Unit 1
PHY 133 2022
What is measuring?
The outcomes of this unit
Physics content
In this unit, we will look at methods to measure various quantities, so
that the measurements are understandable, useful, reliable and accurate
enough for the purpose of measurement and communication.
By the end of this unit you should be able to
1. discuss the necessity of standard units and show how to use them
2. discuss the meaning of a standard and show how it is applied during
any measurement process
3. discuss the meaning and process of calibration against a standard
4. determine (by inspection) and discuss the accuracy of different
measuring devices
5. identify quantities to be measured (such as length, mass, time and
temperature) in different situations
6. calculate compound quantities (such as area, volume, density and
speed), while doing the proper unit conversions, when and where
necessary
7. work cooperatively and efficiently with your fellow students when
doing group work.
Everybody knows how to take (simple) measurements. For example, in
measuring length we take a metre ruler, and then . . . . . . . . . But wait!
Where does the ruler come from? Has it been around always or was it
invented? Do all people use a metre ruler or are there different rulers?
These are the questions to which we want to find answers in this unit. As
we go along, we will also find a precise answer to the question raised in
the title: „What is measuring?‟
You are used to measuring lengths by using a metre ruler. However, the
metre was only introduced about 300 years ago. But before that time
people also needed to measure lengths. They wanted to know the
distance between two towns or the area of a piece of land needed for a
family to grow sufficient food. In the experiment below we will measure
length and area without using a metre ruler.
PHY 133 UNIT 1 2022
worksheet
Experiment 1: Determining the length, width and area of the
classroom
Instructions
Suppose that you are living more than 300 years ago, so you do not
have a metre ruler. Find a way to determine the length, width and area of
the classroom without using any ruler.
Work in groups of 3 or 4 people. First discuss how you can take the
measurements, then take the various measurements. Report your
results in the table below under the heading „Our results‟.
Measurements
Our
results
Other Group
A
Other Group
B
Other Group
C
Length
Width
Area
Discussion of the results
Also write down the results of any three other groups of students in the
table.
Question 1 (write your answers in the box on the next page)
(a) Which group has the best results?
(b) Is Question 1 (a) a fair question? Explain.
(c) What needs to be done so that the measurements can be compared?
Conclusion
Draw a conclusion by answering the next question:
Question 2
(a) Use the answers found in Question 1 to describe exactly what needs to
be done so that measurements can be compared.
(b) Now describe exactly what measuring means.
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PHY 133 2016 UNIT 5
Standard Units
In the above experiment, we have seen that measuring length is
actually expressing a particular length in terms of a number of
units. These units can be a foot, a step, or anything that we wish
to use. But if we wish to compare measurements, we all need to
use the same unit and this unit then becomes the standard unit.
Theory
Question 1
To measure lengths, people in the past often used certain parts of their
Q1
own body as the standard unit. In the experiment above, you have
probably done the same.
Give at least three units used in measuring length, some still used in
some countries today, based on the length of certain parts of the human
body.
A FAMOUS STANDARD UNIT FROM HISTORY – THE ROYAL CUBIT
People have understood the need for standard units since very early
times. A famous example is the Royal Cubit, which was the original unit
for length used by the Egyptians.
The cubit is equal to the length from the elbow to the fingertips.
The Egyptians are famous for their pyramids. These were burial places
for the kings or pharaohs and their next of kin. Pyramids were built by
cutting out large rectangular blocks and piling these on top of each other.
Inside the pyramids are a number of rooms. In the biggest room the
pharaoh was laid to rest. The smaller rooms were meant for his or her
family members. Several other rooms were used to store treasures the
pharaoh would have needed for his life after death.
Pyramids
are
truly
enormous. The biggest
one is over 180 m high.
The Egyptian building
masters realised that in
order to build pyramids
that would last for thoupyramid
sands of years, the
blocks had to be cut out
very precisely. The cubit,
defined as the length
person
palm trees
from the elbow to the
fingertips, lacked precision, as some people are bigger than others. The
standard unit had to be defined more accurately and they did this by
defining the length of a certain block of black granite (a hard type of
rock) as the standard cubit, the so-called Royal Master Cubit.
3
PHY 133 2020 UNIT 1
Pyramids were built more than 5 000 years ago (that is about 3 000
years before Christ was born). So, people have indeed used
standardized units for a very long time.
THE STANDARD UNIT OF LENGTH – THE METRE
In Question 1 above, we saw that there are other standard units
for measuring lengths. The other better-known units, which are
still used in Britain and in particular in the USA, are the yard, the
inch and the foot. In the past, most countries had their own set of
units, which differed from country to country. When trading started
in a big way in the 16th century, different units made trading
Theory
complicated. The need was felt for one international system of
units.
In the 18th century, emperor Napoleon of France, who had conquered
several countries in Europe, took the initiative to get one standardized
unit for length for all countries. This process later developed into an
international organization; the General Conference on Weights and
Measures. The system of units defined by this organization is
SI units
based on the metric system and known by the abbreviation SI
(Système International) which is French for “International System”). SI
units are used by all scientists. In most countries (including South
Africa), the SI units are also used for all daily life practical purposes.
Scientists defined the standard unit of
length, that is, the metre, as follows:
The metre is the distance between two
very thin, parallel lines drawn on a bar of
platinum-iridium kept in a laboratory in
Paris. The bar is kept at a temperature of 0 °C.
Question 2
Q2
Suppose you are a manufacturer of ruler
sticks. What do you need to do
to make sure that the ruler sticks produced
have the right length?
platinum-iridium bar
1 metre
MEASURING MASS
Question 3
Suppose you are living a few thousands years ago and you are the chief
Q3
of village in which nobody knows anything about balances. To help your
people, you want to measure the harvest from year to year, so that you
will know whether there is sufficient food or not.
Describe a measuring procedure and measuring devices that you
could introduce to measure the mass of the harvest from year to year.
The standard unit of mass
Like units for length, there were also many different standard
units for mass. If you ask a person from Britain or the USA for his
weight (actually for his mass1) you may get an
answer: „My weight is four and a half stones.‟
There was also a need for one standard unit to be
used by all countries.
Theory
The standard unit of mass is defined as the mass of a
cylinder made of platinum–iridium alloy that is kept very
safe under a bell jar in a laboratory in Paris in France.
