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Department of Physics
Chapter 5
Skill Session 1 Errors
5.1 Introduction
There will always be some
uncertainty in the readings you take
and therefore data without an
assessment of the associated
uncertainty is as useless as data
presented without units.
Where
instruments specify an accuracy, take
this as the error in your readings; for
digital instruments, assume an error
of plus or minus one digit, unless you
know better. In subsequent Units
you will learn detailed statistical
techniques for the quantification of
random errors through repeated or
related readings. However, here,
only the most basic concepts are
required. These do not lead to the
best estimate of your errors (your
errors are likely to be overestimated), but they will enable you
to get used to the process and
techniques of error analysis without
undue mathematical complexity.
are spaced half a wavelength apart.
An average could be obtained by
taking the difference between the
final and initial scale readings and
dividing by (n/2), but this procedure
neglects all intermediate readings.
5.2.1 How NOT to Proceed
Do
not
average
successive
differences as shown above. In this
 Y1 = Y2 - Y1
Y2 = Y3-Y2
 Y n-1 = Y n - Y n-1
n
Y =Y
i
- Y1
case the average is independent of all
but the extreme readings.
5.2.2 The Correct Method
Take an even number of readings,
and pair them as shown below for n
= 12.
5.2 Averaging
Suppose that you have to measure
the wavelength of a standing wave
(), and also the uncertainty in that
value, by measuring the spacing of n
successive pressure maxima, which
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n
1
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11
10
9
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6
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Note that in this method each reading
is used once only. The data yield 6
independent values of 3. The
variation between the 6 differences
enables the random error to be
estimated.
5.3 Accuracy of the Mean
Value
You will frequently repeat individual
readings, each of which is subject to
a random error, and then calculate a
mean value from 3 or 4 repeats. This
is then your best estimate of the
parameter, A, of interest. Although
there are statistical techniques which
enable you to calculate the likely
error using such an approach, for the
activities in this module, simply
repeat a measurement 3 or 4 times
and then generally take plus or minus
half the spread of readings as the
error, A. You can easily visualize
the spread if you choose to plot a
blob chart, as in Section 4.3.
they claim to be. For example, a
diffraction grating could have only
590 lines per mm rather than the 600
lines per mm quoted, so that
wavelengths measured with this
grating will all be in error by more
than 1%.
Errors of this type are called
systematic, and may sometimes be
difficult to deduce.
In some
experiments, it is possible to
eradicate systematic errors by
calibrating the equipment against
apparatus of known high precision.
5.5 Combining Errors
So far we have discussed how to
estimate the random error A
associated with a quantity A.
Usually, however, the end result
depends on intermediate values B, C,
etc, all of which have their own
uncertainties.
Statistical theory
states that:-
 A   A

A   B   
C 
 B
  C

2
5.4 Systematic Errors
A spread about an average value is
obtained because of random errors,
which arise from factors such as the
impossibility of reading with
complete accuracy a scale marked
with finitely wide divisions. A
different source of error arises if the
divisions on the scale are not what
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2
where A = f(B,C), B = the error in B
and C = the error in C. So, if:-
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A=B+C
A 
B  C 
2
2
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This gives the likely error in A.
However, it is possible that the errors
in B and C are both at a maximum.
Then the error in A would be given
by:-
A = B+ C
Whilst it is generally pessimistic to
assume that all the intermediates will
exhibit their maximum errors
simultaneously, this assumption is
simple to use. Such an approach
yields a value of the largest possible
error, and you should draw attention
to this fact. Then:(i)
Sum
(ii)
Difference A = B - C
A =
B + C
(iii)
Product
A
A
A
A=B+C
=
B + C
A=BxC
=
B + C
B
C
(iv)
Quotient
A
A
A=B/C
=
B + C
B
C
(v)
Constant power
A =
A
(vi)
In general
 A   A

A   B   
C 
 B
  C

5.6 How Many Significant
Figures?
You may be able to measure the
wavelength of a spectral line to 1
part in 10,000 but the value of a
capacitance to no better than 10%.
