SIC~ STY·,
DEALING WITH
UNCERTAINTIES
!
os ranverv 2009
CONTENTS
Introduction
Precision and Accuracy
Uncertainties and Errors
Significant Figures
Dealing with Uncertainties
Repeated Measurements
Repeated Measurements Analysis
Rejecting Data
Counting Measurements
Random and Systematic Uncertainties
Analogue and Digital Measurements
Uncertainties in Graphs
1
2
3
4
:
5
6
7
8
8
9
11
13
,
Graphing Example One
Graphing Example Two
Standard Deviation ±a
14
15
16
Sum and Difference
Product and Quotient
Ignoring an Uncertainty
Powers
Roots
Pure Numbers
Functions
Example Calculation
Advanced Student Work
18
20
.22
22
23
24
25
26
.27
:
Conclusion
27
Sununary of Rules
Problems
Solutions
;
28
30
34
Dr. Mark Headlee, Physics Department
United World College-USA, State Road 65
Montezuma NM 87731-0248
1
Unless you can measure what you are speaking about
and express it in numbers, you have scarcely advanced
to the stage of science.
=Lord Kelvin
is a quantitative science-it
PHYSICS
deals with numbers and measurements.
If we are to
understand the knowledge of physics we must be aware of the nature of measurement.
When the purpose of an experiment is to test a theory, we ask how accurate and how precise the predictions
need to be. If the theory predicts a deflection of light by 12° but
experiment reveals an angle of only 10°, is the theory then confirmed or rejected? The
importance of errors and uncertainties in physics cannot be overstated.
The purpose of Dealing With Uncertainties is to help you appreciate the quality of your
experimental work. This booklet explains the basic treatment of uncertainties as practiced
in the IB Physics curriculum. A more mathematical approach is possible, including statistical analysis, but for a high school survey course the treatment suggested here is more than
adequate.
INTRODUCTION,
There is a basic difference between counting and measuring. My class
has exactly 16 students in it, not 15.5 or 15.4. That's counting, In contrast, a given student
is never exactly 6 feet tal!, nor is she 6.000 feet tall. There is always some limit to the
accuracy and precision in our knowledge of any measured property-the
the time of an event, the mass of a body. Measurements
uncertainty.
Appreciating
student's height,
always contain
a degree of
the uncertainty in laboratory work will help demonstrate
reliability and the reproducibility
the
of the investigation, and these qualities are hallmarks of
any good science.
Why do measurements
always contain uncertainties?
fectly defined and so no measurement
significant
figures; the so-called 'true'
uncertainties,
Physical quantities are never per-
can be expressed
with an infinite number of
value is never reached. There are also hidden
which are part of the measurement
technique itself, such as systematic or
random variations. The resolution of an instrument is never infmitely fine; analogue scales
need to be interpreted, and instruments themselves need calibration. All this adds to the
2
uncertainty of measurement. We can reduce uncertainty but we cannot escape it. In order
to do good science, we need to acknowledge these limits.
PRECISION
AND ACCURACY. To obtain a more reliable result, a physical quantity is
often measured a number of times. For instance, the period of a pendulum may be measured (in seconds) three times to be 3.3, 3.2, and 3.3. There is a relatively small variation or
range of measurements here, and so we say the precision is high. But if the results have a
relatively large range, say 3.3, 2.8, 3.5, then we say precision is low. The precision of a
series of measurements
is an indication of the agreement among repetitive measurements.
Precision can be quantified as the standard deviation of the measured values, but this is not
necessary in IB Physics work. An example of an uncertainty due to the lack of precision
would be the measurement of the thickness of a thin wire using a meter stick. The wire is
only a fraction of a millimeter thick while the meter stick is calibrated in millimeters,
and
so the precision of the measurement will be low.
The number of justifiable
significant figures also expresses a notion of precision. Two
students may have calculated the free-fall acceleration due to gravity as 9.625 m S-2 and
9.8ms-2
respectively.
The former is more precise-there
are more significant figures-
but the latter value is more accurate; it is closer to the correct answer. Precision, then, can
mean the resolution of a measurement, be it meters or millimeters. An amazing example of
a
high
degree
of
precision
qe
= (1.602176487 ± 0.000000040)
is
the
measured
charge
of
x 10-19 C. This is an uncertainty
an
electron,
where
of only 2.5 x 10-6%.
Another example of high precision is the measured equivalence of inertial and gravitational
mass, where
mj = mg
is good to 1 part in 1012, which is an uncertainty of only 10-10 % .
The accuracy of a measurement is its relation to the true, nominal, or accepted value. This
can be expressed as a percentage deviation from the known value. In most cases, the true
value is an experimental value, such as the charge of an electron, and acceptance is based
upon reproducible
measurements.
An example
of an uncertainty
accuracy would be the measured length of a football field using only
due to the lack of
a
single meter stick,
laying it end on end. Although the meter stick has a precision of one millimeter, your over-
3
all measurement
would be far from accurate. In physics, we seek both precision and
accuracy; it is of little value to have a digital stopwatch displaying time in thousands of a
second if the electrical circuitry is running slow. Alan Greenspan, the
U.S. Federal Reserve
Chairman, has commented: "It is better to be roughly right than precisely wrong. "
Consider a hunter shooting ducks. Don't worry; the ducks are plastic. The four figures
sketched below represent combinations of precision and accuracy.
. .
•
..
b
.,
'
~
...
....
'
'
Low Precision
Low Precision
High Precision
High Precision
Low Accuracy
High Accuracy
Low Accuracy
High Accuracy
Precision and Accuracy
We can conclude that an accurate shot means we are close to (and hit) the target but the
uncertainty could be of any magnitude, large or small. To be precise, however, means there
is a small uncertainty, but this does not mean that we hit the target. To be both accurate
and precise means we hit the target often and have only a small uncertainty.
UNCERTAINTIES
uncertainties.
AND/OR
ERRORS.
Some textbooks treat errors as equivalent to
Errors, of course, are mistakes, and you can often find your mistake and
correct it. Human errors arise from carelessness. Perhaps you misread a meter and wrote
down 1.50 instead of 1.05. Repeated readings can reveal this. An error can also be the
difference between an accepted value and an experimentally measured value. Perhaps you
calculated
gravity as (9.6 ±O.1)ms-2
estimated uncertainty is ±O.lms-2
whereas the accepted value is 9.81ms-2.
Your
and gives you a probable range from 9.5 to 9.7 m s".
The error here is the difference between your calculation and the accepted value, an error
of O.2ms-2
.
Clearly, the error was not an explicit part of your uncertainty, and your result
is off due to errors and uncertainties
not yet accounted for. In contrast to errors, an
4
uncertainty is a limit to the precision of a measurement
will keep the terms' error' and 'uncertainty'
SIGNIFICANT
significant
FIGURES.
or calculation. In this booklet I
distinct.