Question 4
Suppose that you are a manufacturer of mass
Q4
pieces. The diagram shows mass pieces of 500 g,
200 g, 100 g and so on. These mass pieces are
used on a balance to measure the mass of objects.
What do you need to do to make sure that the mass
pieces have the right mass?
ASSIGNMENT
1
Many people do not know the difference between the terms mass and weight. In Physics, these terms have
different meanings:
Mass is determined by the amount of substance. The mass of an object does not change when it is taken from
one place to another.
The weight of an object is equal to the gravitational force on the object. Because the gravitational force
depends on the masses of the objects and the distance between them, the weight of an object on Earth is
different from the weight of the same object on the Moon. (We will discuss this in more detail in Mechanics.)
5
PHY 133 2020 UNIT 1
Use the internet (use the Google search engine) and/or the library (start
with Encyclopaedia Britannica) to determine what instruments or
apparatus are used to measure the following quantities listed below; find
the cost of the instruments and the accuracy with which they are able to
measure mass.
a) The mass of the fruit on a large truck on a road, if the mass is
greater than 10 000 kg
b) The mass of vegetables in a supermarket or shop
c) The mass of a single small diamond
Question 5
The process referred to in question 2 and 4 is called calibration. All
Q5
measuring devices have to be calibrated; that is, checked whether they
actually give the correct readings.
Describe a process that we can follow to calibrate electronic balances.
Most countries have a special department, often called the Department
(or Bureau or Institute) of Measures and Standards. The task of such a
department is to check whether measuring devices used in factories and
shops in a country are showing the correct readings; therefore, they
check whether the devices have been calibrated correctly.
MEASURING TIME
In measuring time, we use the second as the standard unit.
How was the second defined until recently?
Information for
reading only
Units are
man made
Currently the second is defined in terms of the vibrations of a
cesium atom. You may have heard of quartz clocks: the
duration of the vibrations of the quartz crystal inside the
watch is constant and is used to drive the electronics that
show you the time. In the cesium clock, the frequency of a
single cesium atom is used and the second can be defined
very accurately.
Did you notice that all units are man made!
FUNDAMENTAL UNITS, LARGER AND SMALLER UNITS
We have now defined the units for length, mass and time, namely
the metre, kilogram and second. These are the so-called
fundamental units or base units.
Now, suppose we want to measure the thickness of a human
hair. It will be clear that the metre is a rather large unit. To
Theory
measure the thickness of a hair, it would be easier if we have a
smaller unit for length. In general, there is a need for other units both
bigger and smaller than the fundamental units. How should we define
these smaller units?
6
PHY 133 2020 UNIT 1
The old units systems were based on
parts of the human body and therefore not
based on any particular number system.
The box shows the system still used in the
USA (and to a lesser extent in Britain).
Non-metric units for length
1 foot = 12 inches
1 yard = 3 feet
1 mile = 1 760 yards
Doing calculations in a non-metric unit system makes calculations very
complicated. (Check for yourself how difficult it is to calculate the area of
a rectangle with a length of 4 yards and 2 feet and a width of 1 yard and
1 foot.)
As you know, our number
system is based on the
number 10. By means of
place values we show the
number of 1‟s, 10‟s, 100‟s
and so on; all as powers of
10 (see box on the right).
Number system – base 10
counting 100‟s
counting 1‟s
563 = 500+60+3
= 5×100 + 6×10 + 3×1
1
0
= 5×10² + 6×10 + 3×10
counting 10‟s
Because our number system is based on the number 10, we want our
system of units to be based on the number 10 as well. Thus, all smaller
1
and larger units should be multiples of 10 or 10 of the fundamental units.
For example, if we want to measure a small length, the metre is too
1
m. Then the
clumsy, so we want to use a smaller unit, for example
1
100
length can be expressed as: L = 3.7 x 100 m. This still does not look
1
good; we prefer to use prefixes for the multiple of 10. For 100 we use
the letter c (“centi-”) and with this we get L = 3.7 cm.
Calculations in a unit system, based on the number 10 (the so-called
metric unit system) are easy. If the length of a room is 4 m and 37 cm,
we can write this as L = 4.37 m. If the width is 2 m and 9 cm, we can
write W = 2.09 m. The area is found by multiplying 4.37 m by 2.09 m.
This is indeed much simpler than having to deal with yards, feet and
inches.
The next table shows the prefixes that you will have to memorize:
Table: Prefixes in the SI system
Power of 10
Prefix
Abbreviation
Example
nanon
ns = 10-9 s
10-9
micro
10-6
μ
μm = 10-6 m
millim
mg =10-3 g
10-3
centic
cm = 10-2 m
10-23
10
kilok
kg = 103 g
Mega
Mg = 106 g = 103
106
M
kg
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PHY 133 2020 UNIT 1
Remarks:
1. Note that both lower case and upper case letters are used (eg. m
and M). You must be precise in writing prefixes correctly.
2. The prefix micro- is abbreviated by the Greek letter μ, which is
different from the letters n or m.
3. Note that the fundamental unit for mass is the kilogram and not ram.
4. Prefixes are not only used with the units for length, mass and time,
but also with units for other quantities. An example is the unit for
frequency, where we often use kHz (kilohertz), which is equal to 103
Hz.
In the above, we have made a good start with developing tools used to
measure things. It seems that measuring things is an important aspect in
doing physics. Let us now look closer at why measuring things is crucial
in physics. This will also help us to understand better what physics is all
about.
The Measurement Summary (page 1) gives an overview of the SI Units
and prefixes.
THE AIM OF PHYSICS
In Experiment 1 we asked: „What is the length of the room?‟ You
could have answered the question by saying: „The room is long‟,
or, „The room is short.‟ But in physics we like to give answers that
are precise and that is why we decided to measure the length of
the room.
Theory
Suppose now that we would ask: Do you like the room? Can you
measure anything to answer this question? It should be clear
that we cannot measure anything to give an answer to this
question.
Physics
deals only
with
„things‟
that can
be
measured
This brings us to a core idea in physics. Physics deals only with
„things‟ that can be measured. In other words, anything that
cannot be measured is not part of physics. Love is not part of
physics, because we cannot measure love. But length is part of
physics, because we can measure length; likewise, colour is
part of physics, because we can measure colour (by measuring
the wavelength of light).