In the first case you can quote a
value to 5 significant figures (eg l =
589.53nm), but in the second case no
more than 2 significant figures are
justified (eg 12mF). Your calculator
will probably deliver 8 significant
figures, but when you quote a result
do not use any more figures than the
experiment allows. Remember that
the overall accuracy is largely
determined by the least accurate part
of the experiment. An extreme
example of this feature occurs in the
measurement of some torsional
properties of a long wire. In this
case, there is no point in measuring
the length with high accuracy,
because the overall accuracy is
dominated by the small radius
(which can vary by as much as 5%),
which appears as the fourth power.
A = Bn
n B
B
A = f(B,C)
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5.7 Exercise
Calculate the most likely value for
the parameter a, and the associated
maximum error, in each of the cases
listed below.
b = 5.9  0.2
d = (3.3 0.4)x10-2
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
c = 11.3  0.7
e = 9.667 0.001
a=b+c
a = b/e
a = b.c
a = (b - c)/d
a = b - c.d
a = sin (37 5)o
a = 2b.c/e
a2= (b - c)/2d
a = c + bsin (37 5)o
a2= b.e sin (b + c)o
Experimental Physics
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Chapter 6
Skill Session 2 Graphical Analysis
6.1 Introduction
Almost all experiments in physics
involve taking instrument readings
and then manipulating the raw data
to derive a quantitative result. This
result is usually the mean obtained
from several readings, so that you
will need to become familiar with
graphical and other methods for
obtaining that mean. There will
always be some experimental
uncertainty in your original readings,
and a consequent error in the values
derived from them, and it is most
important to quote these with all your
results and calculations.
arbitrary units, so will need to be
calibrated against a standard. If,
subsequently, you discover that your
initial readings were incorrect or
inaccurate, do not destroy the record.
Draw a single line through the data
and, if possible, note what went
wrong. Did you, for instance, set the
AVOmeter on an incorrect scale or
was the end point on your mm scale
something other than zero?
6.3 Graphical Representation
In many experiments you will
investigate two quantities that are
linearly related, that is:y = mx + c
6.2 Data Collection
Write down all your readings, and
when you record the data note the
various causes of uncertainty and
how big they are. For example, how
precisely can you read the scale on
the instrument? Is there a zero error?
Is there a spread of readings when
you repeat a measurement? Record
the units in which the readings are
taken, and if these are not SI units,
note that a conversion factor is
required. For example, a meter
might have a scale marked in
Experimental Physics
The gradient m is the change in y for
a given change in x, and the intercept
c is the value of y when x is zero.
This type of relation is best displayed
graphically because, in addition to
the ability to measure c and m
directly, you are able to see the
extent of scatter on individual data
points. You can therefore estimate
the overall uncertainty, and also
observe whether any data are
obviously wrong.
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Always label the divisions on your
graph so that interpolation is as easy
as possible. For example, let 1cm
equal 2 or 5 units rather than 3 or 7.
When the experimental values are far
removed from zero it may be
appropriate to suppress the origin
(x=0, y=0), but if the relation is one
of simple proportionality:y = mx
the origin should provide the most
accurate point on the graph.
However, do not force the graph
through the origin; your measuring
instrument may have a zero error.
Experimental data points should be
spread as uniformly as possible along
the graph. Note that if the relation is
of the form:1
 mx  c
z
and you are plotting 1/z against x,
then you should not take uniformly
spaced values of z.
For each experimental point,
estimate the uncertainty in the
measured value, and draw error bars
to indicate this uncertainty. The use
of error bars enables you to recognise
whether a data point lies so far off
the straight line that a mistake in
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recording must have occurred. You
are now able to draw the best straight
line fit to your data, and also the
worst fit that is compatible with the
experimental uncertainty. This latter
measurement gives the overall
uncertainty, usually known as the
random error, in the gradient. For
highest accuracy the gradient should
be measured over as wide a range as
possible.
For example, where
measuring the change in x you may
not be able to read each end of your
ruler to better than 0.5mm, so it is
better to measure a line 100mm long
than one that is only 10mm long.
