Transcript Measurement - DocLockert.com

Measurement
How far, how much, how many?
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PROBLEM SOLVING
STEP 1: Understand the Problem
STEP 2: Devise a Plan
STEP 3: Carry Out the Plan
STEP 4: Look Back
2-2
Step 1. Understand the Problem
2-3
Step 2. Devise a Plan
2-4
Step 3. Carry Out the Plan
2-5
Step 4. Look Back
2-6
A Measurement
A Number
A Quantity
An implied precision
15
1000000000
0.00056
A Unit
A meaning
pound
Liter
Gram
Hour
degree Celsius
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Implied versus Exact
An implied or measured quantity has
significant figures associated with the
measurement
1 mile
=
1603 meters
Exact - defined
measured - 4 sig figs
An exact number is not measured, it is
defined or counted; therefore, it does not
have significant figures or it has an
unlimited number of significant figures.
1 kg = 1000 grams
1.0000000 kg = 1000.0000000 grams
2-8
Types of measurement
Quantitative- use numbers to describe
measurement– test equipment, counts, etc.
Qualitative- use descriptions without numbers
to descript measurement- use five senses
to describe
4 feet
extra large
Hot
100ºF
2-9
Scientists Prefer
Quantitative- easy check
Easy to agree upon, no personal bias
The measuring instrument limits how good
the measurement is
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Uncertainty in Measurement
All measurements contain some uncertainty.
• We make errors
• Tools have limits
Uncertainty is measured with
Accuracy
How close to the true value
Precision
How close to each other
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Accuracy
Measures how close the experimental
measurement is to the accepted, true or
book value for that measurement
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Precision
Is the description of how good that
measurement is, how many significant
figures it has and how repeatable the
measurement is.
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Differences
Accuracy can be true of an individual
measurement or the average of several
Precision requires several measurements
before anything can be said about it
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Let’s use a golf analogy
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Accurate? No
Precise? Yes
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Accurate? Yes
Precise? Yes
2 - 17
Precise?
No
Accurate? Maybe?
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Accurate? Yes
Precise? We can’t say!
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Accuracy vs. Precision
Synonyms for Accuracy…
2 - 20
Accuracy vs. Precision
Synonyms for precision…
Repeatable
2 - 21
Significant figures
The number of significant digits is
independent of the decimal point.
These numbers
All have three
significant figures!
25500
2550
255
25.5
2.55
0.255
0.0255
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Significant Figures
Imply how the quantity is measured and to
what precision.
Are always dependant upon the equipment or
scale used when making the measurement
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SCALES
0
1
0.2, 0.3, 0.4?
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SCALES
0
0
1
0.2, 0.3, 0.4?
1
0.26, 0.27, or 0.28?
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SCALES
0
0
0
1
0.2, 0.3, 0.4?
0.26, 0.27, or 0.28?
1
1
0.262, 0.263, 0.264?
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Significant figures
Method used to express accuracy and
precision.
You can’t report numbers better than the
method used to measure them.
67.2 units = three significant figures
Certain
Digits
Uncertain
Digit
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Significant figures:
Rules for zeros
Leading zeros are not significant.
0.00421 - three significant figures
Leading zero
4.21 x 10-3
Notice zeros are
not written in
scientific notation
Captive zeros are significant.
4012 - four significant figures
Captive zero
4.012 x 103
Notice zero is
written in
scientific notation
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Significant figures:
Rules for zeros
Trailing zeros before the decimal are not
significant.
4210000 - three significant figures
Trailing zero
Trailing zeros after the decimal are significant.
114.20 - five significant figures
Trailing zero
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How Many Significant figures?
123 grams
1005 mg
250 kg
250.0 kg
2.50 x 102 kg
0.0005 L
0.00050 L
5.00 x 10-4 L
3 significant figures
4 significant figures
2 significant figures
4 significant figures
3 significant figures
1 significant figures
2 significant figures
3 significant figures
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Significant figures
Zeros are what will give you a headache!
They are used/misused all of the time.
Example
The press might report that the federal
deficit is three trillion dollars. What did they
mean?
$3 x 1012
or
$3,000,000,000,000.00
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Significant figures:
Rules for zeros
Scientific notation - can be used to clearly
express significant figures.
A properly written number in scientific
notation always has the the proper number
of significant figures.