You first become aware of uncertainties
when you deal with
figures. There may be no mention of errors or uncertainties
in a given
calculation, but you must still decide on the number of significant figures to quote in your
final answer. Consider the calculation of the circumference of a circle, where c
the radius r
= 4.1 cm.
= 2 tt rand
Enter these quantities into your calculator and press the equal key,
and the solution is given as c
= 25.76106cm.
really mean?
the circumference
Do we know
What does the 0.00006 in the calculation
to 6 one-hundred-thousandths
of a
centimeter? The 2 in 2nr is an integer and we assume it has infinite accuracy and zero
uncertainty, and tt can be quoted to any degree of precision and we assume it is known
accurately. But the radius is given to only two significant figures, and this represents a
limit on the precision of any calculation using this value. The circumference
therefore, to two significant figures, and we can only say with confidence that c
is known,
=
26 ern .
There are some general rules for determining significant figures. (1) The leftmost non-zero
digit is the most significant figure. (2) If there is no decimal point, the rightmost non-zero
digit is the least significant figure. (3) If there is a decimal point, the rightmost digit is the
least significant digit, even if it is a zero. (4) All digits between the most significant digit
and the least significant digit are significant figures. For instance, the number "12.345" has
five significant figures, and "0.00321" has three significant figures. The number "100" has
only one significant
figure, whereas "100." has three. Scientific notation helps clarify
significant figures, so that 1.00 x 103 has three significant figures as does 1.00
X
10-3 , but
0.001 or 1x 10-3 have only one significant figure.
There is no such thing as an exact measurement,
only a degree of precision. Significant
figures, then, are the digits known with some reliability.
Because no calculation
can
improve precision, we can state general rules. First, the result of addition or subtraction
should be rounded off so that it has the same number of decimal places (to the right of the
decimal point) as the quantity in the calculation having the least number
For example: 3.1em - 0.57 em
= 2.53em = 2.5em.
of decimal places.
5
The result of multiplication
or division should be rounded off so that it has as many
significant figures as the least precise quantity used in the calculation. For example:
11.3cm
--
1
3.01s
DEALING
X
6.8cm = 76.84cm2
= 0.3322259Hz
WITH UNCERTAINTIES.
""
76cm2
""0.332Hz
To help understand the technical terms used in our
treatment of uncertainties, consider an example where a length of string is measured to be
24.5 em long, or .e
= 24.5 em.
This best measurement is called the absolute value of the
measured quantity. It is 'absolute' not because it is forever fixed but because it is the raw
measured value without any appreciation of uncertainty. Next, we estimate the absolute
uncertainty
in the measurement,
appreciating that the string is not perfectly straight, and
that at both the zero and measured end of the ruler there is some interpretation of the scale.
Perhaps in this case we estimate the uncertainty to be 0.2 em; we say that the absolute
uncertainty here is !1.e
= 0.2 em
(where A is pronounced 'delta'). A repeated measurement
or a more precise measurement of the string might reveal it to be slightly longer or slightly
shorter than the initial absolute value, and so we express the uncertainty as "plus or minus
the absolute uncertainty,"
which is written as "±". The length and its uncertainty
are
.e ±;:...e = 24.5 em ± 0.2em = (24.5 ± 0.2)cm.
±J:...x
±!1x
±(
x
raw or absolute
uncertainty
relative or fractional
uncertainty
~.100)%
percentage
uncertainty
We now understand the string's length measurement
by saying that there is a range of
probable
= (24.5
values. The minimum probable value is .e min
maximum probable value is .emax
(24.5
± 0.2)
= (24.5 + 0.2)cm = 24.7 em.
- 0.2) em
= 24.3 em
and the
By using the parentheses in
we can easily match the uncertainty value with the least significant figure in
the absolute value. Uncertainty is rarely needed to more than one significant figure. Therefore, we can state a guideline here. When adding experimental
or calculated
values, uncertainties
should be rounded
uncertainty
to one significant
to measured
figure. We
6
might say ±6 or ±0.02 but we should not say ±63.5 or ±0.015. Also, we cannot expect
our uncertainty
to be more
precise than the quantity itself because then our claim of
uncertainty would be insignificant. Therefore there is another guideline. The last significant figure in any stated answer should be of the same order of magnitude
(in the
same decimal position) as the uncertainty. We might say 432± 3 or 3.06 ± 0.01 but not
432 ± 0.5 or 0.6 ± 0.02.
REPEATED MEASUREMENTS. The following is the IB non-statistical
method. Consider
a simple pendulum as it swings forth and back while five students each measure the time
f
for 20 complete cycles or periods T of the pendulum. The following data are recorded:
t]
= 32.45s,
t2
= 36.21s,
= 29.80s,
t3
t4
= 33.66s,
t5
= 34.08s.
The
average
of
the
various measurements is
fave
tl +t2 + ... tn
= -t = -=----=----"n
t
=
t. +t2 +t3 +t4 +t5
=
555
(32.45+ 36.21+29.80+
The range R of the five measurements
measurements:
R
= tmax
-
tenin
33.66+ 34.08)s
=
166.20s
= 33.24s
is the difference between the largest and smallest
= 36.21s -
29.80s
d mi
P 1us an mmus one-half of the range: I:1t
= 6.41s.
The uncertainty of the average is
= ±-R = ±--6.41s = ±3.205s.
2
2
The average and its uncertainty are thus: t ± l1t = 33.24s ± 3.205s . In high school physics
we normally round off uncertainties to one significant digit, so the overall result for the
average can thus be stated: t±l:1t=(33.24±3.205)s",,(33±3)s.
We can reasonably
expect the average time for twenty periods to be between 30 and 36 seconds.
Th e average period is T
t
=-
20
=
33.24s
20
= 1.662s.
.
Twenty penods were added to find the
time for t, and so the uncertainties of each period were added. The reveres to also true: the
uncertainty for twenty periods divided by 20 would be the uncertainty for one period.
7
t1.T = ± b.i = + 3.205s
20
= ±0.16025s
20
The period and its uncertainty are thus:
T±t1.T=(1.662±O.l6025)s,",,(1.7±0.2)s.
We
are reasonably confident that the period lies within the range from 1.5 to 1.9 seconds.
Repeated measurements
REPEATED
MEASUREMENT
The following approach is not part of the IB
ANALYSIS.
curriculum and will not be examined. A statistical approach to a number of measurements
of the same quantity will reveal a non-uniform scatter of values about an average value.
The further away from this average, the less likely you are to find another value. The
distribution
minimum
is something like a bell-curve.
to the maximum
uncertainty.