Quantities
All the „things‟ that we can measure are
called quantities. As we have seen,
Each
to measure a quantity, we need to
quantity
compare it to a standard unit. When
has a
measuring a length, we compare it to
unit
the standard unit for length, which is
the metre. The measured length is then
given as a number times the standard
unit.
8
Understanding the word quantity:
The word „quantity‟ in daily life means
something like „amount‟.
For example: we can talk about a
huge quantity of food, meaning a
huge amount of food. The word
„amount‟ also implies that we can
measure it.
PHY 133 2020 UNIT 1
Different representations are used in physics to describe Nature
The aim of physics is to describe Nature whereby we focus on the nonliving aspects of Nature. To describe something, we need a
representation. We use different representations in Physics. So far, we
have only used numbers, but we can also use graphs, diagrams and
words. As we will see, all these representations are useful in describing
how Nature works and we will use them all as we go along.
Biology, Chemistry and Physics - differences and overlap
The aim of Chemistry is also to describe Nature, but here the
focus is on molecular processes. When chemists investigate why
substances react, they use concepts like electromagnetic fields,
energy and the like, which are all part of Physics. The distinction
between Chemistry and Physics is actually artificial; there is only a
difference in focus. The aim of Biology is also to describe Nature,
Theory
but the focus is on living organisms. Both Chemistry and Physics
play a role in Biology. For example, in investigating how the heart works,
biologists have discovered that there are chemical processes at work
that produce small electrical currents.
The diagram represents the three subjects. The
overlapping parts of the boxes indicate the overlap
between these subjects.
Biology
Physics
Chemistry
Mathematics could be added as a fourth subject and
again there is much overlap between Mathematics and
Physics, between Mathematics and Chemistry and, to a lesser extent,
between Mathematics and Biology.
UNIT CONVERSIONS
Let us now return to the topic of measurement. In many practical
problems, a measurement expressed in a certain unit has to be
converted into another unit. For example, if we want to add two
lengths, say L1 = 32 mm and L2 = 0.53 m, we have to convert L1
into metres (or L2 into millimetres). The following section
discusses how this can be done without making mistakes.
Theory
However, the aim is to develop your understanding and not to
teach recipes.
Note to the student: Aim for understanding
Scientists consider unit conversion as very easy; students seem to disagree, as many
students make endless mistakes. The reason could be that teachers in schools
focussed on procedure - this is how you do it – they did not adequately explain why you
do it.
You cannot do science successfully without understanding. In the whole of this course,
keep on asking yourself: Why am I doing this?
Move away from just following recipes - they are meaningless if you do not understand
why you are doing certain things. The aim of this course is to develop understanding. In
marking your work, marks are given for showing understanding and insight, not for
following recipes.
9
PHY 133 2020 UNIT 1
A discussion at a builders‟ warehouse
Customer:
„I need a plank with a width of 67 cm.‟
Shopkeeper: „You mean, 670 mm?‟
Customer:
„Yes that is right, a width of 670 mm.‟
The shopkeeper converted the width of 67 cm into 670 mm. The
customer agreed that the conversion was correct. The width of the plank
that the customer wanted to buy can be expressed as either 67 cm or
670 mm. The unit conversion was correct, while the width of the plank
remained the same.
This brings us to the main idea in unit conversions:
In a unit conversion, the measurement itself DOES NOT CHANGE.
An example
The length of a piece of wood was measured and found to be L = 32
mm. Express L in cm.
The measurement showed that we needed 32 millimetres to get a length
that is the same as the length of the piece of wood. The question is now:
How many centimetres do we need to get the SAME length again?
A given length of wood
Measurement:
L = 32 mm
32 mm
L = 3.2 cm
Conversion:
1 cm
1 cm
1 cm
0.2 cm
In the conversion, the length of
wood remains the same
The diagram shows the conversion graphically: the same length is
obtained by taking 32 mm or by taking 3.2 cm.
Of course, we don‟t need to draw a diagram to find the answer: the unit
cm is 10 × bigger than the unit mm. If we needed 32 mm for the length of
the wood, we need 10 × fewer cm for the same length of wood, that is,
we need 32/10 = 3.2 centimetres. The next box summarizes this
reasoning:
If the unit becomes 10× larger …
L = 32 mm conversion L = 3.2 cm
. . . then the number becomes 10× smaller
10
PHY 133 2020 UNIT 1
Make sure that you understand the above fully.
You can test your understanding in the next question.
Question 6
Without doing any calculations, determine for each conversion given
Q6
below whether it is right or wrong. Try to do this as fast as possible.
(a) 4.3 mm = 0.43 cm
(b) 0.25 cm = 2.5 mm
(c) 7.2 km = 7200 m
(d) 390 km = 0.390 m
(e) 240 min = 4.0 s
(f) 300 min = 5 h
(g) 12 km = 12 000 000 mm
(h) 1 m2
= 10 000 cm2
In the above question, you had to find how much 1 m2 is bigger than 1
cm2. We can find this in an entirely mathematical way. Two examples:
Example 1: 1 cm2 = . . . . mm2 ?
Answer:
1cm2 (1cm)(1cm) (10 mm)(10 mm) 102 mm2
Example 2: 1 cm2 = . . . . m2 ?
Answer:
1 cm2 1cm x 1cm 
1 m x 1 m  1
1 m2 104 m2
2 x
100
100
10 102
Question 7
Do the same for the following:
Q7
(a)1 m2 = . . . . mm2
cm3
(b) 1 m3 = . . . . mm3
(c) 1 mm3 = . . . .
(d) 1 mm3 = . . . . m3
.cm3
(e) 1 km2 = . . . . m2
(f) 1 km3 = . . .
1 cm = 10 mm
In Physics, we use mathematics a lot. It happens that students can do
physics exercises by using mathematics correctly, without really
understanding the physics itself. In this course, we want to make sure that
the real understanding of the physics itself is never lost in the
mathematics. The purpose of the next question is to show in a nonalgebraic (graphical) way that 1 cm2 = 100 mm2.
Question 8
Use the diagram on the right to explain why
Q8
1 cm2 = 100 mm2.
1 cm = 10 mm
11
PHY 133 2020 UNIT 1
Question 9
How can we show, using a drawing, that
Q9
1 cm3 is equal to 1 000 mm3?
