate what the experimental uncertainty is. The estimation of
experimental uncertainty is absolutely vital in all scientific measurements. Without it,
you can’t draw any meaningful conclusions.
Let us take an example. An opinion poll before the election tells us Fianna Fail will win
42% of the vote. You probably aren’t surprised if they actually win 40% of the vote.
However, the poll will probably also have stated that their prediction has a ‘margin of
error’ of 3%, in which case the prediction and the result are in good agreement. The
actual definition of what we mean by ‘margin of error’ and how far apart prediction and
result may reasonably lie is quite tricky. We will touch on it here but a complete answer
requires a course in probability and statistics.
2
It is a curious paradox that strictly speaking, you can never prove a theory to be 100% correct. You can of course
prove it is false. However all you can say about the truth of a good theory is that it is compatible with all the
experimental evidence – which doesn’t preclude someone doing an experiment in the future that invalidates the
theory!
3
Consider the following: Three surveying companies measure the distance from Dublin
to Cork. The first says it is 245km with a margin of error of 5km. The second says
253km with a margin of error of 1km. The third says it is 254.2km with a margin of error
of 0.1km.
Which measurement is the best and why?
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Given these results, can you come up with a better estimate?
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Let’s further suppose that a fourth company of international repute with the latest and
greatest state-of-the-art equipment measures the distance to be 253.2125km with a
margin of error of 0.0001km. What would you now conclude about the original three
measurements?
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4
Experimental Uncertainties
The example above has most of the elements of a scientific measurement. There is
some ‘true’ value that you are trying to estimate and your equipment has some intrinsic
uncertainty. Thus you can only estimate the ‘true’ value up to the uncertainty inherent
within your method or your equpiment.
Conventionally you write down your
measurement followed by the symbol ± , followed by the uncertainty. Thus the
surveying companies above might report their results as 245 ± 5 km, 253 ± 1 km,
254.2 ± 0.1 km. You can interpret the second number as the ‘margin of error’ or the
uncertainty on the measurement. If your uncertainties can be described using a
Gaussian distribution3, (which is true most of the time), then the true value lies within
one or two units of uncertainty from the measured value. There is only a 5% chance
that the true value is greater than two units of uncertainty away, and a 1% chance that it
is greater than three units
An experimental measurement consists of a central value AND an uncertainty
Why is it poor scientific procedure to quote an experimental result without an
uncertainty?
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How would you interpret an experimental result that hadn’t an associated uncertainty?
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3
For those interested, there is more discussion of this point at the end of the chapter under ‘Gaussian distribution’.
5
How do you find the uncertainty on your result?
Estimating the experimental uncertainty is at least as important as getting the central
value, since it determines the range in which the truth lies. Frequently scientists will
spend much more time estimating the experimental uncertainty than finding the central
value.
To get your final result, you will combine measurements from a number of sources,
each having its own uncertainty. Suppose we call the measurements you make
x ± δ x , y ± δ y , z ± δ z , then the final result, f , is just some combination of the individual
measurements, i.e. f ( x, y , z ) . The difficult question is, what is the uncertainty, δ f , on
your final result? Once you know that, you can report your final answer as f ± δ f .
Finding δ f is often quite difficult because you first have to identify all sources of
uncertainty, then you have to evaluate δ x , δ y , δ z K , before finally combining them in
some way to get δ f . Fortunately, in many cases we can just consider the largest
source of uncertainty in your experiment, and scale it to see the effect on your final
result.4 So for most of the experiments you will do in first year it is sufficient to:
1. Consider the various uncertainties that could have affected your result;
2. Roughly estimate the size of each;
3. Find the source with the largest relative uncertainty;
4. Find the effect of that source on your result.
1. How do you find the sources of uncertainty?
In all the experiments you will have made a number of measurements that are
combined together to produce a final result for some physical quantity. Think about the
various uncertainties that could enter each due to the intrinsic precision of your tools
and changeability of the environment.
Suppose I gave you a 30cm ruler and asked you to measure the length of the labbench. List at least three sources of uncertainty.
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4
This is a result of two things: firstly the Central Limit Theorem which allows us to treat most experimental
measurements as coming from a Gaussian distribution; and secondly the method of combining different sources of
Gaussian uncertainties in which the largest uncertainty dominates (for those interested, see later.)
6
2. How do you find the size of each source of uncertainty?
Use common sense!
If you are reading a scale, how precisely can you read off the gradations?
If a display or instrument is unstable or moves, over what range does it change?
If you are timing something, what are your reaction speeds?
If you are viewing something by eye, with what precision can you line it up?
Sometimes, a good way to estimate the size of a source of uncertainty is to repeat the
measurement a few times and see by how much your reading varies on average.
For each of the sources of uncertainty you wrote down above, make a reasonable
estimate of both the absolute size of the uncertainty and the relative size compared to
the measurement you are making.
Source of
uncertainty
Estimated size of
uncertainty
Typical size of
measurement
Relative size of
uncertainty
3. Find the source with the largest relative uncertainty
In the table above, put an asterisk beside the source with the largest relative
uncertainty.
7
4. Find the effect of that source on your result
You have identified the largest source of uncertainty, but now you must figure out how
that affects your final result. There are two ways to do this: (i) calculate the uncertainty
on your final result by changing the source value by its uncertainty; (ii) plug your
numbers into a formula (but you have to know which formula to use!)
4.1 Method 1: Recalculating your result by changing the source values.
•
•
From your measurements, calculate the final result. Call this f , your answer.
Now move the value of the source up by its uncertainty.
Recalculate the final result. Call this f + .
•
The uncertainty on the final result is the difference in these values: i.e. f − f +
You could also have moved the value of the source down by its uncertainty and
recalculated the final result. You should get the same answer (in most situations).
If you like to express this in mathematics, let x ± δ x be the measurement and f ( x ) the
result you want to calculate, then δ f = f ( x + δ x ) − f ( x ) and your final answer is f ± δ f .
4.2 Method 2: Plug your numbers into a formula.
Here’s a formula that works in most5 cases:
Relative uncertainty in the result = relative uncertainty in the source:
δf
f
=
δx
x
One important exception to this however comes about if you simply add a well known
quantity to x ± δ x . Clearly you will shift the central value by that amount, but in this case
it’s hopefully clear that the uncertainty will remain unchanged. Let’s take an example.
You have about one euro in loose change in your pocket; you estimate you have
1.0 ± 0.2 euros. I give you a 50 euro note. How much money do you have?
The answer is 51.0 ± 0.2 euros. The uncertainty remains the same, and in this case the
relative uncertainty on the final results is less than the relative uncertainty on the
source.
5
To be precise, it works when f = Ax or f = A / x . In the general case when f = f (x ) then δ f =
8
∂f
δx.
∂x
Example.
