The Measurement: SI Units, Conversions, and Experimental Uncertainty

Introduction to Measurement Science

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Science of Matter and
Energy
Jesús Aguado Molina
The measurement
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Index of the Topic

Science of Matter and
Energy
Jesús Aguado Molina

1. INTRODUCTION
2. THE INTERNATIONAL SYSTEM OF UNITS
3. BASE QUANTITIES IN THE SI

Time as a Base Quantity

3.1 Time

Longitude as a Base Quantity

3.2 Longitude

Mass as a Base Quantity

3.3 Mass

1. DERIVED QUANTITIES IN THE SI
2. CONVERSION OF UNITS
3. EXPERIMENTAL UNCERTAINTY
4. SIGNIFICANT FIGURES

Calculations Involving Significant Figures

7.1 Calculations involving significant figures

Rounding Off Numbers

7.2 Rounding Off

Scientific Notation

7.3 Scientific Notation

1. ORDER OF MAGNITUDE

Prefixes and Measurements

8.1 Prefixes and measurements
20 m
1 m
2 m
3 m
4 m
5 m
6 m
30 m
1 m
2 m
3 m
4 m
5 m
6 m
5.40 m
5.42 m
5.4425 m
5.45 m
5.40 m

• The International System Of Units
• more than one measurement
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Introduction to Science of Matter and Energy

1. INTRODUCTION
Science of Matter and
Energy
Jesús Aguado Molina
Measurements taken by different people in different places with different tools must
yield the same result.
5
6
5.40
5.42
5.4425
5.45
5.40
If it is not possible, you have to average all the measurements.
How many numbers after the comma do we have to use?
Comma not
coma
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Numbers After the Comma

How many numbers after the comma do we have to use?
Science of Matter and
Energy
Jesús Aguado Molina
You can say ( ... ) after the comma or
( ... ) beyond the decimal point.

Appropriate Measurement Selection

• Which is the most appropriate
  measurement?
• 9.5 cm
• 9.52 cm
• 9.521 cm
  It depends on both the skill
  of the experimenter and the
  apparatus used
  often can only be estimated
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Using Numbers After the Comma

How many numbers after the comma do we have to use?
Science of Matter and
Energy
Jesús Aguado Molina
We have to use always the same number of numbers after the comma, therefore, the
average of the measurements have to be expressed in the same way:
Significant figure
5/40
5.40
5.42
5.42
5.4425
5.45
5.45
5.40
5.40
Why does it happen?
In this case the numbers "5" and "4" have some degree of confidence
But the last digit "2" or "5" or "O" is an estimate or approximation
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Significant Figures in Science

7- SIGNIFICANT FIGURES
Science of Matter and
Energy
Jesús Aguado Molina
. A reliably known digit (other than a zero used to locate the decimal point)
number
significant figures
2.50
three significant figures
2.503
Four significant figures
0.00130
three significant figures
1.00130
six significant figures
0.0
one significant figure
2300.0
five significant figures
2300
four significant figures
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Measurement Sensitivity and Significant Figures

Science of Matter and
Energy
Jesús Aguado Molina
. The number of significant figures in a measurement, such as 2.531 (m), is equal to the
number of digits that are known.
. In this case the numbers "2", "5" and "3" have some degree of confidence
. But the last digit "1" is an estimate or approximation.
. As we improve the sensitivity of the equipment used to make a measurement, the number of
significant figures increases
· 2.53 ±0.05 (m): 3 significant figures
3 (m): 1 significant figure
2.531 (0.005)
(m) 4 significant figures
3 ±5 (m): 1 significant figure
1 significant figure
Be careful: the number of significant figures of the number has nothing to do with the
number of significant figures of the experimental uncertainty (it is always one)
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Finding Significant Figures and Experimental Uncertainty

Find out the S.F. and experimental uncertainty
Science of Matter and
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Jesús Aguado Molina
· 0.0 :
· 0.1:
· 1.01 :
· 0.01:
· 1.10:
· 0.10:
· 1.11:
· 0.11:
· 0.001 :
· 1.001 :
· 0.011:
· 1.0
· 1.1:
· 1.011:
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Calculated Significant Figures and Experimental Uncertainty

Find out the S.F. and experimental uncertainty
Science of Matter and
Energy
Jesús Aguado Molina
· 0.0 :
1 significant figure±0.5
· 1.0
2 significant figure ±0.5
· 0.1:
1 significant figure± 0.5
· 1.1:
2 significant figure ± 0.5
· 0.01:
1 significant figure± 0.05
· 1.01 :
3 significant figure ± 0.05
3 significant figure
+ 0.05
· 0.10:
2 significant figure± 0.05
· 1.10:
· 0.11:
2 significant figure± 0.05
· 0.001:
1 significant figure± 0.005
.
1.001:
· 0.011:
2 significant figure± 0.005 . 1.011;
· 1.11:
3 significant figure ± 0.05
4 significant figure ± 0.005
4 significant figure ± 0.005
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Calculations Involving Significant Figures

7.1 Calculations involving significant figures
Science of Matter and
Energy
Jesús Aguado Molina
3.1
·
4.1 = 12.7
3.12 .
4.1 = 12.79
3.12 . 4.12 =12.85
. When combining measurements with different degrees of accuracy and precision, the accuracy of
the final answer can be no greater than the least accurate measurement.
3.1 . 4.1 = 12.7
3.12 . 4.1 = 12.8
3.12 . 4.12 =12.85
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Addition and Subtraction Rules