For the present it is normally
adequate to derive the uncertainty by
finding the difference between the
gradients of best and worst straight
lines. This procedure represents the
worst possible case, so that the
quoted error is larger than that which
would
be
encountered
most
frequently. In later units you will
learn how to use a more precise
statistical technique known as least
squares fitting, but for this Unit the
maximum uncertainty approach is
adequate. Note that using the worst
fit straight line means that the
fractional error in a small intercept
can be very large.
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6.4 Use of Logarithmic Graph
Paper
In several experiments, you will
encounter quantities that vary
exponentially:-
log 5.01 is 0.700, so the point y =
5.01 is 7/10 of the way up the
ordinate between the values 1 and
10.
An example:-
y=Ae bx
y = 16 when x = 15
y = 1.5 when x = 0
where e is the exponential function
and A and b are constants. Taking
natural logarithms to the base e:-
log(16)  1.204
log(1.5)  0.176
(log y )  1.03
1.03
 Gradient 
 0.069
15
 log y  log A  0.069 x
ln y = ln A + bx
Taking natural logarithms to the base
10:log y = log A + cx
ln y  ln A 
where c is equal to b/2.303. For an
expression of the above type we can
use semi-log or (log-linear) graph
paper, which has a normal linear
scale along the x-axis, but the
ordinate
is
divided
up
logarithmically, so that each power
of 10 (a cycle) corresponds to the
same length of scale. For instance, if
1 to 10 occupies 10cm, then 10 to
100 also occupies 10cm, as does 100
to 1000. Within each cycle the
divisions
are
progressively
compressed towards the upper end.
If you plot y on this logarithmic scale
the points are distributed in the same
way as if you had plotted log y on
ordinary graph paper; for example,
Experimental Physics
0.069
x
2.303
If a relation is of the form y = Axn,
then:log y = log A + n log x
A straight line graph of gradient n is
then obtained by using log-log graph
paper, in which both axes are divided
up logarithmically. The gradient n is
found very simply by measuring a
convenient base line parallel to the x
axis (say 5cm long). If the two
equivalent y values are 15cm apart,
than n has the value of 15/5 = 3.
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6.5 Exercises
The following set of data relate the
parameters a and b, where it is
known that:a  Ab  B
Graphically derive values for A and
B, including the associated errors.
b/s  1s
21
33
40
48
52
61
67
a/mA10mA
13
36
39
46
57
64
73
The following set of data relate the
parameters a and b, where it is
known that:1
B
 2
aA b
Graphically derive values for A and
B, including the associated errors.
b/s  1s
5.2
7.5
8.8
10.1
10.6
11.7
12.8
a/mA10mA
557
842
1205
1436
1640
1820
2380
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The following set of data relate the
parameters a and b, where it is
known that:a  A exp 
b
B
Derive values for A and B, including
the associated errors, for the two
data sets given below. In each case,
plot graphs using linear paper and
log-linear paper.
b/s  1s
23
50
89
121
183
210
265
a/mA20mA
508
385
270
173
116
62
48
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Chapter 7
Skill Session 3 Data Acquisition
7.1 Introduction
In this skill session you will be
concerned with making simple
measurements,
assessing
uncertainties and analyzing data.
The exercises have been chosen so
that the physics is straight forward,
enabling you to concentrate on
laboratory procedure.
group member should end up with
the same average length.
7.3 Data Acquisition and
Variability
Objective: To gain experience of
making laboratory notes and writing
an abstract, by determining the focal
length of each lens and an estimate
the uncertainty in each value.
7.2 Variability
Objective: To gain experience of
making laboratory notes and writing
an abstract, through finding the mean
length of the straws in a bottle.
You are provided with a bottle
containing straws which have been
cut to various lengths; in statistical
terminology this represents the
population.
The object of the
exercise is to obtain a value for the
mean length of the straws. Each
member of the group will need to
select a sample and, from this, an
average length for the straws (in the
sample) can be obtained. Assuming
that each sample is really
representative of the population, each
Experimental Physics
The focal length of a lens can be
measured by using it to project an
image of a distant object onto a
screen. From the equation:1 1 1
 
F l l
where F is the focal length, l is the
position of the object and l' is the
position of the image in which
positive is from left to right. With l
set to infinity, l' = F, that is, the
distance between the lens and the
screen then gives the focal length.