Let’s try an example of this and do it three ways: first by common sense, then by the
first method above, and finally by the second method.
Suppose I bet 1 euro on a horse with odds of 10-1.
How much will I win if the horse wins?
Suppose now that I bet my jam-jar of 1-cent coins on the horse.
I think there are about 100 coins in the jar (in fact I think there
are 100 ± 5 coins in the jar). How much would I win?
±
Now lets try Method 1. Fill in the table below.
Number of coins
100
Amount bet (є)
x=
105
x + δx =
Amount won (є)
f =
f
+
Difference wrt. f
0
δf =
=
So f ± δ f =
±
Finally Method 2.
x ± δ x = 1.00 ± 0.05 є. So the relative uncertainty is
δx
x
=
f
=
δf
So the relative uncertainty
And since the central value of what I expect to win is
f =
That means
δf =
So f ± δ f =
±
9
Systematic uncertainties
So much for the intrinsic precision of your experiment. However consider the following
example where a group of doctors attempt to measure the height of a patient. The first
says the patient is 2.00 ± 0.01 m, the second says 1.99 ± 0.01 m, the third
says 2.02 ± 0.01 m. All are pretty happy that the patient is within a centimetre or so of
being two metres tall. However it’s only after the patient departs that a nurse asks
whether or not the patient was wearing shoes when the measurements were taken –
and nobody can remember.
This is an example of a different source of uncertainty called a systematic uncertainty.
Although the precision of the doctors’s measuring procedure was about 1cm, there is an
additional common error to all their measurements if the patient was wearing shoes.
You might like to discuss what the correct procedure is in dealing with errors like this.
Note that statistical uncertainties as discussed earlier get smaller the more
measurements you make, but systematic uncertainties do not.
On way to report on the above example is to quote an additional uncertainty
corresponding to the typical height of people’s shoes (say 5cm). Thus the first doctor
could quote their result as 2.00 ± 0.01 ± 0.05 m, the second as 1.99 ± 0.01 ± 0.05 m and the
third as 2.02 ± 0.01 ± 0.05 m. When you see two uncertainties written down, the first is
the statistical uncertainty and the second is the systematic uncertainty.
Systematic uncertainties are the bane of the experimentalist’s life. It is usually easy
enough to assign a statistical uncertainty but how do you deal with systematic errors?
How do you know they are there? In the example above, without the presence of the
astute nurse the doctors would have overlooked a systematic effect and their results
would be inaccurate.
If you do suspect some source of systematic to be at work, the correct procedure is to
remove it if possible, or else assign an additional uncertainty due to it. In the above
example the doctors could repeat the measurement by inviting the patient back, and
being a bit more careful second time around. Alternatively, if they recalled that the
patient had indeed worn shoes, they could correct their result by the height of an
average shoe, and then included their estimate of an ‘average shoe’ as a systematic
uncertainty.
In most cases you can combine the statistical and systematic uncertainties together to
end up with one overall uncertainty which is your estimate of the ‘margin of error’.
10
How would you record the following experimental measurements?
(i)
A digital voltmeter that says 1.04V
(ii)
A digital voltmeter that says 1.04V but the last digit
flickers to 3, then to 2, then back to 3, then to 4
(iii)
A mechanical voltmeter that reads half way
between the 2V mark and the 2.2V mark.
(iv)
The reading is as in (iii) but when you disconnect
the voltmeter you notice it doesn’t return to zero but
to -0.5V
(v)
The reading is as in (iii) but the demonstrator tells
you that there is a calibration error of 0.2V
(vi)
You forgot to measure the temperature in the lab
and now you need it as part of a calculation. What
value would you use?
You and your lab partner measure the time it takes
a ball to drop using a digital stopwatch. You shout
‘Go’, press the stopwatch and your partner drops
the ball. You stop the stopwatch when the ball hits
the ground. The watch reads 1.07 seconds.
(vii)
How does the uncertainty on a source propagate through to the final answer in these
cases?
(i)
Ohm’s law is V=IR. I measure a voltage of 10.0 ± 0.1 V
and a current of 1 ± 0.1 A. What is the resistance?
(ii)
Boyle’s Law says PV=constant and for a particular
apparatus in the lab the constant is 100 Pa.m3. If the
volume is 10 ± 0.1 m3, what is the pressure?
(iii) The height of a person wearing shoes is 2.00 ± 0.01 m.
The height of their shoes is 0.05 ± 0.01 m. What is the
bare-foot height of the person?
(iv) What is the volume of a cube of side 1.0 ± 0.1 m?
11
Uncertainties in the First Year Laboratories
You are expected to apply this treatment of uncertainties to all your experiments in first
year. Specifically:
• When you make a measurement, also make an estimate of the uncertainty.
• Identify the largest source of uncertainty.
• Propagate this through to your final answer.
• When you are asked to measure the slope or intercept of a line from data, quote
the associated uncertainty on the slope and intercept as well. (This will come out
automatically in the graph-plotting software provided you have input the
uncertainty on the sources.)
In an experimental subject, a number means nothing unless
accompanied by its uncertainty.
12
Practical Example 1: Now let’s put this to use by making some very simple
measurements in the lab. We’re going to do about the simplest thing possible and
measure the volume of a cylinder using three different techniques. You should compare
these techniques and comment on your results.
Method 1: Using a ruler
The volume of a cylinder is given by πr2h where r is the radius of the cylinder and h its
height.
Measure and write down the height of the cylinder.
Don’t forget to include the uncertainty and the units.
h=
±
Measure and write down the diameter of the cylinder.
d=
±
Now calculate the radius.
(Think about what happens to the uncertainty)
r=
±
(Show your workings)
Calculate the radius squared – with it’s uncertainty!
r2 =
Finally work out the volume.
V=
±
13
±
Method 2: Using a micrometer screw
This uses the same prescription. However your precision should be a lot better.
Measure and write down the height of the cylinder.
Don’t forget to include the uncertainty and the units.
h=
±
Measure and write down the diameter of the cylinder.
d=
±
Now calculate the radius.
(Think about what happens to the uncertainty)
r=
±
(Show your workings)
Calculate the radius squared – with it’s uncertainty!
r2 =
Finally work out the volume.
V=
±
14
±
Method 3: Using Archimedes’ Principle
You’ve heard the story about the ‘Eureka’ moment when Archimedes dashed naked
through the streets having realised that an object submerged in water will displace an
equivalent volume of water. You will repeat his experiment (the displacement part at
least) by immersing the cylinder in water and working out the volume of water displaced.
You can find this volume by measuring the mass of water and noting that a volume of
0.001m3 of water has a mass6 of 1kg.
Write down the mass of water displaced.
±
Calculate the volume of water displaced.
±
What is the volume of the cylinder?
±
Discussion and Conclusions.
Summarise your results, writing down the volume of the cylinder as found from each
method.