Draw a big cube, much bigger than the one
on the right. Draw as many cubes of 1 mm3
as you can. You cannot draw all 1 000
cubes, because the drawing will get cluttered
and it will take too much time to draw all the
small cubes.
Use your drawing to explain that there are
1 000 little cubes of 1 mm3 in the big cube of
1 cm3.
Question 10
In the following, first state by how much the unit on the right is smaller or larger
Q10
than the unit on the left, then state by how much the number has to be multiplied
or divided, and then write the answer. The first row is given as an example:
0.73
“New” unit “New”
0.73 mm =
cm =
0.73 mm = . . .
10bigger number 10
10
cm
smaller
0.073 cm
(a) 0.25 m = . . . mm
(b) 0.72 kg = . . . g
(c) 0.39 g = . . . mg
294 cm2 = . .
(d)
.mm2
0.930 m2 = . . .
(e)
mm2
880 mm3 = . . .
(f)
cm3
(g) 24 min = . . . h
2.0 m3 = . . .
(h)
mm3
Unit conversions, using an entirely mathematical approach
We do not encourage you to use the approach shown below to do unit
conversions. The approach is entirely mathematical and does not
develop a feel for the physics and is also slow. However, mathematics is
a powerful tool in physics and the approach is an interesting addition to
the approach followed so far.
Consider the following conversion: 880 mm3 = . . . cm3
As explained above, in a correct conversion the measurement remains
the same; that is the left hand side must be equal to the right hand side.
So the problem can also be written as an equation: 880 mm3 = x cm3,
which has to be solved for x. This requires standard mathematics, as we
will show:
Start with the equation:
880 mm3
= x cm3
12
PHY 133 2020 UNIT 1
Replace prefixes by their numerical values:
m)3
Work out the brackets:
Divide both sides by m3:
Solve for x:
880 (10-3 m)3
= x (10-2
880 x 10-9 m3
880 x 10-9
= x 10-6 m3
= x 10-6
x
=
880 109
106
= 880 x 103
= 0.880
Thus:
880 mm3
= 0.880 cm3
Check some of your answers in question 10 by using this mathematical
approach.
COMPOUND UNITS AND CONVERSIONS WITH COMPOUND UNITS
Many quantities in physics are made up of two or more quantities.
A familiar example is speed, which is defined as the distance
covered in a unit of time. The unit of distance is the metre, the
unit for time is the second, and therefore, the unit of speed is
metre/second or m/s.
Theory
A unit made
up of two or
more
fundamental
units is
called a
compound
unit
Conversions of compound units (like m/s) are a little bit harder
than conversions of single units. Again, it is possible to do
these conversions following a mathematical approach.
However, we prefer you to develop a „feel‟ for what is
happening. Take your time in answering the following
questions. The purpose is to develop a feel for these
conversions. A correct feel is extremely important in the rest of
this course. It is also extremely useful in chemistry.
Question 11
The density of a substance is defined as the mass of a unit volume of
Q11
that substance. The density of iron is 7.9 g/cm3.
(a) What does this actually mean? Complete the sentence:
The mass of . . . . . cm3 of iron is . . . . . . . . .
We want to convert the density from g/cm3 into kg/m3. So, instead of
giving the mass of 1 cm3, we must give the mass of 1 m3. Secondly, we
do not want to express the mass in g but in kg. Therefore, we are
dealing with 2 unit conversions. To understand what we are doing, you
have to do these two conversions one after the other.
(b) If 1 cm3 has a mass of 7.9 g, then what is the mass of 1 m3?
(c) Now express the result found in (b) to find the mass of 1 m3 in kg.
(d) Is the density found in (c) still the same density that was given?
(e) Explain why the number in (c) is much bigger than before.
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PHY 133 2020 UNIT 1
A mathematical approach to do unit conversions for compound units
For those of you who feel strong in mathematics, we will present an
entirely mathematical approach in which we use no physics at all.
(Physicists do not really like it.)
The equation:
7.9 g/cm3
= x kg /m3
Replacing prefixes by their numerical values:
7.9 g
(102 m)3
3
= x 10 g / m 3
7.9 g
106 m3
Work out the brackets:
= x 103 g /
m3
7.9
Divide by g and multiply by m3:
10
3
Divide by 10 and simplify:
Answer:
kg /m3
= x 103
6
x
7.9 g/cm3
= 7.9 x 103
= 7.9 x 103
Question 12
Do the following conversions:
Q12
(a) 0.84 g/cm3
= . . . . . . . . kg/m3
3
(b) 1.6 mg/mm
= . . . . . . . . g/cm3
(c) 0.035 kg/cm3
= . . . . . . . . g/mm3
(d) 4.2 N/cm2
= . . . . . . . N/m2
(e) 67 m/s
= . . . . . . . km/h
In the above, we have discussed what „measuring‟ means and we
looked in detail at units and unit conversions. Another issue in taking
measurements is accuracy; how can we take good measurements. In
the next experiment we will make a start in investigating how to take
good measurements.
Uncertainty
Introduction
Two students A and B are given a pendulum and a
stopwatch. The task is to accurately measure the
time (called the period) to complete one oscillation
of the pendulum. Here follows the discussion
between the two students:
Student A: „First take the mass piece
(the
pendulum “bob”) a couple of centimetres
to the left. Then let go and allow it to
T=?
swing from left to right. Then start the
stopwatch when the bob is on the left
and stop it when it is again back on the left‟.
Student B: „I agree with you to let it swing first, but I disagree with trying
to measure the period of one oscillation. I think we must
14
PHY 133 2020 UNIT 1
measure the time it takes to complete more oscillations, say
10 oscillations.‟
What do you think is best? Which procedure will give more accurate
results?
In the experiment we will try to find an answer to this question.
Measuring procedure A
Measure the period of 1 oscillation according to the method of student A.
Each student in your group must measure the period twice. Record the
measurements in the table.
Student 1
Student 2
Measurement Measurement Measurement Measurement
1
2
1
2
1 period
(s)
Measuring procedure B
Measure the period of 10 oscillations according to the method of student
B. Each student in your group must do this twice. Record all the
measurements in the table. From each measurement, calculate the time
for one period. Record the answer to these calculations in the last row of
the table:
Student 1
Student 2
Measurement Measurement Measurement Measurement
1
2
1
2
10 period
(s)
1 period
(s)
Analysis
Carry out an analysis of the results by answering the questions below. While
doing so, keep
in mind what we are actually trying to find out.