Repeated
individual
measurements
It is thus unlikely that the range from the
values
represents
the reasonable
usually do not increase
range
of
the range but do help
reinforce a mean value. Therefore, we want to reduce the probable range of uncertainty and
give more credit for the values near a mean value. Consider the average or mean .emean of
four independent
.e2
measurements
= 136 em ± 2em,
.e
=
mean
.e3
.e +.e
1
2
= 142em
of a certain
± 1em, and .e4
.e,
length
= 144em
where
.e = 140cm
1
± 2cm,
± 5em .
+ ... + .ell = 140em + 136 em + 142em + 144em = 140.5 em
4
n
An expression of the uncertainty of the mean may be found by dividing the range (the
difference between the largest and the smallest values) by the number of measurements.
..
= .e
Uncertaznty tn Mean
The mean and its uncertainty
correct form, as
<t :
mal(
nun
n
= 144 em -136em = 2.00 em
4
are now expressed as .e ± t1..e
.e±b..e = (I41±
2 )cm.
= 140.5 em
± 2.00 em or,
III
Notice how the range of absolute values goes
beyond the uncertainty of the mean. We can improve upon the uncertainty range of an
average
value
but
there
is no
guarantee
of improving
accuracy
with
repeated
8
measurements.
Moreover, we cannot improve on the precision of any measurements.
series of measurements
A
with 3 significant figures cannot yield a mean value with 4 or 5
significant figures. We should limit n ~ 8. This simplified method is not the same as the
statistical calculation of mean deviation (where you find the average of the differences
between each value and the mean).
Number Line Showing Uncertainty Rangefor
Mean and Raw Values
Mean Range
Raw Value Range
I
135
1~6
1~7
1~8
1~9
1~OJ
Raw Mean = 140.5 em
Mean to 3 Significant Figures = 141 em
REJECTING
J
1~1
DATA. Sometimes one measurement
I
142
I
143
I
144
I
145
in a series of repeated measurements of
the same quantity appears to disagree with the pattern of the others. When this happens,
you must decide whether the suspect measurement resulted from some mistake and should
be rejected, or was a bona fide measurement and should be used with all the other data. For
example, imagine you make measurements of the period of a pendulum and get the results
(in seconds) of3.8, 3.5, 3.9, 3.9, 3.4, and 1.8. The value 1.8 is noticeably different from the
others, and we must decide what to do with it.
If you cannot find the cause for the suspect result you must then decide whether or not to
reject it by examining the other measurements.
In our example, the best estimate of the
period of the pendulum is significantly affected if we reject the suspect 1.8 seconds. The
average of all six measurements
is about 3.4 seconds, whereas that of the first five is 3.7
seconds. The decision to reject data is ultimately a subjective one, but one which requires
careful judgment.
Sometimes drawing a graph helps you appreciate the degree of how
inconsistent a given measurement
is. The best straight-line graph should be drawn using
only the consistent data points.
COUNTING
MEASUREMENTS.
Some experiments involve counting events that occur at
random but with a definite average rate. For example, in a sample of radioactive material,
each individual nucleus decays at a random time, but there is a definite average rate at
9
which we would expect to see decays in the whole sample. We can try to measure this
average rate by observing how many decays occur within some finite interval, such as one
minute. Suppose we find that N decays have occurred after one minute. Because the decays
occur at random, we can ask just how reliable N is as a measure of the expected average
number of events. The answer is that the uncertainty in N is
±IN,
and so the average
number of events is N ± .IN . If we count 15 decays from a radioactive sample over a time
of one minute, we would conclude that, on average, our sample undergoes
15 ±.J15 = 15± 3.8972 = 15 ± 4 decays per minute. The probable range for any given
minute would be from 11 to 19 decays per minute.
RANDOM
AND SYSTEMATIC
UNCERTAINTIES.
A series of measurements may be
recorded with the greatest possible precision but the numbers will often differ. For
example, in measuring the thickness of a piece of wire with a micrometer screw gauge, the
wire may not be ofuniforrn thickness, the jaws of the gauge may be closed with a different
tightness each time, or there may be temperature variations between measurements. There
are a variety of reasons for imperfect measurements.
The most common uncertainties are random uncertainties, and these occur as deviations
from a normalized value. Repeated measurements of the same quantity are just as likely to
be either smaller or larger than the previous measurement. Random variations could be
caused by such things as slight changes in atmospheric pressure, room temperature, supply
voltage, or by changes of friction. Uncertainties in
instrument readings can reveal random fluctuations. The range of random uncertainties depends
in part on the skill of the experimenter. Two
different students doing the same laboratory
experiment often get slightly different results. As
the skill of the experimenter increases, so the
random
errors and uncertainties
should be
reduced. However, the basic random uncertainties
are part of the measuring process itself and cannot
be eliminated.
10
An uncertainty in a measurement
may also be due to a constant error. When a series of
measurements is consistently shifted in one direction we say there is a systematic
error in
these results. An experiment may contain a systematic fault when, for instance, an ammeter
is not correctly zeroed and so gives consistently low readings. A stray magnetic field may
offset all measurements,
or the background count in a radioactivity measurement needs to
be accounted for. A spring balance may not have been correctly zeroed and so resulting
measurements
are incorrect. Repeated measurements
will have no effect on systematic
errors but graphing your data can reveal the consistent shift in the data. To eliminate
systematic
error introduced
by a measuring
instrument,
we must calibrate
it using
standards that are known to be of high precision and accuracy.
There is also the possibility of a theoretical meaning to systematic shifts in data. Here, the
"error" is replaced by a physical representation of the shifted data. For instance, in
measuring volume and pressure in a gas law experiment, there is always some 'dead' space
in the pressure gauge (and this is actually part of the volume under investigation).
electrical
circuit there is the internal
resistance
of the battery
In an
cell. In a dynamics
experiment there may be unaccounted friction.
1;/
y
&7>
Cf
". .r ' Theory
x
Mistake
Systematic
x
Y
r
0/
Random
x
Mistakes or personal errors are clearly inconsistent with the rest of the data.
Systematic errors are consistent but shifted away from the theoretical line.
Random uncertainties,
the most common and inevitable limit to our data,
reveal a statistical scatter.
There is the possibility of an error inherent in some digital equipment. Occasionally digital
measuring devices will introduce an uncertainty called quantisation
by the conversion
representations
of measurements
error. This is caused
from analogue (or a continuous
of that signal. A number of measurements
signal) to digital
are made over a time interval.
11
The signal source may vary during the sampling time. As a result, the true signal and the
measured signal may differ. This is especially true if the source signal varies at a frequency
higher than the sampling frequency. In odd cases, a beat frequency might occur and totally
wrong result can obtain. In the vast majority cases, however, this does not happen, and the
sampling is representative of the source itself.
ANALOGUE AND
DIGITAL
MEASUREMENTS. Reading
analogue
scales
requues
interpretation. Measuring the length of a pencil against a ruler with millimeters divisions
requires judgments
about the nearest millimeter
or fraction of a millimeter.