±
±
±
Comment on how well they agree, taking account of the uncertainties.
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6
In fact this is how the metric units are related. A litre of liquid is that quantity that fits into a cube of side 0.1m and
a litre of water has a mass of 1kg.
15
Can you think of any systematic uncertainties that should be considered? Can you
estimate their size?
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Requote your results including the systematic uncertainties.
±
±
±
±
±
±
What do you think the volume of the cylinder is? and why?
±
My best estimate of the volume is
±
because ______________________________________________________________
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16
Practical Example 2: Measure the length of the lab bench.
Without using any explicitly calibrated equipment (e.g. a ruler), estimate the length of
the lab bench and give a reasonable uncertainty.
Briefly describe your technique here:
Tabulate the raw measurements you have made here together with their uncertainties.
17
Justify the size of the uncertainties on the raw measurements.
Show explicitly how you calculated the final answer from your measurements and how
you calculated an uncertainty on your final answer.
Quote your final result:The length of the bench is:
18
±
Gaussian Uncertainties
Here is some background to help in understanding experimental uncertainties which
may prove useful to the interested student.
As discussed earlier, quoting an experimental number with no uncertainty is all but
useless, particularly if we want to compare data to theory. However, even if we quote
an uncertainty as well, it is important to know what we mean by ‘uncertainty’. Since an
experimental measurement is really giving us information on a range of consistent
values, we should be trying to describe this by some function rather than a single value,
or a single value and an uncertainty.
As soon as you notice this, you realise that probability theory becomes very important.
For a given true value which you are trying to find, there are a range of measurements
you could get, each of which has a certain probability. You are most likely to get an
answer close to the truth, but sometimes you will happen, by chance, to be a bit further
off. We will quantify the probability of being off by a given amount, presently.
There are a few important probability distributions that arise naturally in nature.
Binomial statistics are obeyed by coin tosses. Goal scoring in football matches follow
Poisson statistics. But most of the time you can forget about all these and just consider
the Gaussian or Normal distribution. The reason for this is the Central Limit Theorem
which I will somewhat imprecisely summarise as saying that in the long run everything
looks like a Gaussian.7 So to understand how uncertainties propagate from your
measurements through to your answer, you really only have to know how Gaussian
distributions behave.
Given an (unknown) true value,
Probability of
experimental
values
will
be
obtaining
probabilistically distributed about it
experimental
at shown in this plot. The x-axis is
value
rescaled into units of experimental
One unit of
uncertainty.
Spend a little time
experimental
looking at this and appreciating what
uncertainty (σ)
it means
Straight away you can see that
about 68% of the time your
measurement will be within 1σ of the
truth, 95% within 2σ and 99% within
True value
3σ. For the mathematically inclined,
the
Gaussian
distribution
is
described
by
the
equation:
⎡ 1 ⎛ x − xtrue ⎞ 2 ⎤
1
exp ⎢ ⎜
⎟ ⎥
Experimental values P ( x ) =
2π
⎣⎢ 2 ⎝ σ ⎠ ⎦⎥
7
More precisely, the Central Limit Theorem states that if you create a random variable from a sum of independent
random variables, the expectation value of the sum is the sum of the separate expectation values, and the variance of
the sum is the sum of the separate variances. Furthermore, as the number of independent random variables
increases, you get closer and closer to a Gaussian distribution. Since experimental uncertainties usually are the
result of a series of different effects, their sum can therefore usually be modelled quite well be a Gaussian.
19
Once this distribution is known, you can work out how the uncertainty on the final
answer is related to the uncertainties on the individual sources. If you have two
independent sources with associated uncertainties, x ± δ x , y ± δ y , which your final
answer,
f
is a function of, then the uncertainty on
f
is δ f =
∂f
∂f
δ x ⊕ δ y where the
∂x
∂y
symbol ⊕ is a specially sort of addition called adding in quadrature: a ⊕ b = a 2 + b 2 .
Much of the time this simplifies to one of the following cases.
Case 1: Multiply an experimental measurement by a constant.
If
f = Ax then δ f = Aδ x
Case 2: Add or subtract two experimental measurements
If
f = x + y then δ f = δ x ⊕ δ y
Case 3: Multiply or Divide two experimental measurements
If
f = xy then
δf
f
=
δx
x
⊕
δy
y
Case 4: A functional dependence of an experimental measurement.
If
f = f ( x ) then δ f =
20
∂f
δx
∂x
21
Name:___________________________
Date:___________
Student No: ________________
Demonstrator:_________________________
Plotting Scientific Data
In many scientific disciplines, and
particularly in physics, you will often
come across plots similar to those shown
here.
Note some common features:
• Horizontal and vertical axes;
• Axes have labels and units;
• Axes have a scale;
• Points with a short horizontal
and/or vertical line through them;
• A curve or line superimposed.
Why do we make such plots?
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Why are there horizontal and vertical axes?
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Why are they labelled?
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Why do they have a scale?
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What do the points represent?
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Why do they have short vertical or horizontal lines through them?
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Why is there a superimposed line or curve?
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How close to all the points should the line pass?
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When can you say that theory and experiment are in good agreement?
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24
Comment on the agreement of theory and experiment in each of these plots.
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25
We will now perform a series of increasing complex examples culminating in a data set
which is typical of what you will produce in the laboratory, requiring that you both
present the data clearly and use it to estimate a physical parameter. The ability to do
this with ease and to understand what you are doing and why, is essential to
successfully completing the practical laboratories.
Example 1: Simple linear dependence with graph produced by hand.
We start with a very simple, perhaps ‘obvious’, example.
The following simple data relate to the speed of a car as it accelerates from rest. Take
a look at the data and answer the questions below.
Speed (ms-1):
1
3
5
7
11
Time (s)
0
1
2
3
5
Describe in words what you notice about the relationship between speed and time?
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Can you write this as an equation relating speed (s) and time (t) ?
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26
Graph the data below. Choose a scale that is simple to read and expands the data so it
is spread across the page. Label your axes.
27
Algebraically a straight line can be described by y = mx + c where x and y refer to
any data on the x and y axes respectively, m is the slope of the line (∆y/∆x), and c is
the intercept (where it crosses the y-axis).
Suppose that theoretically I tell you that the data should be consistent with a straight
line. Superimpose the best straight line you can draw on the data.
Work out the slope of this line.
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What is the intercept?
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Compare the slope and intercept to the formula you hypothesised when you first saw
the data.
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28
Example 2: More realistic linear dependence with graph produced by hand.
You probably will never come across experimental data as in example 1, since there it
is implied that both the speed and time have been perfectly determined. Experimental
data will have uncertainties associated with the measurement process and these must
be recorded, displayed in your plots, and correctly assessed when making fits to the
data.
Consider now this data relating to the speed of a car as it accelerates.