Experiment 3: Question 1
(a) Look at the results of measuring procedure A. Are there any differences
in the results?
(b)
Is it possible to repeat the experiment, taking perhaps better
measurements, so that there will be no differences at all?
(c)
Look at the results of measuring procedure B (the last row). Are there
any differences in the results?
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PHY 133 2020 UNIT 1
(d)
Compare the differences in the results of procedures A and B. In which
procedure are the differences smaller?
(e)
Which measuring procedure is better? Explain.
Experiment 3: Question 2
(a) What causes the differences in measuring procedure A?
(b) The differences in measuring procedure A are called measuring errors.
Are these errors mistakes caused by the students, or are they caused by
the way we carried out the measurement?
Experiment 3: Question 3 - conclusion
(a) In taking measurements, there are always measuring errors which we can
not avoid. However, by following good measuring procedures, we can
make the measuring errors smaller. In measuring procedure . . . . . . (fill in
the letter) the procedure followed is better than in measuring procedure . .
. . . (fill in the letter) because the . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
............................ ...............................
... ...
(b) Give this experiment a proper title
Measuring error
WHAT IS “MEASURING ERROR”?
Can we a use a bathroom scale to measure the weight of a pencil? Can we use
a wristwatch to measure the time for an object to fall from a table down to the
ground? Of course we can, but the results will be very inaccurate; or as we say
in Physics, the measuring error will be large. In this unit, we will see what
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PHY 133 2020 UNIT 1
causes measuring errors and how to estimate the “size” of the measuring
errors.
Question 13
Q1
Six students measured the time for a ball to roll down an incline and they
obtained the following results:
Student
1
2
3
4
5
6
Time (s)
2.34
2.44
2.39
2.02
2.40
2.37
(a) Plot the measurements on a numbers line:
(b)
Which measurement is not acceptable? Why not?
(c) Is this measurement different from the other measurements due to a
measuring error or due to a mistake?
(d)
What was probably the cause?
(e)
Do you now understand the title of Investigation H? Explain.
Question 14
Q2
(a) What is the main cause for getting measuring errors in the investigation
above - are the measuring errors caused by the stopwatches or . . . . . . ?
(b) We can often make the measuring error smaller by using more advanced
measuring tools. What kind of equipment should we use in the experiment
above?
ESTIMATING THE MEASURING ERROR IN EXPERIMENT
INTRODUCTION
After doing an experiment, we normally have to write a report and present the
measurements. Suppose that some students performed the experiment above.
Each of them has measured the time, they have calculated the average time
and in the report they write that the time was 3.03 s. Now a reader of the report
knows the average time, but he does not know how accurately the students
were able to measure the time. Were the values for the time as measured by
the students far apart or very close? So, apart from reporting on the average
time, we also have to report on how accurately we were able to measure the
time.
The next box shows a simple method to find an estimate ("indication") of the
uncertainty (or measuring error):
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PHY 133 2020 UNIT 1
A simple method to estimate the uncertainty
1.
Rank the values as measured in the experiment, starting with the smallest one.
2.
Represent them on a numbers line.
3.
Ignore the measurement(s) that are clearly too small or too large.
4.
Calculate and then indicate the average value on the numbers line.
5.
Calculate the difference between the average value and the smallest value.
6.
Calculate the difference between the largest value and the average value.
7.
Take the value that is the larger of the two values calculated in Steps 5 and 6.
8.
This value is called the uncertainty (or measuring error).
9.
The measurement is now reported as:
measurement = average value ± uncertainty
(e.g. time = 1.57 0.04 s)
Note: There is a more sophisticated method to calculate the uncertainty, called the
least-squares method. This method gives a better (and more reliable)
estimate of the uncertainty, but it is beyond the scope of this course. Note
decimal places.
Question 15
Use the values from Investigation H and
Q3
(a) apply the above procedure to calculate the uncertainty, and
(b) present the measurement in the format:
measurement = average value ± uncertainty
THE MEANING OF THE MEASURING ERROR OR UNCERTAINTY
The measuring error or uncertainty is an indication of the “spread” in the
measurements. If the spread is large, then the measurements were not very
precise. If the spread is small, then the measurements were precise. When
measurements are not precise, it does not mean that the measurements were
bad; it only means that the measuring procedure is not able to give very
precise results. When we report a result like
t = 4.62 ± 0.30 s,
it means that the measurements ranged from 4.32 s to 4.92 s. Hence, we do
not know the true value exactly.
We
• cannot claim that the time is 4.62 s
• can only claim that the true value is somewhere between 4.32 s and 4.92 s.
In other words, the measuring error leads to uncertainty and the value 0.30 s in
the result above is therefore called the uncertainty.
Note that in this example it is actually better to write: t = 4.6 ± 0.3 s. Given the
fairly large measuring error of 0.30 s, the last digit in 4.62, that is the 0.02, has
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PHY 133 2020 UNIT 1
no meaning. We are not sure about that value at all. If we are not sure about
the value of a certain digit (the digit 2 in 4.62), we should not write the digit.
Question 16
Q4
Two groups of 5 students each measured the time required by a
ball to drop down from a building to the ground. Each student
used a stopwatch to measure the time:
Results for group 1 (in s)
group 1
4.24 4.33 4.36 4.28 4.34
Results for group 2 (in s)
4.38 4.30 4.10 4.24 4.48
(a) Calculate taverage and the measuring error for each group.
(b) What is the range of possible values for ttrue value as found by
group 1 and as found by group 2?
(c) Which group produced the most accurate results? Explain your answer.
Question 17
A metal ball A is dropped from a building and the time needed to reach
the ground is measured by using stopwatches, as in our experiment
above. The same is done for a heavier metal ball B.
Q5
Measurements for dropping times: ball A: 3.47 s and ball B: 3.48 s.
The conclusion was drawn that ball A falls quicker than ball B.
Is the conclusion right or wrong? Explain.
Question 18
Q6
Two students are competing in a cycling race. As only one bicycle is
available, they raced the bike over a distance of 400 m one after the other.
The times for both students were measured using a handheld stopwatch, as
in the experiment above. The times needed to cover the distance were:
Student A: 44.2 s and student B: 44.4 s.