For an
analogue scale, we can usually detect with confidence one-half the smallest division at
both the measured end and the zeroed end. We can say that our measurement
has an
uncertainty of plus or minus the smallest division. If the scale divisions are large, we can
often interpolate even smaller divisions.
Digital readouts
are not scales but are displays of integers, such as 1234 or 0.0021. Here,
no interpretation or judgment is required, but we should not assume there is no uncertainty.
There is a difference between 123 and 123.4, and so digital readouts are limited in their
precision by the number of digits they display. A voltage display of 123 V could be the
response to a potential difference of 122.9 V or 123.2 V, or any voltage within a range of
about one volt. Although there is no interpolation with a digital readout there is still an
uncertainty. The displayed value is uncertain to at least plus or minus one digit of the last
significant figure (the smallest unit of measure). There may be, of course, other reasons
affecting the uncertainty
that would increase this value, such as the calibration,
the
sampling method, a systematic error or a linearity problem, or just the range of repeated
measurements.
Analogue:
Bathroom Scale
Least Count on Scale
Best Estimate
= 1 lb
= 144 lb
Uncertainty is ± 0.5 lb at zero,
and± 0.51b at Weight.
W
± flW
= (144 ± 1)lb.
Probable Range: 143 lb to 145 lb
12
Digital: Stopwatch
Time Interval T = Range
Time Interval
=
T
= 16.26s
Least Count on Scale
T ±!1T
=
0.01 s
= (16.26 ± o.oi),
Start and Stop Time Readings:
Probable Range: 16.25 s to 16.27 s
T
= [52'38"
42 ]-[52'22"
16]
= 16" 26
Repetition: An Average Time
Data for three trials of a ball rolling down a 1.00 m inclined plane.
Distance
slm
~S=
Average Time
fave I s
Time
f
±O.Olm
!1t
Is
= ±O.Ols
~tave
6.28
6.39
1.00
= ±0.06s
6.33
6.31
tave
= tl + t2 + t3 =
3
At
ave
=
Range
2
(6.28 + 6.39 + 6.31 )8
3
= tmax -
2
tmin
= (6.39
tave ± Atave
= 6.3266666
- 6.28)8
2
= 0.055s
= (6.33 ± 0.06)s
Probable Range: 6.27 s to 6.39 s.
s ""6.33s
""0.06s
13
UNCERTAINTIES
IN GRAPHS.
Too often students will draw a graph by connecting the
dots. Not only does this look bad, it keeps us from seeing the desired relationship of the
graphed physical quantities. Connecting data-point to data-point is wrong. Even when
there is no measurable uncertainty, you should mark data points with small circles or
crosses. When a graph includes uncertainty limits, or uncertainty bars, the best straight
line more easily connects or nearly connects to the data regions. With an uncertainty bar,
the data 'point' becomes a data 'area.'
Data Point with x and y Uncertainty Bars
y+Lly
x-Ay
x+Llx
1
1
1-
y-Lly
Uncertainty
Area Range
Because there is not such thing as an infinitely precise data
point, you should never mark a data point on a graph with just
a dot. You should use a small circle, or, if relevant in size,
draw an uncertainty bar.
We can easily sketch the best-fit line and the two worst (the minimum gradient and the
maximum gradient) acceptable lines by using the extremes of the uncertainty bars. We
only look at the first and last data points to draw these slopes or gradients. The graph thus
performs the function of averaging the data for the best fit and determining the uncertainty
range by the maximum and minimum gradients.
Y
Uncertainty Bars for x values only.
Minimum Gradient
Best Straight Line
x
14
Graphing
Example
One. Consider
an experiment with a uniformly
accelerated body
where a graph of speed against time is plotted. Three gradients are constructed: the best
straight line, the minimum and the maxim gradients (both based on the first and last data
points and an uncertainty in the speed of ±0.4ms-1).
The uncertainty in the time is too
small to plot. The relevant equation here is v = u + at where the speed v is given at time t,
the acceleration is a while the speed at time zero (the initial speed) is u. First we determine
the acceleration and its uncertainty.
Speed vs. Time
Lnc~r Fit fer: Data Set I Speed
» mt+b
m (Sicce): 1.20 m s"/s
t> (Y·intcrcept): "'.10 nl s···
Correlation;
0.996
RMSE:0.2.50
v
Une"r Fit fe.r: Data Set I M;n
Y'"" tnt·i-b
m (Slope):
1.G9
b (Y'lntercep!): 4.29'
Correatioo: 1.00
RMSE: 0
linc~r Fit fer: Data Set I M2X
y" mr-b
m (Sfcpe)~ 1~3S
b (¥-Intercept): 3.23
Correlation: 1.00
RMSE:0
1-----~-------r------~----~------------_,------~-----_r
e
z
6
8
4
Time (s)
The acceleration is represented by the gradient m (slope) as determined by the graphing
program. The acceleration and its uncertainty are a ± ~ a .
a
This is an uncertainty
A
± lJ.a = aBest
+ Range
-
-----
2
a
aMin
= aBest ± -"'=--=
Mox
-
2
of about 8%. We assume a symmetrical
uncertainty above and below the best straight-line value.
relationship
for the
15
The best estimatedy-intercept
is given as Y
= 4.10ms-
line we find a maximum y-axis intercept of YM""
gradient line we find a minimum
1
.
Looking at the minimum gradient
= 4.29 m S-I , and
y-axis intercept
of
YMin
looking at the maximum
= 3.23ms-
l
.
Assuming a
symmetrical range around the y-intercept we can determine its uncertainty as follows:
+ Ll Y A
YBcst
-
_
YBest
+
YMax 2
YMin _
-.
(410
ms
-I)
+ (4.29 -2 3.23) ms -I -- (4 . 10 -.
+ 053) ms -I
-
Rounding off the uncertainty to one significant figure we get
YBest
± L\y
= (4.1
± 0.5)ms-
1
•
This is an uncertainty of about 12%.
Graphing Example Two. The graph the gradients are used to determine resistance and
the experimental uncertainty range in resistance. The graph is of current against voltage.
·1
w~-;i '
Correl~tio~:.LQ.-
!
I
. I
J
..
I
I
..
Linear Fit For: Data Set:~1aximum '
y = mx-b
. -'------'---/~f-:r'--"m(Slope): 1,1
b(Y-ln,ercept): ·0.47
. 1_ ".
Linear Fit For: Data Set:Minimum
y-mX+b
m(Slope): 0_9
b(Y-lntercept): 0.3
c.or~~la!ion:1..._ .
linear Fit For: Data Set:Current I
y = mx+b
m(Slope): 1.0 mAN
b(Y·lntercept): -0,013 m,'
Correlation:l.0
1
e
YOlt6~e Y (Y)
10
The computer generates the best-fit line, in this case with gradient m = l.OmA V-I _ The
resistance R is then calculated using this value as follows:
16
V
L1V
1
1
R=-=-=-=-=
I
M
M
m
1
I =l.OxlO
10-3 AV-
1.0 X
3
Q=l.OkQ
L1V
The minimum and maximum experimental values of resistance are now calculated.