Time (s)
Speed (km h−1)
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
16 ± 6
18 ± 6
37 ± 6
44 ± 6
58 ± 6
62 ± 6
64 ± 6
70 ± 6
99 ± 6
This time the linear relationship (if it exists) is much less obvious and you will need to
plot the data or do some further analysis to see it.
The values for the speed of the car now have an associated experimental uncertainty.
The values for the time do not. This does not mean that there is no experimental
uncertainty in the time measurement; rather that the relative uncertainty is much smaller
for time than for speed, and so the scientist has decided that they are negligible
compared to the dominant speed uncertainties.
When plotting this data the usual convention is to place a point at the experimentally
determined value and to extend a line through this point, the length of the line
corresponding to the estimated uncertainty.
Make a plot of this data.
Don’t forget to label your axes.
29
30
Is there evidence for a linear relationship between time and speed?
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Superimpose the ‘line of best fit’ on the graph above.
What is the slope of this line.
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What is the intercept?
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31
Theoretically I now hypothesise that speed (v) and time (t) are related by the equation
v = v0 + at
where v0 is the initial speed and a is the acceleration of the car.
What is the best value for the acceleration of the car?
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How well do you know this value? What is the uncertainty on the acceleration?
How can you determine it?
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What was the initial speed of the car (at time t=0)?
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How well do you know this value?
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Predict the speed of the car after 13 seconds.
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32
Example 3: Take your own data in the laboratory. Plot it and work out the density of a
material.
The density of a material is defined as its mass divided by its volume: ρ =
Thus, the volume of a material is proportional to its mass: V =
M
ρ
M
.
V
.
In the laboratory you will experimentally check whether this relationship holds, and if it
does, work out an experimental value for the density of a material. The data you will
take is simple but there are a lot of subtle points about identifying sources of uncertainty
in the data, interpreting your data to prove or disprove the hypothesis, and working out a
value for the density of the material (with its corresponding uncertainty).
Experimental Procedure
Fill the cylindrical vessel until its about 20% full. The radius of the cylinder will be given
to you, and can be assumed known to high precision. Measure the height of the
material within the cylinder using a ruler. Estimate the uncertainty on this
measurement. Now work out the volume of material you have given that the volume of
a cylinder is πr h . Calculate the uncertainty on this volume. Finally measure the
mass of material using the precise electronic balance.
2
Now repeat these measurements filling the vessel to about 40%, 60%, 80% and 100%
of its capacity. Record your results in the table below.
Data
Height
Uncertainty on
Height
Volume
Uncertainty on
Volume
Mass
Graph
Plot your results on the next page with mass on the x-axis, and volume on the y-axis.
Select an appropriate scale and label the axes. Include error bars.
33
34
Data Analysis
Your theoretical hypothesis is that V =
M
ρ
or that there is a linear relationship between
mass and volume.
From your graph, is there evidence for a linear relationship?
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Superimpose the ‘line of best fit’ on your graph.
What is the slope of this line?
What is the intercept?
Theoretically, what do you expect the slope to represent?
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Theoretically, what do you expect the intercept to be?
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35
Comment on how well your experimentally determined intercept agrees with your
theoretically expected intercept.
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From the slope of your graph, work out a value for the density of the material.
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How could you determine the correct uncertainty on this value?
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36
Example 4: Relieving the tedium....and improving our precision
In the examples above we have somewhat causally referred to the ‘best fit’ through the
data. What we mean by this, is the theoretical curve which comes closest to the data
points having due regard for the experimental uncertainties.
This is more or less what you tried to do by eye, but how could you tell that you indeed
did have the best fit and what method did you use to work out statistical uncertainties on
the slope and intercept?
The theoretical curve which comes closest to the data points having due regard for the
experimental uncertainties can be defined more rigorously8 and the mathematical
definition in the footnote allows you to calculate explicitly what the best fit would be for a
given data set and theoretical model. However, the mathematics is tricky and tedious,
as is drawing plots by hand and for that reason....
We can use a computer to speed up the plotting of experimental data and to improve
the precision of parameter estimation.
In the laboratories a plotting programme called Jagfit is
installed on the computers. Jagfit is freely available for
download from this address:
http://www.southalabama.edu/physics/software/software.htm
Double-click on the JagFit icon to start the program. The working of JagFit is fairly
intuitive. Enter your data in the columns on the left.
•
•
•
Under Graph, select the columns to graph, and the name for the axes.
Under Error Method, you can include uncertainties on the points.
Under Tools, you can fit the data using a function as defined under
Fitting_Function. Normally you will just perform a linear fit.
Note that when you fit the data, a box will open with the values for the fitted parameters.
It will also give you a value for the reduced chi-squared χ 2 / N which is an indication of
the goodness of fit to your fitting hypothesis. This value should be about 1: values from
0.5 to 2 are reasonable. If your χ 2 / N is very much bigger than 1, something is wrong.
Check your data. Perhaps you have a typo or you have not allowed for a significant
source of error. Alternatively the fitting hypothesis may be incorrect. More unusually
Technically, if your data points are given by ( xi , yi ) with uncertainties σ i on yi , and you have
a theoretical function f that relates x to y via y = f ( x; a) where a are free parameters, then the
8
2
⎛ y − f ( xi ; a ) ⎞
⎟⎟ .
best values for a are found by minimising the quantity χ = ⎜⎜ i
σi
⎠
⎝
If you want to know more about this equation, why it works, or how to solve it, ask your
demonstrator or read about ‘least square fitting’ in a text book on data analysis or statistics.
37
2
χ 2 / N is very much smaller than 1, the most likely explanation being that your
uncertainties are over estimated.
1. Input the data from Example 2 into JagFit.
2. Plot the graph and fit a linear fit using the drop down menu.
3. Put labels on the x- and y-axes.
4. Choose an appropriate scale so that the data is clearly visible.
5. Suppose that theoretically I hypothesise that speed (v) and time (t) are related by
the equation
v = v0 + at
where v0 is the initial speed and a is the acceleration of the car.
6. What value do you get for the acceleration of the car?
7. What is the uncertainty on this value?
8. What was the initial speed of the car at time t=0?
9. What is the uncertainty on this value?
10. How do these values compare to those you found in Exercise 2?
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11. What is the value for the reduced chi-squared?
12. Does this indicate a good or bad fit?
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38
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Print out your graph and attach it here.
39
Example 5: The swinging pendulum.
It’s not just straight lines you can fit to data. Under Fitting Function you can see there
are options to fit the data with polynomials, power laws and exponentials. Furthermore,
scientists routinely fit more complicated theoretical functions to the data using
essentially the same technique.
In this example you will work out the acceleration due to gravity from data obtained from
a swinging pendulum. This will be done in two different ways: first you will fit a power
law dependence to the data; and second you will recast your data in order to fit a
straight line. Needless to say, you ought to get the same result for the acceleration due
to gravity.