Who is the winner? Are we sure?
Question 19
Q7
To measure the times in an official swimming contest, electronic timing
devices are used. To test the reliability of the timing system, four
electronic stopwatches were used to measure the time for one
swimmer. The following times were recorded for the swimmer: 44.04 s;
44.03 s; 43.98 s and 44.00 s.
(a) What is the spread in the values for the time?
(b) What is the uncertainty of the electronic timing devices?
(c) In testing the devices, why do we measure the time for one swimmer?
(d) This electronic equipment is now used in a race. The time measured for
swimmer A was 42.38 s and for swimmer B 42.44 s. Who is the
winner? Can we be sure?
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PHY 133 2020 UNIT 1
group 2
Question 20
Q8
A diary puts milk into one-litre cartons. A government department takes
measurements regularly to check whether the diary actually puts the right
amount of milk into the cartons. This is done by taking a sample
measurement as follows: From a whole load of cartons, one carton is taken
out at random and the amount of milk in the carton is measured using a
measuring cylinder. The measuring cylinder used by the department has an
uncertainty of 0.005 ℓ.
(a) One day the department measures an amount of 0.997 ℓ. Given the
uncertainty of the measuring cylinder, the actual amount of milk could
be a bit more or a bit less. Show on a number line the range of values
that are actually possible.
(b) Should the department issue a warning?
(c) A few weeks later they measured 0.993 ℓ of milk. What is the range of
possible value for the actual volume now?
(d) Should the department issue a warning?
SYSTEMATIC ERRORS
Demonstration experiment:
Catching a banknote
Instructions
Person A keeps holds a banknote. Another person B keeps his/her thumb
and forefinger slightly open with the banknote in between but not touching
the banknote (see the drawing). When person A let go person B must try
to grab the banknote. Person A should not give any warning when letting
go.
(The experiment becomes really interesting when A and B make a bet. B
may keep the banknote if he/she can grab it, but B must pay A the same
amount of money if he/she fails to grab it).
Observations
Perform the experiment and write down your observation.
Conclusion
In the above experiment, you have probably found that person B finds
it difficult, if not impossible, to grab the banknote. This is not surprising, if we
realize what B has to do. Let us consider a bit of Biology:
The eyes register the image of the falling banknote
Via the nerves from the eyes to the brain to brain realizes the
banknote is falling.
The brain thinks quickly; the banknote is falling - I must grab it.
It sends a signal via the nerves to the muscles of the fingers to
grab the note.
On receiving the signal, the muscles contract.
Reaction time:
the time needed
to respond
All these actions require a bit of time. Measurements show that
the reaction time of human beings is between 0.2 s and 0.3 s.
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PHY 133 2020 UNIT 1
hand A
banknote
In the experiment, the fingers close after about 0.2 s and by then the banknote has
already fallen too far.
Reaction time plays an important role in driving:
Suppose you drive at a speed of 120 km/h (33 m/s) and you see a man on the
road in front of you. Before you even touch the brakes, you have already
covered a distance of about (0.3 s)(33 m/s) = 10 m. (Braking itself requires a
much greater distance; we will discuss this in mechanics). With some alcohol in
your blood, the reaction time becomes much longer.
Question 21
Q9
Students are asked to measure the time needed by
a trolley to cover the distance AB. The trolley is
released at point A, but the person releasing the
trolley does not indicate when he/she is going to
release the trolley.
A
B
(a) What is more difficult for the students: To start
the stopwatches at the right time or to stop the stopwatches? Explain
your answer.
(b) Are the times as measured by the students too short or too long as
compared to the true time? Explain your answer.
(c) The type of measuring error referred to in question (b) is called a
systematic measuring error. Explain the name systematic.
Question 22
An educator in a Grade 8 class asks her learners to measure her
length. She positions herself against a metre ruler with a length of
Q10
2 m, as shown in the drawing. The learners measure her length
by standing in front of her and looking just over her head at the
metre ruler. Note that the educator is pretty tall.
(a) Will the learners find the correct length of their educator?
(b) If not, will the value of the length be too large or too small?
(c) What should the learners do to avoid this error?
The Demonstration Experiment showed us that people need time to respond.
In the measuring procedure followed in Question 9, this resulted in values for
the time, which are always too small. In Question 10, learners measured the
length of their educator. In this case, the measurement produces values, which
are always too large.
In both examples, the measuring error is called a systematic error.
Compare this to measuring the length of a block of metal with a ruler or vernier
callipers. Here there are also errors, but of a different type. Suppose we find
that the length is 45.3 mm. The true value might be 45.34 mm. Other students
may find a length of 45.4 mm for the same object. Because our measurements
produce values that are either a bit too small or a bit too large, we call the error
in these measurements random errors.
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PHY 133 2020 UNIT 1
Random
errors and
systematic
errors
The next diagram is a graphical summary of the concepts random error
and systematic error. The short vertical lines represent measurements
taken in certain experiments. The arrows show the actual or true value
of the quantities that we wish to measure:
Only random error
measurements spread
around true value
true value
Random & systematic
most/all measurements
are too large
Random & systematic
most/all measurements
are too small
true value
true value
Question 23
Look at a spring balance:
Q11
Spring balances measure the force exerted on the hook.
If there is no force exerted on the hook, the reading
should be 0 N. To get a correct 0 N reading, the scale
can be shifted up or down; this is called the zero
adjustment.
0N
Systematic measuring errors occur if the zero adjustment
is incorrect. In the drawing on the right, two of the spring
balances show an incorrect zero adjustment.
1N
(a) Which two spring balances show an incorrect zero
adjustment?
A
B
C
(b) If spring balance A were used to measure a force,
would the reading be too small or too large?
(c) What type of measuring error is made when using A?
(d) What type of measuring error is made when using C?
Question 24
Which type of measuring error would you consider being more serious: a
systematic measuring error or a random measuring error?
Q12
Question25: Parallax
The drawing shows a speedometer
of a car. In older cars, the pointer is
Q13
a couple of millimetres above the
reading scale.
60
90
120
30
150
(a) A person sitting to the left of the driver
takes a reading. Will his/her reading
0
22
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PHY 133 2020 UNIT 1
be correct, too small, or too large? Explain.
(b) Readings taken by looking at a scale at an angle cause systematic
errors and this type of error is also called parallax. In which of questions
9, 10 and 11 above were systematic errors made due to parallax?