Rm,. = _1_ =
~
Rrru-n
mmin
= _1_ =
mmax
0.9
1
= 1.1kQ
10-3 A V-I
X
1
1.1 x 10-3 AV-I
The resistance uncertainty is then M = ±
R
-R -
mal(
mm
= 0.9 kQ
(1.1-0.9)kQ
=±
2
= ±O.1kQ.
2
The overall resistance and its uncertainty are therefore: R ± M = (1.0 ± 0.1) kQ .
STANDARD
_DEVIATION
IN
GRAPHS.
Determining
the uncertainty
in a graph's
gradient and y-axis intercept using the minimum and maximum gradients based on the first
and last error bars often yields an excessive uncertainty. This method does not reflect the
scatter of all the data. Richard Feynman once said that there is a reason why the first and
last data points are the first and last! The method of standard
deviation (which is beyond
the scope of the IB curriculum) accounts for all the data points, and calculates an absolute
uncertainty, denoted with the lower case Greek letter sigma,
The standard
deviation
(5.
of a probability distribution (normally a bell-shaped curve) is a
measure of the spread of its values. It is defmed as the square root of the variance. To
understand standard deviation, keep in mind that variance is the average of the squared
differences between data points and the mean. Variance is tabulated in units squared.
Standard deviation, being the square root of that quantity, therefore measures the spread of
data about the mean, measured in the same units as the data. More formally, the standard
deviation is the root mean square (RMS) deviation of values from their arithmetic mean.
See http://en.wikipedia.orglwiki/Standard
_deviation.
Statistically, the best straight-line gradient, m, with the uncertainty
68% of the measurement
lie within this range. For ±2
Sometimes in science ±2
(5
accurate.
(5
of ± (5, tells us that
its 95%, and for ±3
(5
its 99.7%.
is used but in this booklet we suggest that ± (5 is sufficiently
17
With Vernier's LoggerPro
software the standard deviation uncertainties
for the gradient
and the intercept are for the y-axis scatter only with the assumption that the uncertainty in
the x-axis is negligible compared to that in the y-axis. Normally in IB physics we
need only
to graph the larger uncertainty bars for two quantities, so with LoggerPro you would graph
the bars on the y-axis only.
Consider the following set of data where the uncertainty is determined with the minimum
and maximum
gradients (using two different error bar values) and then again using
standard deviation.
Sample data of time t and speed v for a
uniformly accelerating object. The
uncertainty in time is negligible and the
uncertainty in the speed is considered
both as ±O.4111S-1 and ±O.2111S-1 • The
Data Set
Ti me
(5)
gradient of the speed against time graph
represents acceleration a, and the
y-axis intercept represents the initial
speed u, in the equation v = u + at.
"1
1
,-..
sccec
Max
Min
(>11 ;")
H)
4.9B
2.0
6.78
3.0
7.75
4.0
8.89
5.0
1G.40
6.0 .1
11.34
7.0
12.30
4.5B
5.36
12.70
11.90
Linear Fit fer: Data Set I Saeed
mt •.b
.
II ~
m (SiC!}C): 1.Z05 ./- 0.0471
a In s'l/s
b (Y-fntercept): 4.10 I +/- O.Z 11 G In S'l
Correlaticn: 0.9962
R~·iSE:O.2t9G
10 ~
."
VI
~
'-"'
"0
(IJ
(IJ
o,
Vi
5
-I
I
.j
e if-I
o
----.------rj---..-----..,.------.-----r-----.-------,2
4
Time (5)
6
8
18
The graphing software tells us the best straight-line gradient and its standard deviation are
mBest = (1.205 ± 0.04718 )ms-2 and
YBestlntercept
= (4.101 ± 0.2110 )ms-l . The following table
summarizes the two methods. Uncertainties are rounded off to one significant figure.
Minimum & Maximum Gradient Method
=
:,~>="'$'P'm'
=
SPEED UNCERTAINTY·
m
±O.4
"
Standard Deviation
Method
-.
SPEED UNCERTAINTY
±O.2 m
5-1
S-l
,.
a
± l1a ""(1.2 ± 0.I)ms-2
a ± l1a ""(1.21 ± 0.07)ms-Z
or about 8%
or about 6%
u±l1u",(4.1±O.5)ms-1
,u±l1u",,(4.1±O.3)ms-1
a ± l1a ""(1.21 ± 0.05)ms-Z
or about 4%
u±l1u",(4.1±O.2)ms-1
L,~~~o~::~_"_L=b~~~~_:'~~~~"~-=_
SUM AND DIFFERENCE. Let us say you want to make some calculations
of a rectangular
metal plate. The length L is measured to be 36 mm with an estimated uncertainty
of
± 3mm, and the width W is measured to be 18 mm with an estimated uncertainty of
± 1mm . How precise will your calculations of the circumference
and area be when you
take into account the uncertainties?
We
assume
L = (36
the
parallel
lengths
and
the
parallel
widths
are
identical,
where
± 3) mm and W = (18 ± 1)mm. The circumference is the sum of the four sides.
Cabsolute
= L+ W +L+ W = 36mm+ 18mm+ 36mm+ 18mm= 108mm
To find the least probable circumference, you subtract the uncertainty from each measurement
and then add the four sides. The minimum
Lrnin =(36-3)mm=
33mm
and Wrnin =(18-1)mm=17mm.
lengths
and widths
The minimum
would
be
circum fer-
ence is Crnin and is calculated as follows: Crnin = Lrnin + W min + Lrnin + W rnin = 100 mm. To
find the maximum probable
circumference,
you add the uncertainty
to each length and
width, and then you add the four sides together, where Lmax = (36 + 3)mm = 39mm
where Wmax = (18 + l)mm = 19mm.
and
19
The range from maximum to minimum is the difference of these two values.
Crange = Cmax - Cmin
= 116mm
-100mm
= 16mm
This range includes both the added and the subtracted uncertainty values. The absolute
value lies midway,
circumference,
so we divide the range in half to find the uncertainty
in the
!:1C.
!:1C
= Crange = 16mm = 8mm
2
This is correctly expressed as !:1C
=±
2
8 mm , and the circumference is now written as
C ±!:1C
= (108 ±
8)mm.
We can generalize this process. When we add quantities, we add their uncertainties.
The chances are that some of the uncertainties
are 'plus' and others 'minus'
and so a
statistical approach to this would reduce the uncertainty range, but for most ill Physics
work this worst probable range is satisfactory. Of course, the accepted value could still lie
outside this range. The ± range has a confidence level of 68%.
What about subtracting quantities? Subtraction is the same as addition except we add a
negative quantity, A + B = A + (- B)
=A-
B . And what about the uncertainties?