The simple pendulum is an example of a mechanical system that exhibits periodic
motion. It consists of a particle-like bob suspended by a light string of length L, that is
fixed at the upper end. Application of Newton’s second law to this system gives an
expression for the period T of the oscillation, i.e., the time taken by the pendulum to
undergo a complete cycle. The period is given by
T = 2π
L
.
g
(Eq.1)
The above formula tells you that the period of a simple pendulum depends only on the
length of the string, L, and the acceleration due to gravity, g. Note that the period is
independent of the mass of the bob.
In order to test the above formula, a simple pendulum is set up in the laboratory. During
the experiment, the length of the string is varied and the time taken for 50 complete
oscillations is recorded. Each measurement is repeated 5 times in order to estimate the
uncertainty in the measured period. A summary of the results obtained is given in the
table below.
Length (m)
L
Period (s)
T
0.05
0.456 ± 0.008
0.10
0.645 ± 0.008
0.20
0.906 ± 0.007
0.40
1.270 ± 0.006
0.60
1.556 ± 0.006
0.80
1.789 ± 0.004
1.00
2.009 ± 0.005
1.20
2.191 ± 0.006
1.40
2.366 ± 0.004
1.60
2.540 ± 0.003
40
Method 1
Using Jagfit, plot the above data using ‘Length’ as the independent variable on the xaxis and ‘Period’ as the dependent variable on the y-axis. Use a third column to
introduce the uncertainties in the measured period. Plot the error bars associated with
this variable, label the axes and scale the graph appropriately so that the data are
clearly seen.
Let us consider Eq. 1 again. We can re-write this equation as
⎛ 2π ⎞
⎛
⎞
⎟ L = ⎜ 2 π ⎟ L0.5 .
T = ⎜
⎜ g⎟
⎜ g⎟
⎝
⎠
⎝
⎠
(Eq. 2)
This formula is of the form
y = a xb
(Eq. 3)
which is the equation for a ‘power function’.
Compare (Eq. 2) with (Eq. 3). If we plot the period T as a function of the length of the
pendulum L, we expect the data to be represented by a ‘power law’, where
a =
b =
and
Using Jagfit, fit the data to a power function. To do this, select the Power Law Fit within
the Fitting function menu.
What values do you get for a and b ?
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What is the value for the reduced chi-squared, χν2/N ?
What does this tell you about the fit?
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41
Print out your graph and attach it here.
42
Let us now compare the values you obtained with those predicted by the theory.
Write down the value for b? Is this compatible with 0.5? Should it be? Comment.
Value of b: ___________ ± ___________
Compatible?: __________
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Write down the value for a. From this, derive a value for the acceleration due to gravity,
g. Is this value what you expect? (Note that the most precise experiments measure g
at sea level to be between 9.780 ms-2 and 9.785 ms-2, depending on your location.)
Value of a: __________ ± ___________
Value of g: __________ ± ___________
Compatible?: __________
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43
Method 2
We will now use a little algebra to recast Eq. 2 so that we can perform the more usual
straight line fit.
⎛ 2 π ⎞ 0.5
⎟ L . Take the natural logarithm of both sides and show
T = ⎜
Eq. 2 said that
⎜ g⎟
⎝
⎠
that you can write this equation in the linear form y = mx + c where y represents the
natural log of T and x represent the natural log of L.
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What is m?
What is c?
Using JagFit, make a plot of ln T on the y-axis against ln L on the x-axis.
Think about what you will do to the uncertainties on T.
Make a straight-line fit to the data and record the values for the slope and intercept
below.
Slope =
Intercept =
What is the value for the reduced chi-squared, χν2/N ?
What does this tell you about the fit?
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44
Print out your graph and attach it here.
45
From the slope and intercept work out the value for the acceleration due to
gravity.
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g=
Does this agree with your previous determination? Should it?
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This example has shown you that there is more than one way to plot your data in order
to extract the physical quantity. Much of the time you can recast your data so that it has
a linear dependence which allows you fit a straight line. Having manipulated the
theoretical formula, take care that you know how to extract the physical parameter from
the slope and intercept of your straight line.
46
47
Name:___________________________
Date:___________
Student No: ________________
Demonstrator:_________________________
Newton’s Second Law.
Introduction
r
r
F = ma ,
a force causes an acceleration and the size of
Newton’s second law states
the acceleration is directly proportional to the size of the force. Furthermore, the
constant of proportionality is mass.
This experiment has two parts. In the first part you will apply a fixed force, vary the
mass and note how the acceleration changes. In the second part you will measure the
acceleration due to the force of gravity.
Apparatus
The apparatus used is shown here and
consists of a cart that can travel along a
low friction track. The cart has a mass of
0.5kg which can be adjusted by the
addition of steel blocks each of mass
0.5kg. String, a pulley and additional
masses allow forces to be applied to the
carts.
Take care to ensure that the track is
completely level before starting the
experiments
Investigation 1: Check that force is proportional to acceleration and show the constant
of proportionality to be mass. Calculate the acceleration due to gravity.
The apparatus should be set up as in the
picture. Attach one end of the string to
the cart, pass it over the pulley, and add
a 0.012kg mass to the hook.
48
The weight of the hanging mass is a force, F, that acts on the cart. The whole system
(both the hanging mass mh and the cart mcart) are accelerated. So long as you don’t
change the hanging mass, F will remain constant. You can then change the mass of
the system, M, by adding mass to the cart, noting the change in acceleration, and
testing the relationship F=ma.
If F=ma and you apply a constant force F, what do you expect will happen to the
acceleration as you increase the mass?
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There remains the problem of working out the acceleration of the system.
Recall the kinematics formula
s = ut +
1 2
at , where s is distance, u the initial velocity,
2
t is time, and a acceleration.
If the cart starts from rest write down
an expression for ‘a’ .
Place different masses on the cart. Using ruler and stopwatch measure s and t and
hence the acceleration a. Fill in the table below.
mh
(kg)
M=
mcart
mh+mcart
(kg)
(kg)
s (m)
0.012 0.5
t (s)
s (m)
0.012 1.0
t (s)
s (m)
0.012 1.5
t (s)
±
a (m/s2)
M.a (N)
±
±
±
±
±
±
±
±
±
±
±
49
From the numbers in the table what do you conclude about Newton’s second law?
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Analysis and determination of the acceleration due to gravity.
If Newton is correct, F = ( m h + mcart ) ⋅ a
You have varied mcart and noted how the acceleration changed so let’s rearrange the
formula so that it is in the familiar linear form y=mx+c. Then you can graph your data
points and work out a slope and intercept which you can interpret in a physical fashion.
F = (mh + mcart ) ⋅ a ⇒
m
1 1
= mcart + h
a F
F
So if Newton is correct and you plot mcart on the x-axis and 1/a on the y-axis you should
get a straight line.