0.003210
=
3.210 x 10-3
Four Significant
Figures
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A comparison of masses
Compare the mass of a block of wood
that was taken on 4 different balances.
Mass of a block of wood
1
1.35
grams
2
1.653
grams
3
1.40
grams
4
1.5115
grams
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Average mass calculation
Mass of a
1
2
3
4
block of wood
1.35
grams
1.653
grams
1.40
grams
1.5115
grams
Average
1.48
grams
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Experimental Error
The accuracy is measured by comparing the
result of your experiment with a true or
book value.
The block of wood is known to weigh exactly
1.5982 grams.
The average value you calculated is 1.48 g.
Is this an accurate measurement?
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Percent Error
Indicates accuracy of a measurement
experim ent
al  literature
% error
 100
literature
your value
accepted value
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Percent Error
A student determines the density of a substance to be
1.40 g/mL. Find the % error if the accepted value of
the density is 1.36 g/mL.
% error
1.40 g/m L 1.36 g/m L
1.36 g/m L
 100
% error = 2.94 %
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Scientific Notation
Is used to write very, very small numbers or
very large numbers
Is used to imply a specific number of
significant figures
Uses exponentials or powers of 10
large positive exponentials imply numbers
much greater than 1
negative exponentials imply numbers
smaller than 1
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Scientific notation
• Method to express really big or small
numbers.
Format is
Mantissa x Base Power
Decimal part of
original number
Decimals
you moved
We just move the decimal point around.
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Scientific notation
If a number is larger than 1
• The original decimal point is moved X
places to the left.
• The resulting number is multiplied by 10X.
• The exponent is the number of places you
moved the decimal point.
• The exponent is a positive value.
1 2 3 0 0 0 0 0 0 = 1.23 x 108
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Scientific notation
If a number is smaller than 1
• The original decimal point is moved X
places to the right.
• The resulting number is multiplied by 10-X.
• The exponent is the number of places you
moved the decimal point.
• The exponent is a negative value.
0. 0 0 0 0 0 0 1 2 3 = 1.23 x 10-7
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Scientific notation
Most scientific calculators use scientific
notation when the numbers get very large or
small.
How scientific notation is
displayed can vary.
x10n
It may use
or may be displayed
using an E or e.
1.44939 E-2
cos tan
CE
ln
7
8
9
/
log
4
5
6
x
1/x
1
2
3
-
x2
EE
0
.
+
They usually have an Exp or EE
button. This is to enter in the exponent.
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Examples
378 000
3.78 x 10 5
8931.5
8.9315 x 10 3
0.000 593
5.93 x 10 - 4
0.000 000 40
4.0 x 10 - 7
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Expand
1 x 104
10,000
5.60 x 1011
560,000,000,000
1 x 10-5
0.000 01
5.02 x 10-8
0.000 000 0502
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Significant figures and calculations
Addition and subtraction
Report your answer with the same number
of digits to the right of the decimal point as
the number having the fewest to start with.
123.45987 g
+ 234.11
g
357.56987 g
357.57
g
805.4
g
- 721.67912 g
83.72088 g
83.7
g
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Significant figures and calculations
Multiplication and division.
Report your answer with the same number
of digits as the quantity have the smallest
number of significant figures.
Example. Density of a rectangular solid.
251.2 kg / [ (18.5 m) (2.351 m) (2.1m) ]
= 2.750274 kg/m3
= 2.8 kg / m3
(2.1 m - only has two significant figures)
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Significant figures
and calculations
An answer can’t have more significant figures
than the quantities used to produce it.
Example
How fast did the man run
if he went 11 km in
23.2 minutes?
speed
= 11 km / 23.2 min
= 0.47 km / min
0.474137931
cos tan
CE
ln
7
8
9
/
log
4
5
6
x
1/x
1
2
3
-
x2
EE
0
.
+
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How many significant figures?
What is the Volume of this box?