It would
make no sense to subtract uncertainties because this would reduce the uncertainty value;
we might even end up with zero uncertainty. Hence we add the uncertainties.
There is a
general rule for combining uncertainties with sums and differences. Whenever we add or
subtract quantities, we add their absolute uncertainties. This is symbolized as follows.
± LlA) +(B ± LlB) = (A + B) ±(LlA + LlB)
Sum
(A
Difference
(A ± LlA) -(B ± LlB)
= (A -
B) ±(LlA + LlB)
20
PRODUCT AND QUOTIENT. Next we calculate the area of the metal plate. The absolute
area is the product of length and width.
Aabsolute =L absolute xW absolute = 36mmx18mm=648mm2
The minimum probable area is the product of the minimum length and minimum width .
.<1
.L~
•
= Lnun. xW.rrun = 33mmx 17mm = 561mm2
The maximum probable area is the product of the maximum length and maximum width.
= Lmax xWmax = 39mmx19mm=
~ax
741mm2
The range of values is the difference between the maximum and minimum areas.
Arange
=
~ax
-
= 741mm2 - 561mm2 = 180mm2
~n
This range is the result of both adding and subtracting uncertainties
to the lengths and
widths, and the absolute value is midway between these extreme values. Therefore, the
uncertainty
M
in the area is just half the range; it is added to and subtracted from the
absolute area in order to find the probable limits of the area.
~A =
A...nge
2
= 180mm
2
= 90mm2
2
We can now express the area and its uncertainty to two significant figures.
To further develop the handling of uncertainties,
let us now calculate the percentages
uncertainties in both the absolute length and absolute width measurements.
~L
~L%=-100%=
L
~W
~ W % = -100%
W
3mm
36mm
lmm
= --100%
18mm
100%= 8.33%
= 5.55 %
of
21
Adding these two percentages give us 8.33%+ 5.55%
= 13.88%.
It is no coincidence that
when we take this percentage of the absolute area that we get the same value of uncertainty
we got when we calculate the extreme values.
13.88%x
The
calculated
area
= 0.1388
Aabsolute
is now
x 648mm2
= 89.9424mm2
A±6.A=(648±89.9424)mm2
",,(6.5±0.9)x102mm2•
Expressed with a percentage of uncertainty, the area is written as A ± 6. A % .
A ± 6.A% = 648mm2 ± 13.88% ""650mm2 ± 14%
The only apparent difference between finding the extreme values and using percentages
is
that using percentage is easier; it requires fewer calculations. Moreover, using percentages
will simplify our work when dealing with complex equations involving
variations
of
product and quotient, including square roots, cubes, etc. Although some examples may
show a slight difference between the calculation of extremes and the use of percentages,
most of this difference is lost when rounding off to the correct number of significant
figures. However, when making higher-level calculation (such as cubes and square roots),
any slight difference between the two methods may become noticeable,
and the use of
percentages will yield a smaller uncertainty range. This, of course, is desirable.
When multiplying
quantities,
we add the percentages
of uncertainties
to find the
uncertainty in the product. Dividing two quantities is the same as multiplying one by the
reciprocal of the other, such that ~
= A(
~).
This means we should add the percentages
of uncertainties when we divide. You might think that you could subtract percentages of
uncertainties when dividing but then you would be reducing the effective uncertainty and
you might end up with zero or even negative uncertainty. This is not acceptable. Therefore,
the rule of products and for. quotients is one and the same. We add the percentages of
uncertainties when we find the product or quotient of two or more quantities. This
rule is symbolized as follows.
22
Product
(A ± ~ A) X (B ± ~ B) = (A X B) ± [( ~AAJ 00 ) % + ( ~:
Quotient
A±~A = A ±[(~A100J%+(~B
B±~B
B
A
B
z=-
Forexample,findzwhere
z±~z=
x
x±~x =
y±~y
z±~z=-
22
431
and
y
x±~x=22±1
22±1
431±9
=
± (4.545% + 2.089%)
quarter
AN UNCERTAINTY.
of
4.8%+0.9%
the
other,
you
100J%]
y±~y=431±9.
22 ±[(_1 100)%+(~100)%]
431
22
431
z ± ~z = 0.05104 ± 0.00338""
IGNORING
and
100 ) %]
= 0.05104
± 6.633%
(5.1 ± 0.3) x 10-2
If one of the percentage uncertainties
may
ignore
the
smaller
percentage.
is ess than a
Fa
""4.8% and not 5.7%. A statistical approach would show that
example,
certainties
are not as bad as the combining rule suggests, but statistical methods are beyo
of most high school student's work. So, when one percentage is significantly
naller than
the other we can ignore it. I say a quarter somewhat arbitrarily but when takin
significant
figures into account, this suggestion is justified.
POWERS.
Squaring a number is the same as multiplying it by itself, A 2
uncertainties,
we have
(A ± ~ At = (A ± ~ A) (A ± ~ A).
=
Where the uncertai
expressed as a percentage, we can see that this percentage is added to itsel
percentage of uncertainty in A 2
(A±~A%r
X
A. With
ty in
A
is
to find the
•
=(A±~A%)(A±~A%)=A2
±2~A%
Often the range of uncertainty is greatly reduced by using the method of perce tages compared to the method of calculating upper and lower limits. Consider the nut
er line for
(x±~xr
upper and
where
x±~x=36±4.
The percentage range is ±I44,
while th
23
lower limits when calculated by the maximum and minimum probable values of x are
-272 and +304.
NUMBER LINE:
1152
1
x2
= 362 =1296
Percentage Range
1440
I
Raw Value Range
1024
1
1600
I
I
I
I
I
I
I
I
1000
1100
1200
1300
1400
1500
1600
Similarly, when you cube a quantity you add the percentage of uncertainty of the quantity
to itself three times, because A3
= A x A x A.
Hence (A
r
± ,1.A% = A3 ± 3,1.A%. We can
generalize the adding of percentages rule for any power, An
= A 1+A2 + ... + An'
First, we
find the percentage of uncertainty in A, which is ,1. A%, and then we add this to itself n
times in order to find the percentage of uncertainty in A" . The rule for any
nth
power is
symbolized as follows.
dh Power
(A
± ,1.Ar = A" ±
For instance, if x is known to 3% then
X2
n( ,1.AA100% ) = A" ± n,1.A%
is known to 6%, or x5 is known to 15%, and
so on. Using percentages reduces the uncertainty range when compared to calculating raw
values of maximums and minimums.
ROOTS. Taking the square root of a number is the same as taking that number to a power
of one-half,
Fx = x1, where,
Previously,
mathematical
for instance,
operations
.J36 = 36! = 6.
increased
We have to be cautious here.
the overall uncertainty.