In terms of the algebraic quantities above,
what should the slope of the graph be equal to?
In terms of the algebraic quantities above,
what should the intercept of the graph equal?
Plot the graph and see if Newton is right. Do you get a straight line?
What value do you get for the slope?
What value do you get for the intercept?
50
Print out your graph and attach it here
51
Now here comes the power of having plotted your results like this. Although we haven’t
bothered working out the force we applied using the hanging weight, a comparison of
the measured slope and intercept with the predicted values will let you work out F.
From the measurement of the slope, the constant force applied can be calculated to be
From the intercept of the graph you can calculate F in a different fashion. What value
do you get?
You’ve shown that F is proportional to a and the constant of proportionality is mass. If
the force is the gravitational force, it will produce an acceleration due to gravity
(usually written g instead of a) so once again a (gravitational) force F is proportional to
acceleration g. But what is the constant of proportionality? Are you surprised that it is
the same mass m? (By the way, a good answer to this question gets you a Nobel prize.)
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So since that force was just caused by the mass mh falling under gravity, F= mhg, you
can calculate the acceleration due to gravity to be
Comment on how this compares to the accepted value for gravity of about 9.785 m/s2
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52
Investigation 2: Acceleration down an inclined plane.
The apparatus should be set
up as in the picture. Raise one
end of the track using the
elevator. The carts can be
released from rest at the top of
the track.
A cart on a slope will
experience the force of gravity
which causes the cart to
accelerate and roll down the
incline. The acceleration due
to gravity is as shown in the
schematic. The component of
this acceleration parallel to the
inclined surface is, g.sinθ and
this is the net acceleration of
the cart (when friction is
neglected).
Note
the
acceleration depends on the
angle of the incline.
Vary the angle of the incline (repeat for at least four different angles) and measure the
1
acceleration of the cart using s = ut + at 2
2
How will you measure the angle of the incline?
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53
Summarise your results in this table.
Angle (radians)
Distance (m)
Acceleration (ms-2)
Time (t)
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
±
Since the acceleration a = g sin θ , there is a linear dependence on the sine of the
angle. Make a graph with sin θ on the x-axis and a on the y-axis.
What value do you expect (theoretically) for the slope?
What value do you expect (theoretically) for the intercept?
What value do you obtain (experimentally) for the slope?
What value do you measure for the intercept?
Comment on your results.
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54
Print out your graph and attach it here
55
Name:___________________________
Date:___________
Student No: ________________
Demonstrator:_________________________
Waves and Resonances
Introduction
This experiment produces waves in a stretched wire and looks at how the frequency of
vibration is related to the tension of the wire, its length and the mass per unit length.
The phenomenon of resonance is used in the investigation.
Using Newton’s laws, it can be shown9
that the velocity, v, of a symmetrical pulse
on a string, as shown on the right, is
related to the tension of the string, T, and
the mass per unit length, µ, by
v= T
µ
(Eq. 1)
Thus the velocity of a wave along a string depends only on the characteristics of the
string and not on the frequency of the wave. The frequency of the wave is fixed entirely
by whatever generates the wave. The wavelength of the wave is then fixed by the
familiar relationship:
v = fλ
(Eq.2)
Example: A string on a bass guitar is 1m long and held under a tension of 100N. If the
string has a mass of 10g, what is the velocity of a wave on the string?
∆l experiences a tension on each end. The horizontal components cancel and
a net vertical force of 2T sinθ ≈ 2Tθ = T∆ l R acts. Newton says this force = mass times
acceleration. The mass is µ∆l . Looking at the system from a frame where the pulse is at rest (and the
9
The portion of the wire
wire moves with velocity v), the portion of wire moves along the indicated circle and the centripetal
acceleration is
v2 R .
Thus T∆ l
R = ( µ∆l )(v 2 R) and the result follows.
56
If a wave with a frequency of 80Hz is sent into the string, what will its wavelength be?
If a wave with a frequency of 50Hz is sent into the string, what will its wavelength be?
A resonant frequency is a natural frequency of vibration determined by the physical
parameters of the vibrating object. There are a number of resonant frequencies for a
string which are the multiples of its length that allow standing waves to be formed. Thus
λresonant = 2l / n .
where n=1,2,3...
What is the lowest resonant frequency (the fundamental) of the bass guitar string
referred to above?
What happens if a wave with a frequency which is the same as the resonant frequency
enters the string?
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What happens if a wave with a frequency which is different to the resonant frequency
enters the string?
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57
Experimental Procedure
In this experiment we will attempt to confirm Eq.1 above by sending waves of different
frequency into the wire and identifying the resonant frequency.
The apparatus is shown above and consists of a stretched string held under tension by
a hanging weight. In this configuration, the tension equals the force exerted by gravity.
The wire passes over a bridge which defines a node thus changing the length in which a
wave can resonate. A small horseshoe magnet should be placed half-way between the
bridge and the pulley. An AC generator is connected to either side of the wire and can
send electrical signals of selected frequencies down the wire.
Choose eight different positions of the bridge (remember to move the magnet too), and
find the lowest frequency at which maximum vibrations (resonance) occurs. This can
be observed by placing a folded piece of paper on the wire and noting when it gets
thrown off by the vibrations. Record your data below.
Resonant Frequency
(Hz)
Length
(m)
58
1/Length
(m-1)
Wire length (m)
Mass/ length
µ (kg/m)
Eq.1&2 can be combined to give
f =1
T
λ
µ
.
When this is at the resonant frequency
f resonant = n
2l
T
µ
(Eq. 3)
Plot the resonant frequency against 1/l.
What should the slope be equal to algebraically and numerically?
What is the slope of a straight line fit to your data?
Comment on whether you consider Eq.3 has been verified and whether a good
agreement exists between theory and experiment.
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59
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Now place the bridge in a fixed position. Vary the tension in the wire by adding or
removing weights, and find the lowest frequency at which maximum vibrations occur.
Resonant Frequency
(Hz)
Frequency squared
(s-2)
Wire tension
(N)
Wire length (m)
Mass/ length
µ (kg/m)
Plot the square of the resonant frequency against the tension.
What should the slope be equal to algebraically and numerically?
What is the slope of a straight line fit to your data?
60
Comment on whether you consider Eq.3 has been verified and whether a good
agreement exists between theory and experiment.
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If you have time, set up a resonant position and write down the frequency. Now
increase the frequency until you see resonance again, and write down this frequency.
What do you notice? With reference to Eq.3, explain what is happening.
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Print out your graph and attach it here
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Print out your graph and attach it here
63
Name:___________________________
Date:___________
Student No: ________________
Demonstrator:_________________________
Investigation into the Behaviour of Gases
and a Determination of Absolute Zero
Introduction:
The gas laws are a macroscopic description of the behaviour of gases in terms of
temperature, T, pressure, P, and volume, V. They are a reflection of the microscopic
behaviour of the individual atoms and molecules that make up the gas which move
randomly and collide many billions of times each second.