TWO TYPES OF NUMBERS: EXACT NUMBERS AND NON-EXACT NUMBERS
If there are two people in the room, then we say the number of people is 2. This
is an exact number, because there cannot be 1.95 or 2.3 people in a room. If
three more people enter the room, then there are 2 + 3 = 5 people in the room.
The number 5 is an exact number.
If we measure the length of a room and we find L = 4.0 m, then the number 4.0
is not an exact number. If we would use a more precise measuring tool, we
might measure the length to be L = 4.04 m. The number 4.0 in L = 4.0 m is the
result of a measurement and it is not exact.
All numbers used to express the results of measurements are non-exact
numbers.
In the sciences, we use both types of numbers. It would have been nice to use
a different notation for exact numbers and non-exact numbers; for example,
underlining non-exact numbers. Then we could see the difference clearly; for
example: there are 5 people in the room and the length of the room is: L = 4.0
m. However, this is not done and you must determine from the context of the
problem whether you are dealing with an exact number or a non-exact number.
Perhaps you think that it does not matter whether a number is exact or nonexact. In the next section we will explain that it does matter. Make sure that you
understand the above; many students make mistakes because they think that
there is no difference between exact and non-exact numbers.
USING SIGNIFICANT FIGURES TO SHOW THE ACCURACY OF MEASUREMENTS
On page 3 we used the spread in the measured values to determine the
uncertainty in a measurement and the measurement was gives as:
measurement = average value ± uncertainty.
Another way to present measurements is by expressing the measurements
with the correct number of figures. This will be explained below.
Suppose we carry out two experiments in which we measure the length of a
room.
Measurement A: taking steps
Each student measured the length of the room by taking big steps of roughly
1 m. They found that they needed four steps to walk from one side to the other
side. How should they report on the result of the measurement?
The method is very inaccurate; the students only know that the length is
roughly four metres. It is clear that the best way to report is: L = 4 m.
Measurement B: use a long measuring tape
The students then measured the room using a measuring tape. They found a
length L = 4.28 m.
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PHY 133 2020 UNIT 1
Measurement A and B have different accuracies. The difference in accuracy is
shown by the number of significant figures used to report the measurements.
The box below shows the results again.
Measurement A
Measurement B
L=4m
L = 4.28 m
1 significant figure
3 significant figures
In Measurement A we got information about the number of metres, but there
was no information about the next figure (tenths of a metre), no information
about the next figure (hundredths of a metre, or cm) and so on.
In Measurement B we got information about the number of metres, we also got
information about the number of tenths of a metre (there were 2), and even
about the number of centimetres (there were 8). There was no information
about the number of millimetres and the following smaller units.
Thus in Measurement A only one figure was known; in Measurement B three
figures were known. We call the figures that we know significant. Thus in
Measurement A there is one significant figure; in Measurement B there are
three significant figures.
The number of significant figures used to report measurements shows the
accuracy of the measurements; more significant figures mean that the
measurement was more accurate.
So we have two ways of showing the accuracy of measurements:
■
We can report a measurement by giving the value and the
uncertainty, or
■
we can give the value of the measurement with the correct
number of significant figures.
Question 26
In two different experiments, the mass of the same block of metal was
measured:
Q14
Experiment 1: m = 3.4 kg.
Experiment 2: m = 3.400 kg.
The measurements are reported correctly.
Are the two measurements (m = 3.4 kg and m = 3.400 kg) equal? Explain
your answer.
USING SIGNIFICANT FIGURES TO REPORT THE PERIOD OF A PENDULUM
Suppose that we measured the period of a pendulum four times. The following values
for the time were found:
Time (s)
4.20
4.30
4.44
4.38
4.23
Calculating the average time, we find that Taverage = 4.31 s. If we don’t think further, we
could report this value: Tpendulum = 4.31 s.
However, the spread in the values is large, from 4.20 s to 4.44 s. Calculating
the uncertainty, we find a value of 0.13 s (check). This means that the true
24
PHY 133 2020 UNIT 1
value can be anywhere between: Taverage + 0.13 = 4.44 s and Taverage - 0.13 =
4.18 s. The ange of values is shown in the next diagram:
The true value is located somewhere in the
range indicated by the double-headed arrow
Let us look at the value Taverage = 4.31 s. Given the uncertainty, which digits do
we know for sure, which digit has some uncertainty, and which digit is
unknown? All values in the range above have a 4 as the first digit; therefore,
we know for sure that the first digit must be a 4. The second digit ranges from 1
to 4, so we know something about it - it is between 1 and 4, but there is some
uncertainty. The third digit is completely unknown.
The digits we know must be reported, the digit where there is some uncertainty
we can still report, the digit we do not know anything about must not be
reported.
So, if we want to use significant figures to present the result of the
measurement, we write:
Taverage = 4.3 s.
We do not write T = 4 s, because we have information about the next figure, so,
we should not throw it away. In rounding off, we keep what we know and we
delete what we don‟t know at all.
Rounding off must be done in a sensible way.
If we want to present the uncertainty as well, the best way of writing the
measurement is:
Taverage = 4.3 ± 0.2 s
It often happens that we have to perform calculations with numbers that are the
result of measurements and hence not accurately known. For example, if you
want to calculate the speed of a car, you need to divide the distance covered
by the time required. Both distance and time are found by taking
measurements and are given to a certain number of significant figures.
What is the accuracy of the speed given the accuracy of the distance and time?
This is the topic that we will discuss below. Before we do this, we must first
discuss a certain way of writing down numbers, called the scientific notation.
SCIENTIFIC NOTATION
First, we will explain scientific notation and then we will show how scientific
notation can be used to express measurements with the correct number of
significant figures. In this, we will assume that you know how to work with
powers of 10.
Some examples:
25
PHY 133 2020 UNIT 1
Number given
The number in scientific notation
234.56
2.345 6 x 102
1 002.10
1.002 10 x 103
0.459
4.59 x 10-1
0.000 450
4.50 x 10-4
Check that each number on the left has the same value as the corresponding
number on the right.
In scientific notation numbers are written as follows:
a x 10n, where a is a number between 1 and 10
Question 27
Express the following numbers in scientific notation (without changing
the units):
Q15
(a)
4 309 m
(b)
0.034 5 mm
(c)
399 001 μm
(d)
0.000 02 kg
WHAT IS THE USE OF THE SCIENTIFIC NOTATION?