But taking the
square root of a number yields a much smaller number, and we would not expect the
uncertainty to increase to a value greater than the square root itself. Also, our combination
of uncertainties must be consistent in a way that when we reverse the process we end up
with the same number and with the same uncertainty we started with. For instance,
if;?
must equal x, and the same with the uncertainties. Therefore it seems fair to say that taking
x to the one-half power will cut the uncertainty in half. Then, when you square the
24
square root and propagate the uncertainties, you end up with the original uncertainty and
the original absolute value.
With an argument similar to square roots, we can say that the cube root will reduce the
percentage
of uncertainty
to one-third,
or the fourth root reduces the percentage
uncertainty to one-fourth. In general, where
reduces the percentage
nth
of uncertainty
For ~A
Root
± f:.A , we
!.fA = A; , we can state a rule. The nth root
by lIn. This is symbolized as follows.
find
ifA ± !..(f:.A
n
PURE NUMBERS.
pure numbers
of
100%)
= ifA ± f:.A%
n
A
Often calculations involve numbers such as 2, )Iz, i'j, or tt . These are
and have no uncertainty associated with them. If you were to measure the
thickness of 100 sheets of paper to be (26
a single sheet is (0.26
± O.01)mm
± l)mm
then you could say that the thickness of
. This is reasonable because if you reversed the process
and you measured one sheet as (0.26
± O.01)mm
and then multiplied by 100, you would
end up with the same value and uncertainty as measuring 100 sheets. The general rule for
propagating
multiply
uncertainties
when using
or divide by pure numbers
pure numbers
you multiply
is straightforward.
or divide the absolute
When
you
uncertainty
by the pure number.
For example, when you measure the diameter of a circular object and want to calculate the
radius, where r
=:!...
2
and d
± f:.d
= (4.6 ± 0.2)cm,
you find that dividing by 2 will cut the
.
uncertainty in half. Therefore r±f:.r=(2.3±0.1)cm.
have a circumference
A circle with this radius will then
c = 2 n r and an uncertainty f:. c = 2 n f:. r as follows.
c ± f:.c = 2n(r ±f:.r) = 2n{2.3
c ± f:.c
= 14.4513cm
± O.l)cm
± 0.6283cm;::: (14 ± l)cm
25
FUNCTIONS.
You must not assume that the adding percentage rule is applicable to all
calculations. There are certain mathematical functions (such as reciprocals, logaritluns and
trigonometric
functions) where, if the uncertainty is more than a few percent, we should
use the absolute uncertainty to compute upper and lower limits that then determine the
uncertainty
±~x%;
in the function value. Why? The percentages
are necessarily
symmetrical,
the same uncertainty value ~x is added and subtracted from the absolute value to
find the probable range. A more accurate description of the range of uncertainty with, say,
the reciprocal function, is that it is not symmetrical.
Consider the reciprocal
of x where
y
= (1/x)
and x
± ~x = 0.36 ± 0.06. Calculating
percentages, the uncertainty range is ±0.46 on either side of y
= 2.78.
In contrast, calculating the minimum and maximum values of y gives 0.40 below and 0.56
above the absolute value of y, which is an asymmetry with a range shifted upwards. To
appreciate the asymmetry here we need the following rule. The uncertainty range for the
reciprocal function is found by calculating the upper and lower limits using minimum
and maximum raw values of the denominator. We should not use percentages.
This is
especially true when drawing uncertainty bars on graphs because it is the range of probable
values that allows you to draw the best straight line and we don't want this shifted
symmetrically
when
in fact the
correct
range
is asymmetrical.
However,
when
uncertainties are small (a few percent or less), we may assume that the percentage of the
uncertainty in x is the same as in y, where y is the reciprocal of x.
A similar asymmetry in raw value uncertainty distribution occurs when taking logarithms.
This asymmetry
is especially
important
when we graph functions
because
when we
attempt to draw a best straight line we need to pay attention to the larger side of the
uncertainty
for a given data point. Logarithms,
however, have the asymmetry
in the
opposite direction (shifted to a lower range) compared to the reciprocal function. When
calculating the uncertainty range of a quantity that is expressed in logarithms, you
should find the uncertainty range using minimum and maximum raw values.
The trigonometric
functions of sine, cosine and tangent are examples of pure functions.
The uncertainty range here can be calculated using minimum and maximum values. Try an
26
example of large uncertainties
and show yourself that using percentages yields a greater
uncertainty range than calculating minimum and maximum absolute values. To find the
refractive index n using a measured critical angle fJ with the equation n
sin( fJ - L\fJ),
should make three calculations,
trigonometric
sin( fJ + L\fJ), and
sine.
= l/sine,
you
When using
functions it is best to determine the uncertainty by calculating upper
and lower limits using absolute values.
EXAMPLE
CALCULATION.
The
following
illustrates
what
you
might
do when
calculating gravity from a simple pendulum experiment. The pendulum length and period
are measured as
.e = (2.54 ± 0.02)m
Theory tells us that
T = 2n:{!.
and T
= (3.2 ± 0.1)s
.
First, then, we solve for gravity and then we find the
percentages of uncertainties.
L\.e%
= 0.02m
100%
= 0.7874%
2.54m
O.ls
I1T%=--100%=
3.2s
Period Squared Uncertainty
3.125%
= 2 X I1T% = 2 x
3.125%
= 6.2500%
When we divide the length by the period squared, we add the percentages of uncertainties,
0.7874% + 6.2500%.
But here we recall the general rule that if one of the percentages is a
quarter or less than the other, we can ignore the smaller percentage. Therefore, we can say
that the uncertainty
in this problem is 6.2500 %. We now change the percentage
of
uncertainty into an absolute value, and then express the calculated value of gravity to two
significant figures.
g
= 9.792498
m s ? ± 0.612031 m s ?
:. g ± I1g:=:: (9.8 ± 0.6)ms-2
27
ADVANCED STUDENT WORK. The rules and suggestions in this booklet relate to the IE
Physics curriculum. When doing more specialized work, however, such as an extended
essay in physics or a more developed class lab, you may want to use more sophisticated
methods, including
statistical analysis. When combining two uncertainties,
±~ A and
±~ B, it was suggested that we simply add the absolute uncertainties. There are four
combinations here: +~A with +~B, +i1A with -i1B, -i1A with +i1B, and -i1A with
-~ B. This means that there is a 50% chance that combining uncertainties
will actually
reduce the overall uncertainty. The rule of adding is really for the worst possible case.
Here, then, are two advanced rules that refine the combination of uncertainties.
For sum and difference, the resultant uncertainty is the square root of the sum of the
squares of the individual uncertainties.
This improves upon the uncertainty range (i.e., it
reduces the range) when compared to simply adding the absolute uncertainties.
For product and quotient, the fractional uncertainty is the square root of the sum of the
squares of the individual fractional uncertainties. This ratio can be turned into a percentage
by multiplying it by 100%. This improves the uncertainty range by reducing it.