Their speed is related to the temperature of the gas through the average value of the
kinetic energy. An increase in temperature causes the atoms and molecules to move
faster in the gas.
The rate at which the atoms hit the walls of the container is related to the pressure of
the gas.
The volume of the gas is limited by the container it is in.
Using the above description to explain the behaviour of gases, state whether each of
P,V,T goes up (↑), down (↓) or stays the same (0) for the following cases.
P
V
T
A sealed pot containing water
vapour is placed on a hot oven hob.
Blocking the air outlet, a bicycle
pump is depressed.
A balloon full of air is squashed.
A balloon heats up in the sun.
From the arguments above you can see that at constant temperature, decreasing the
volume in which you contain a gas should increase the pressure by a proportional
amount.
PV = k .
Thus P ∝ 1V or introducing k, a constant of proportionality, P = k V or
This is Boyle’s Law.
Similarly you can see that at constant volume, increasing the temperature should
increase the pressure by a proportional amount. So P ∝ T or introducing κ , a
constant of proportionality,
P = κT
or
P =κ .
T
64
This is Gay Lussac’s Law.
These laws can be combined into the Ideal Gas Law which relates all three quantities
via PV
constant.
= nRT
where n is the total number of moles of gas and R is the ideal gas
Experiment 1: To test the validity of Boyle’s Law
The equipment for testing Boyle’s Law
consists of a volume of gas in a tube
and a bicycle pump that can be used to
compress the gas using hydraulic fluid.
A gauge is attached to measure the
pressure. The height of gas in the tube
can be measured using the scale
situated behind the tube.
Procedure
•
•
•
•
•
•
Loosen the valve to de-pressurise the reservoir to atmospheric pressure. Record
the height of gas in the tube.
Connect the pump to the apparatus and pump to the maximum gauge pressure
using sharp strokes.
The absolute pressure of the system is the gauge pressure plus atmospheric
pressure (take this as 1.013 x 105 N m-2 if you can’t measure it directly in the
laboratory). Note that the gauge is calibrate in units called ‘bars’; you need to
convert this to the S.I. unit of pressure, the pascal (Pa), noting 1 bar = 105 Pa.
Allow the levels to settle and calculate the volume of gas in the tube given by
πr2h, where h is the height of air and the inner radius of the tube is 0.003 m.
De-pressurise the reservoir in small steps, typically 0.5 bar, by slowly unscrewing
the valve retaining nut until the required pressure is reached, then rotating it
clockwise to hold the pressure. Wait a few seconds to allow the oil level to settle.
Tabulate your results below.
65
Pressure on
gauge (Nm-2)
Height of gas
(m)
Total pressure on
gas (Nm-2)
Volume of gas
(m3)
±
±
±
±
±
±
±
±
±
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±
±
±
±
±
±
±
±
±
±
±
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±
±
±
±
±
±
Compare Boyle’s Law: P = k
V to the straight line formula y=mx+c.
What quantities should you plot on the x-axis and y–axis in order to get a linear
relationship according to Boyle’s Law? Explain and tabulate these values below.
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x-axis
y-axis
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Print out your graph and attach it here
67
What should the slope of your graph be equal to?
What should the intercept of the graph equal?
Make a plot of those variables that ought to give a straight line, if Boyle’s Law is correct.
Comment on the linearity of your plot. Have you shown Boyle’s Law to be true?
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What is the value for the slope?
What is the value for the intercept?
Use the Ideal Gas Law to figure out how many moles of air are trapped in the column.
(The universal gas constant R=8.3145 J/mol K)
Number of moles of gas =
68
Experiment 2: To test the validity of Gay Lussac’s Law
The equipment for testing Gay
Lussac’s Law consists of an enclosed
can surrounded by a heating element.
The volume of gas in the can is
constant. A thermocouple measures
the temperature of the gas and a
pressure transducer measures the
pressure.
Procedure
Plug in the transformer to activate the temperature and pressure sensors. Set the
multimeters at the red marks where one is designed to read temperature in centigrade
and the other must be multiplied by 010 in order to get pressure in pascals.
Connect the power supply to the apparatus and switch on. Start with everything at
room temperature, and commence heating the gas. up to a final temperature of 100 C.
Immediately switch off the power supply.
While the gas is heating, take temperature and pressure readings at regular intervals
(say every 5 C) and record them in the table below.
Temperature
(C)
Pressure of gas
(Nm-2)
Temperature
(C)
Pressure of gas
(Nm-2)
±
±
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Gay Lussac’s Law states P = kT and by our earlier arguments heating the gas makes
the atoms move faster which increases the pressure. We need to think a little about our
scales and in particular what ‘zero’ means. Zero pressure would mean no atoms hitting
the sides of the vessel and by the same token, zero temperature would mean the atoms
have no thermal energy and don’t move. This is known as absolute zero.
The zero on the centigrade scale is the point at which water changes to ice and is
clearly nothing to do with absolute zero. So if you are measuring everything in
centigrade you must change Gay Lussac’s Law to read P = k (T − Tzero ) where Tzero is
absolute zero on the centigrade scale.
Compare P = k (T − Tzero ) to the straight line formula y=mx+c.
What should you plot on the x-axis and y–axis in order to get a linear relationship
according to Gay Lussac’s Law?
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What should the slope of your graph be equal to?
What should the intercept of the graph equal?
How can you work out Tzero?
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Make a plot of P versus T.
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Print out your graph and attach it here
71
Comment on the linearity of your plot. Have you shown Gay Lussac’s Law to be true?
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What is the value for the slope?
What is the value for the intercept?
From these calculate a value for absolute zero temperature.
Absolute zero =
±
72
73
Name:___________________________
Date:___________
Student No: ________________
Demonstrator:_________________________
Electrons in Atoms : The Spectrum of Atomic
Hydrogen
Introduction
The aim of this experiment is to observe the spectrum of hydrogen, to explain the
spectrum in terms of the motion of electrons between discrete energy levels in the
hydrogen atom and to evaluate a constant that is fundamental to calculating the
wavelengths of the lines in the spectrum of hydrogen. The spectral emission lines that
are seen in the visible region of the spectrum (red through violet) belong to the Balmer
series and the positions of these lines may be determined using a spectrometer. From
the observed wavelengths, the Rydberg formula may be applied and the Rydberg
constant obtained.
Background theory and historical perspective
At the turn of the last century measurement of the wavelengths of spectral lines in the
light given off by low pressure hydrogen gas in an electrical discharge gave very
consistent, highly repeatable results. The emission of light came to be understood in
terms of electrons moving between discrete energy levels. The wavelengths of the lines
fitted a very regular pattern. It was found that all of the wavelengths, λi , could be
calculated by using the Rydberg formula:
1/λ i = R{(1/nf)2 - (1/ni)2}.