2.1 m
2.351 m
18.5 m
Volume = length x width x height
= (18.5 m x 2.351 m x 2.1 m)
= 91.33635 m3
= 91 m3
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Scientific Notation (Multiplication)
4
5
(3.0 x 10 ) x (3.0 x 10 ) =
9
9.0 x 10
5
4
(6.0 x 10 ) x (2.0 x 10 ) =
9
12 x 10
9
But 12 x 10 =
10
1.2 x 10
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Scientific Notation (Division)
6
2.0 x 10
=
4
1.0 x 10
1.0 x 10
2
2.0 x 10
4
6
=
0.50 x 10
-2
2.0 x 10
-3
= 5.0 x 10
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Scientific Notation Add & Subtract
4
4
(2.3 x 10 ) + (4.1 x 10 ) =
6.4 x 10
4
*Exponent must be the same!*
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Scientific Notation (+ and -)
5
3
3
3
3
5
(1.400 x 10 ) + (3.200 x 10 ) =
(140.0 x 10 ) + (3.200 x 10 ) =
143.2 x 10 = 1.432 x 10
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Rounding off numbers
After calculations, the last thing you do is
round the number to correct number of
significant figures.
If the first insignificant digit is 5 or more,
- you round up
If the first insignificant digit is 4 or less,
- you round down.
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Rounding off
If a set of calculations gave you the following
numbers and you knew each was supposed to
have four significant figures then -
2.5795035 becomes 2.580
1st insignificant digit
34.204221 becomes 34.20
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Measurements
Many different systems for measuring the
world around us have developed over the
years.
People in the U.S. rely
on the English System.
Scientists make use of
SI units so that we all
are speaking the same
measurement language.
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Units are important
45
has little meaning, just a number
45 g
has some meaning - mass
45 g /mL
more meaning - density
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Units
Metric Units
prefix
unit.
One base unit for each type
of measurement. Use a
to change the size of
Some common base units.
Type
Name
Mass
gram
Length
meter
Volume
liter
Time
second
Temperature
Kelvin
Energy
joule
Symbol
g
m
L
s
K
J
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Metric prefixes
Changing the prefix alters the size of a unit.
Powers of Ten http://micro.magnet.fsu.edu/primer/java/scienceopticsu/powersof10/index.html
Prefix
giga
mega
kilo
hecto
deca
base
deci
centi
milli
micro
nano
Symbol
G
M
k
h
da
d
c
m
 or mc
n
Factor
109
106
103
102
101
100
10-1
10-2
10-3
10-6
10-9
1 000 000 000
1 000 000
1 000
100
10
1
0.1
0.01
0.001
0.000 001
0.000 000 0012 - 62
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Mass - the quantity of
matter in an object.
Weight - the effect of
gravity on an object.
Measuring mass
Since the Earth’s gravity is
relatively constant, we
can interconvert between
weight and mass.
The SI unit of mass is the
kilogram (kg). However,
in the lab, the gram (g) is
more commonly used.
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Temperature
Units of
measurement
Fahrenheit,
Celsius,
Kelvin
Method of
measurement
2 - 65
Derived Units
Quantity
Area
Volume
density
speed
acceleration
Force
Pressure
Energy
Definition
Derived Unit
length x length
m2
length x length x length m3
mass per unit volume
kg/m3
distance per unit time m/s
speed per unit time
m/s2
mass x acceleration
kg m/s2
N
force per unit areakg/m s2
Pa
force x distance
kg m2 / s2
J
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Measuring volume
Volume - the amount of space that an object
occupies.
• The base metric unit is the liter (L).
• The common unit used in the lab is the milliliter
(mL).
• One milliliter is exactly equal to one cm3 & cc.
• The derived SI unit for volume is the m3 which
is too large for convenient use.
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Density
Density is an intensive property of a substance
based on two extensive properties.
Density =
Mass
Volume
Common units are g / cm3 or g / mL.
Air
Water
Gold
g / cm3
0.0013
1.0
19.3
cm3 = mL
g / cm3
Bone
1.7 - 2.0
Urine
1.01 - 1.03
Gasoline 0.66 - 0.69
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Example.
Density calculation
What is the density of 5.00 mL of a fluid if it
has a mass of 5.23 grams?
d = mass / volume
d = 5.23 g / 5.00 mL
d = 1.05 g / mL
What would be the mass of 1.00 liters of this
sample?
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Example.
Density calculation
What would be the mass of 1.00 liters of the
fluid sample?
The density was 1.05 g/mL.
density = mass / volume
so
mass
= volume x density
mass
= 1.00 L x 1000 ml x 1.05 g
L
mL
= 1.05 x 103 g
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