To express big and small numbers in an easy way
1.
One mole contains
602 200 000 000 000 000 000 000 particles.
It much easier to write that there are 6.022 x 1023 particles in one mole.
The mass (mH) of one hydrogen atom is
mH = 0.000 000 000 000 000 000 000 000 001 67 kg.
It is much easier to write: mH = 1.67 x 10-27 kg.
2.
To express measurements in the correct number of significant figures
Example 1
Suppose we measure a mass and we find: m = 4.2 kg. The measurement is
presented in 2 significant figures, meaning that the balance used was not very
accurate.
Does 4.2 kg
= 4 200 g?
Does 4.2 kg
= 4 200.0 g?
Now we are asked to present the measurement in grams. You may think
that this is simple: 4.2 kg = 4 200 g. It may not be clear at first, but 4.2 kg
actually has the same number of significant figures as 4 200 g. Let us look
at this a little closer:
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PHY 133 2020 UNIT 1
In the original measurement, m = 4.2 kg. This way of writing the measurement
means that there is no information about the tens of grams, the grams and
tenths of a gram, meaning that we were not able to measure these digits. If,
after the conversion, we write m = 4 200.0 g, then we imply that we have
measured thousands of grams (there were 4), hundreds of gram (there were
2), tens of grams (there were 0), grams (there were 0) and tenths of grams
(there were 0). So, in writing m = 4 200.0 g, it would imply that the mass was
measured much more accurately than writing m = 4.2 kg.
Therefore, 4.2 kg 4 200.0 g, because the implied accuracy is different.
Question 29
(a) How many significant figures are in the number 4 200.0?
Q16
(b) Based on the above discussion, how could one write a measurement
where the thousands (4), hundreds (2), tens (0), units (0), tenths (0)
and hundredths (0) were measured accurately?
(c) What are the differences in precision between the following reported
measurements: 4 200.00 g; 4 200.0 g; 4 200, g and 4 200 g?
(d) How would you indicate a measurement where the thousands (4),
hundreds (2) and tens (0), but not the units, were measured accurately?
(e) Does 4.2 kg = 4 200 g?
Note that we are doing science here and not mathematics. In mathematics, we
normally deal with exact numbers, while in the sciences we deal with exact and
non-exact numbers. The number 4.2 in m = 4.2 kg is a non-exact number and
we have to be careful in applying the mathematics in the usual way.
To convert m = 4.2 kg into grams while keeping the accuracy the
same, we can use the scientific notation. The correct conversion
is: m = 4.2 kg = 4.2 x 103 g. The original measurement and the
measurement after conversion both have two significant figures
(see the diagram).
m = 4.2 kg = 4.2 x 103 g
2 significant figures in both
Example 2
A marine biologist tried to estimate the number of seals on a rocky outcrop at
sea. She estimated that there were 150 seals.
She could not count the seals, because they were moving and diving into the
water and coming out again. So, the number 150 is an estimation. She could
be wrong by 10 seals or so. In her report she wrote: No of seals = 150 ± 10.
Another way of writing this is by using significant figures. Writing 150 in
scientific notation, we get 1.50 x 102. The biologist could not count the seals
properly, therefore, she was uncertain about the units place (the last figure), so,
we should write: Number of seals = 1.5 x 102.
Question 30
Q17
Convert the following measurements to the indicated unit. Express the
final answer in scientific notation, with the correct number of significant
figures.
(a) 4 309 m
= . . . . mm
(b) 0.034 5 mm
=....m
(c) 397.0 μm
= . . . . cm
(d) 0.000 200 kg
=....g
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PHY 133 2020 UNIT 1
FINDING THE NUMBER OF SIGNIFICANT FIGURES IN A GIVEN MEASUREMENT
Consider the measurement m = 0.000 200 kg in Question (d) above. To answer
the question, you have probably written the measurement in scientific notation:
m = 2.00 x 10-4 kg. Writing it in scientific notation, we can see immediately that
the measurement has 3 significant figures.
Can we see that also in the original measurement m = 0.000 200 kg?
In this number, we have two types of zero‟s; zero‟s that are needed
to get the right place value and zero‟s that show the accuracy of
the measurement. The diagram on the right illustrates this.
m = 0.000 200 kg
for place
The 2 zero‟s at the end are given to show that these digits Zeros
2 trailing zeros
value
were measured, thus, together with the 2, it follows that
there are 3 significant numbers in the measurement. The
measurement in scientific notation was: m = 2.00 x 10-4 kg, that is, also 3
significant figures. We have found that also for the original measurement if we
consider only the trailing zeros as significant. Note that writing measurements
in scientific notation makes matters easier, as we do not have to worry about
the different types of zero‟s.
Question 31
Consider each of the numbers in the next table and do the following:
Q18
 Determine the number of significant figures in each measurement
 Write in scientific notation and determine the number of significant
figures.
 Carry out the unit conversion and determine the number of
significant figures. The answer should also be given in scientific notation.
The first row is given as an example:
Measurement
0.034 m
4.30 kg
2
22 m
34.0 μm
72 h
428 min
3
300 cm
No of
sign.
figures
2
In
scientific
notation
-2
3.4 x 10 m
No of
sign.
figures
2
Unit conversion
1
3.4 x 10 mm
. . . . . . . .g
2
. . . . . . . . mm
........m
. . . . . . . . min
........h
3
. . . . . . . . mm
No of
sign.
figures
2
MATHEMATICAL OPERATIONS WITH SIGNIFICANT FIGURES
Introduction
A horse has a mass 658 kg. A fly with a mass of 1 g lands on its back. What is
the total mass of the horse and the fly?
Incorrect solution
1 g = 0.001 kg, so, the total mass becomes: mtotal = 658.001kg (check).
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PHY 133 2020 UNIT 1
We started with a mass of the horse given with 3 significant figures and a mass of the
fly given with 1 significant figure. Now we get an answer that has 6 significant figures.
This must be wrong. The reason for this incorrect result is that we dealt with the
numbers as if they were ordinary exact numbers; we just added the numbers. In this
section, we will see that we cannot simply apply the normal mathematics when dealing
with numbers that are not exact.
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PHY 133 2020 UNIT 1