You are provided with a number of
lenses (labelled A to D). Each
member of the group should use the
above technique to measure the focal
length of each lens. By combining
your data, what do you think is the
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most probable focal length for each
lens? In addition, use a blob chart to
represent the variability in your
measurements. How accurately do
you think you know the focal length
of each lens?
in response to a known voltage, V,
then from V and i, the resistance can
also be found. Thus, an oscilloscope
can be used to measure voltages
(directly), currents (indirectly) and
resistances (indirectly).
7.4 The Oscilloscope
First, use the oscilloscope to set the
power supply to a suitable value. By
applying Ohm's Law to the circuit
shown:V  i ( RB  R )
Objective: To gain experience of
using an oscilloscope, making
laboratory notes and writing an
abstract, by determining a value for
an unknown resistance and an
estimate the uncertainty in this value.
An oscilloscope gives a visual
display of voltage against time and it
is therefore often used when
analyzing AC circuits. However, it
is equally useful as a means of
measuring DC voltages. If it is
possible to measure the DC voltage
across a known resistor then, using
Ohm's Law, an oscilloscope can be
used to measure current. Finally, if it
is possible to measure the current, i,
flowing through an unknown circuit
VB  iRB
V  VB 
R
VB
RB
1 R 1
1
 .

VB V RB V
(NB This schematic diagram is
intended only to provide you with
some initial ideas). Measure the
voltage, VB, across the resistance
box, RB. Vary RB and, by plotting a
suitable graph, find the unknown
resistance, R, and the uncertainty in
this value.
ie
R
Resistance
Box
Voltage
Generator
Experimental Physics
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Oscilloscope
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Chapter 8
Skill Session 4 Experimental Design
8.1 Introduction
In this skills session the activities are
concerned
with
using
experimentation to achieve predetermined objectives, where the
precise method is not specified. You
will use the equipment that is
provided to obtain a value for g, the
acceleration due to gravity, and to
measure the capacitance of a
capacitor. How you achieve these
goals is largely up to you but, as in
all experiments, your objective is to
employ good laboratory practice to
obtain the most accurate results that
your equipment will allow within the
available time (about 80mins each).
Before
you
start
taking
measurements, plan out what you
intend to do so as to work as
efficiently as possible. That way you
obtain the best possible results with
the minimum of effort. Discuss your
plan with a demonstrator before
starting
8.2 Acceleration due to
Gravity
Experimental Physics
Objective: To
obtain
a
measurement for the acceleration due
to gravity and the uncertainty in g.
Equipment: Timers, string, weights,
rulers, retort stands, wooden blocks,
assorted wooden planes, graph paper.
Useful Equations:
v = u + gt
t
2
for an object in free fall, where v is
the final velocity, u is the initial
velocity, g is the acceleration due to
gravity, t is the time and h is the
distance fallen.
h = (u + v)
l
g
for a simple pendulum where t is the
period of oscillation and l is its
length.
  2
2.8 s 2
t =
gh
2
for a sphere rolling down an inclined
plane where s is the length travelled
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along the inclined plane, h is the
height dropped and t is the time
taken.
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8.3 Capacitor Discharge
too rapid a discharge (too low a
resistance on the box) may damage
the AVOmeter.
Objective: Obtain the capacitance of
an unknown capacitor and the
uncertainty in this value.
Useful Equations:
Equipment:
Circuit
diagram,
voltage source, resistance box,
switch, leads, AVOmeter, timer,
graph paper.
You should connect up the circuit
diagram shown below such that,
when the switch is in one position,
the voltage source charges the
capacitor and, in the other, allows it
to discharge through the resistance.
Overcharging the capacitor (too high
a voltage on the voltage source) or
 =RC
I = I 0 exp(-
t
)
RC
where  is the time for the current to
fall from its initial value, I0, to I0/e, R
is the resistance in the discharging
circuit and C is the capacitance of
capacitor and e is equal to 2.718.
R
Voltage
Source
Experimental Physics
Avo
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