Addition and Subtraction
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1.21342 - 1.040?
·The second number, 1.040, has four significant figures. Three after the comma (or
beyond the decimal point)
·The first number, 1.21342, has six significant figures. Five after the comma
.According to the rule stated for the addition and subtraction of numbers,
the difference can have only three significant figures after the comma
·Sum the numbers, keeping three significant figures after the comma:
.1.21342 - 1.040 = 0.17342 (incorrect)
.1.21342 - 1.040 = 0.173
Examples: If we compute the sum
. 123 + 5.35, the answer is 128 and not 128.35.
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Multiplication and Division Rules

Multiplication and Division
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. The same principle governs the use of significant figures in multiplication and
division: the final result can be no more accurate than the least accurate
measurement.
How many significant figures do we have to use in this case?
. 5.4 = 20
· 2.5 . 2 = 5
· 2.5 . 2.0 = 5.0
· 52 : 27 = 1.925925926 ~ 2
· 225: 134 = 1.679104478 ~ 2
14Many of the numbers in science are the result of measurement and are therefore known only to
within a degree of experimental uncertainty
The experimental uncertainty is
expressed by: ±
5.40 ± 0.05
5.42 ± 0.05
5.45 ± 0.05
5.40 ± 0.05
5.4175 ± 0.05
155.40
5.40
5.4175 ± 0.05
5.42
5.42
5.45
5.45
5.40
5.40
5.42 ± 0.05
When the answer to a calculation contains too many significant
figures compared to the experimental uncertainty, it must be
rounded off
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Experimental Uncertainty in Measurements

6. EXPERIMENTAL UNCERTAINTY
Science of Matter and
Energy
Jesús Aguado Molina
. Many of the numbers in science are the result of measurement and averaged measurements
are therefore known only to within a degree of experimental uncertainty
. The magnitude of the uncertainty, which depends on both the skill of the experimenter
and the apparatus used, often can only be estimated
. For example, a table is 2.50 m long, it is saying that its length is close to, but not exactly,
2.50 m. The rightmost digit, the 0, is uncertain
2.51 (m)
2.49 (m)
The average: 2.50 but it is necessary ....
The experimental uncertainty is expressed by: ±
In this case the correct measurement is: 2.50 - 0.05
What is the experimental uncertainty? 3 (m)
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Calculating Experimental Uncertainty

Science of Matter and
Energy
Jesús Aguado Molina
Calculate the experimental uncertainty of the following averaged
measurements :.
2.503 ±. . .
... (m)
● 12321 ±.
.. (m)
· 3.02 ±.
.. (m)
· 456.654378 ±.
.(m)
● 100.00 ±.
(m)
● 3 士
(m)
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Calculated Experimental Uncertainty Values

Science of Matter and
Energy
Jesús Aguado Molina
Calculate the experimental uncertainty of the following averaged
measurements :.
2.503 ± ...
.. (m) ±0.005 (m)
● 12321 ±.
.(m) ±5 (m)
● 3.02 ±,
(m) ±0.05 (m)
· 456.654378 ±.
(m)
±0.000005 (m)
● 100.00 ±.
(m) ±0.05 (m)
● 3 士
(m) ±5 (m)
191 - The
International
System Of Units
2- more than
one
measurement
3-Significant
figure
5.40
m
5.42
5.45
5.40
4-experimental
uncertainty
5- To average
the numbers
6-To round it off
5.40 ± 0.05
5.42 _ 0.05
5.4175 ± 0.05
5.42 ± 0.05
5.45 ± 0.05
5.40 ± 0.05
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5.4
5.42
5.45
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Measurement Accuracy and Ease of Use

. The smaller the "measurement pattern" the more accurate the
measurement
. The bigger the "measurement pattern" the easier the measure
Science of Matter and
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The distance between Madrid and Barcelona :
by car: 620 km = 620000 m = 620000000mm
The table is 150 cm long
150 cm =1.50 m =< 0.0015 km
Which one is the most
accurate measurement?
Which one is the most
accurate measurement?
Which one is the easiest to use?
620 km = 620. 106mm
When we work with very large or very small numbers, we can
show significant figures more easily by using scientific notation.
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Importance of Measurement Standards

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Jesús Aguado Molina
Which do you think is better to design something?
· The same standard (papers A-4, pendrives, .... )
· Different standards (papers for big measurements, coins for small ones)
a standard must be defined: (SI) International System
. If we are going to report the results of a measurement to someone who wishes to
reproduce this measurement, a standard must be defined
. The wall is 2 meters tall and our unit of length is defined to be 1 meter,
we know that the height of the wall is twice our basic length unit.
(important to be "unit=1, Sistema decimal")
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The International System of Units (SI)

2. THE INTERNATIONAL SYSTEM OF UNITS
Science of Matter and
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Jesús Aguado Molina
In science, it is important to use a consistent set of units. In 1960, an international committee established a set of standards for
the scientific community called SI (for Système International). There are seven base quantities in the SI system.

Base Quantities in the SI System

3. BASE QUANTITIES IN THE SI
There are seven base quantities/magnitudes
in the SI system. They are length, mass, time,
electric current, thermodynamic temperature,
amount of substance, and luminous intensity, and
each base quantity has a base unit.
Quantities are often called magnitudes:
To measure distance, we use length magnitudes

Base Quantities and Units Table

BASE QUANTITIES (Base Units)
You always have to use the base
quantities!
Base quantities
/ dimensions
Symbol for
dimension
Base units
Symbol
length
L
meter
m
mass
M
kilogram
kg
time
T
second
s
electric current
I
ampere
A
temperature
Θ
kelvin
K
amount of
substance
N
mole
mol
luminous
intensity
J
candela
cd
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