CONCLUSION. Because scientific theories are simplified approximations
of a complex
reality, an appreciation of the precision and accuracy of experimental results is an essential
part of the experimental process. The LB. Physics Course Guide tells us that students are
to gain an understanding
of what it means to determine a value, and the confidence that
one can associate with that value. Since physics is a quantitative
fundamental level, this is an essential skill.
science at its most
28
SUMMARY
OF
BASIC
RULES
Repeated Measurements
For a number of repeated measurements, we find the average or mean.
The uncertainty in the mean is plus or minus one-half of the range between
the maximum value and the minimum measured value.
Sum
(A±~A)+(B±~B)=(A
+B)±(~A+~B)
Difference
(A±~A)-(B±~B)=(A
-B)±(~A+~B)
Product
(A± ~A)x
A±AA
B±~B
Quotient
nthPower
(B
(A
± ~A)"
± ~B)
= (Ax
= A ±[(
B
= An ± n(
B)
[( ~AA100 )
%+ ( ~:
bo )%+(
A: 100
~AA10b )%=An
±n~A%
AAI
A
)%]
100 )
%]
29
nth
Root
For VA
± L1 A , we find ifA ± -1 (L1A)
-100
n
A
%
L1A%
= ifA ± -It
Gradients in Graphs
The gradient of the best straight-line of a graph = mBest and the minimum
and the maximum gradients based on the uncertainty range of the
first and last data points are mMax and mMin.
Stating Uncertainties
Experimental uncertainties should be rounded off to one significant figure. The least
significant figure in a stated answer should be of the same order of magnitude
(in the same decimal position) as the single digit uncertainty value.
g ± L1g = (9.81734 ± 0.0217)ms-2
-7
:.
g ± L1g = (9.82 ± 0.02 )ms-2
30
UNCERTAINTY
1.
PROBLEMS
Four students measure the same length of string and their results are as follows:
-fl
= 38.6cm,
-f2
= 38.7cm,
-f3
= 38.6cm,
-f4 = 38.5cm
What is the mean and its absolute uncertainty for the length of string?
2.
What is the width and its absolute uncertainty of the object being measured in the
sketch below?
Units =mm
3.
The analogue voltmeter below measures a voltage on a scale of zero to 5 volts. What
is the measured voltage and what is the absolute uncertainty shown here?
31
4.
The digital stopwatch was started at a time to
swings of a simple pendulum to a time of t
=0
and then was used to measure ten
= 17.26 s .
If the time for ten swings of the pendulum is 17.26 s what is the minimum absolute
uncertainty in this measurement?
What is the time (period T) of one complete pendulum
uncertainty?
swing and its absolute
5.
Add the following lengths: 12.2cm + 11.62cm + 0.891cm
6.
Add the following distances and express your result in units of centimeters (em).
1.250 x 1O-3m
7.
+ 25.62cm + 426mm
Average speed is the ratio of distance to time:
v
= !.. .
t
What is the average speed if s
8.
Given two masses, m1
m1
9.
= 4.42m
= (100.0 ± O.4)g
+ 1n2' and what is their difference,
= 3.085s?
and t
and
In, -
1112
1112'
= (49.3 ± 0.3)g,
what is their sum,
both expressed with uncertainties?
With a good stopwatch and some practice, you can measure times ranging from about
a second up to many minutes with an uncertainty of 0.1 second or so. Suppose that
we wish to find the period r of a pendulum with r ""0.5 s. If we time one
oscillation, we will have an uncertainty of about 20%, but by timing several
successive oscillations, we can do much better.
If we measure the time for five successive oscillations and get 2.4 ± O.ls , what is the
final answer (with an absolute uncertainty) for the period? What if we measure 20
oscillations and get a time of 9.4 ± O.ls ?
32
10.
(s I cm) of an object under
A computer interface is used to measure the position
(a I em s-2)
uniform acceleration
as a function of time (t). The uncertainty
in the
time measurement is very small, about /).t = ±a.OOOls, and so you can ignore it,
while the uncertainty in the distance is significant, where b..s= ±O.lcm . Here are the
data.
TimeL
Time
tis
Distance
s/cm
2
t I S2
0.0000
0.0493
0.1065
0.1608
0.1989
0.0
0.1
0.5
1.1
1.7
= mt?
The motion of the body can be described by the equation s=(l/2)at2
where
the constant m = 0.5 a. A graph of distance against the square of time should have a
slope of "m". Acceleration is determined by first finding the slope or gradient, where
a = 2 x slope. Use the above data and graph distance against time squared, construct
uncertainty bars for the distance points, and then determine the best straight line and
solve for the acceleration. Do not include the origin in your data points.
Distance against Time Squared
s / cm
2.0
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33
11.
In an optical experiment a deflected ray of light is measured to be at an angle
()= (23 ± 1) o. Find the sine of this angle and then determine the minimum and
maximum acceptable values of this experimental
experimental result in the form sin () ± t::. (sin ()) .
value.
Next,
express
the
12.
On your graph above (in problem 10) draw the minimum and maximum slopes on
graph and determine the range of acceleration based on these two extreme slope
values.
13.
Density is the ratio of mass to volume, where p
= m/V.
What is the density of a
3
material if m = 12.4kg and V = 6.68m ?
Determine the least uncertainty in the
mass and in the volume, and then calculate the uncertainty in the density value.
Express the uncertainty in the density p as an absolute value ±t::.p, as a fractional
ratio
14.
15.
± ( f1p/ p ) , and
as a percentage ±t::.p%.
What is the uncertainty in the calculated area of a circle whose radius is determined
to be r = (14.6 ± 0.5)cm?
An electrical resistor has a 2% tolerance and is marked
R
= 1800.Q.
What is the
range of acceptable values that the resistor might have? An electrical current of
I=(2.1±0.1)rnA
flows through the resistor. What is the uncertainty in the
calculated voltage across the resistor where the voltage is given as V
16.
An accelerating object has an initial speed of u
of
t::.t
1
v=(28.8±0.2)ms- •
= (4.2 ± 0.1) s.
The
Acceleration
time
is
= (12.4 ± 0.1)ms-1
interval
defined
for
as
this
= I R?
and a final speed
change
a = (v - u)/ f1t.
in
speed
Calculate
is
the
acceleration and its absolute uncertainty.
17.
What are the
r=(21±I)mm?
18.
Frequency and period are related as reciprocals. What are the period and its absolute
uncertainty when the frequency of 1.00 X 103 Hz is known to 2%?
19.
Einstein's famous equation relates energy and mass with the square of the speed of
light, where E = m c2• What is the percentage of uncertainty and the absolute
uncertainty of the energy for a mass m = 1.00 kg where the speed of light is
c
= 3.00 X 10
8
volume
ms "?
and
its uncertainty
for
a sphere
with
a radius
of