(1)
Here, nf and ni are both integers (called the principal quantum numbers) with ni > nf. R
is a constant known as the Rydberg constant. These numbers were originally fitted to
the series without any understanding of their physical meaning. However, most
physicists felt that something basic must be inherent in such a simple equation. Niels
Bohr was one of them. Having worked with Rutherford, he was aware of the nuclear
model of the atom. In 1913 he postulated:
(1)
(2)
(3)
(4)
that the electron in the hydrogen atom is in circular motion (in orbit)
about the nucleus (which he assumed to be stationary),
that it can only exist without radiating electromagnetic energy in certain
"allowed" states of definite energy,
that these were specified by quantised values nh/2π where h is
Planck’s constant and n is an integer,
that transitions between these "allowed" states can occur only if photon
emission or absorption occurs and the photon energy must be equal to
the energy difference between the initial and final states. The photon
energy (E) is given by
Ei – Ef = hf.
(2)
74
Based on this model, we can state that the number ni refers to the initial (or upper) level.
The values for nf are lower than those of ni, as they refer to the lower (or final) level
involved in the electronic transition which results in the emission of light. The value of nf
is 2 for the Balmer series, as all of the electronic transitions terminate with the electron
in the n = 2 level. The first three of these series (electrons “falling” to n = 1, n = 2 and n
= 3) are illustrated in Fig.2 below. The balmer series involves eletrons moving between
ni = 3,4,5,6,7….. to nf = 2. Three transitions in this series are shown below.
Identify the series of transitions (Lyman, Balmer or Paschen) that occur at the longest
wavelength. Explain your reasoning.
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Calculate the photon energy of a photon in the red region of the spectrum
(λ = 658
8
nm), using equation 2 above and the fact that the the speed of light (3 × 10 ms-1) is
related to the wavelength (λ) and frequency (f) by the equation c = λf.
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Using these postulates, Coulomb's law, and Newtonian mechanics, Bohr was able to
derive the above equation from first principles and express the Rydberg constant in the
form:
me 4
R=
8cε o2 h 3
(3)
Calculate the value of R, given the following values:
m, the mass of the electron = 9.1 × 10-31 kg.
e, the charge on the electron = 1.6 × 10-19 C.
εo, the permittivity of a vacuum = 8.8 × 10-12 C2N-1m2
h, Planck’s constant = 6.6 × 10-34 Js.
Fill in the table below and use equation 1, and the value for R to calculate λ for the
numbers ni = 3,4,5 and nf = 2. Don’t forget your units!
ni = 3
ni = 4
ni = 5
(1/nf)2 - (1/ni)2
λi
The results of the Bohr Theory were not perfect, but they were close enough to point the
physics community in the proper direction to more fully develop the new quantum
physics.
The purpose of this experiment is to use the spectrometer to observe that angles at
which different wavelengths (colours) of light are diffracted and to calculate those
wavelengths. This will allow you to verify the value of the Rydberg constant.
76
Procedure:
The experimental arrangement and a schematic of the spectrometer is shown below.
The hydrogen discharge tube (light source) is positioned in front of the entrance slit of
the spectrometer.
The spectrometer is adjusted as follows. (1) The entrance slit is set vertical. (2) Adjust
the length of the collimator till the mark on the inner barrel is just visible. At this setting
the light emerging will be collimated. (3) Set the grating perpendicular to the collimator
and with the groves vertical, (4) Align the telescope with the collimator when an image
of the entrance slit will be seen. (5) Adjust the eyepiece till the crosswire is in sharp
focus, (6) adjust the length of the telescope till the image of the slit is in sharp focus,
(7) finally point the spectrometer directly at the discharge tube so that the light through
the entrance slit is as bright as possible.
A vernier scale is an auxiliary sliding scale used to more accurately read the values on a
fixed main scale. Its purpose is to allow accurate readings rather than estimations,
between the smallest graduations on a fixed scale. This vernier scale has 30
graduation marks. Each division is 1/30 of the smallest division on the main scale or
one minute. To use the vernier scale, read the main scale to the last certain digit which
is the graduation just below the zero on the vernier scale ie 136o . The mark on the
vernier scale that directly lines up with a graduation mark on the main scale is 8’ and
finally the reading is 136o + 8’ = 136o 8'
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With the collimator and telescope in a straight line, the light viewed is the zero order line
(m = 0) from the diffraction grating. All of the colours from the lamp are viewed
simultaneously at the zero order. The telescope arm is rotated to the left and using the
vernier the anglar positions, θL , of the blue, green and red lines in the first and second
order are determined and tabulated.
Colour
ni
Purple
5
1/ni2
m
θL
θR
θm = (θL - θR) /2 λ i
Mean λ i 1/λ i
1
2
BlueGreen
4
1
2
Red
3
1
2
The telescope arm is returned to the straight through position and rotated to the right
and the procedure above is repeated to obtain θR for each line. Record the data in the
table above. Thus, the angular position of each spectral line with respect to the direction
of the incident light is given by:
θm = (θL - θR) / 2
(4)
This enables a correction to be made for any slight deviation from the perpendicular of
the diffraction grating to the optic axis of the spectrometer. Note all angular readings
must be taken in the same window (i.e. on the same side) of the spectrometer.
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Analysis & theory of operation of the spectrometer
The wavelengths of the visible emission lines of the hydrogen spectrum are recorded
using a spectrometer equipped with a diffraction grating. In passing through the grating,
light of different wavelengths (colours) is diffracted at different angles. The diffraction
grating formula is given by:
mλ =d·sinθm
(5)
where λ is the wavelength of the spectral line being observed, d is the groove
spacing of the diffraction grating, m is the order of the spectrum, θm is the
corresponding angle at which the mth order spectrum is observed. The 1st order (m = 1)
occurs at the smallest angle with respect to 90o to the grating surface, the 2nd order
occurs at larger angles and so on. Which colour do you expect to be diffracted through
the largest angle? Explain your reasoning.
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From your data and using Eqn. 5, calculate the wavelengths of the visible lines in the
spectrum of hydrogen.
Create a plot of 1/λi against 1/ni2 (i = 3, 4, 5).
Print out your graph and attach it here
80
By re-arranging Eqn. 1 in the form y = mx + c as
1/λ i = -R/ni2 + R/nf2
(6)
the value of R can be found from the slope of the above graph.
Write the value for the slope of the graph.
Write the value for the intercept of the graph.
Write the value of R calculated earlier.
Write the value of R estimated from the slope of the graph.
The intercept of this graph will give R/nf2.
Write the value of R estimated from the intercept of the graph.
Compare the values you obtained based on your measurements with the calculated
value of R and comment on the accuracy of your measurements.
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