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Measurements
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Measurements
A very concrete methods of dealing with
the description and understanding of nature
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Measurements
A very concrete method of dealing with the
description and understanding of nature
Measurements give credibility to the
interpretation of;
•Theories
•Laws
•Principles
•Hypothesis
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Measurements
A very concrete way of dealing with the description
and understanding of nature
Measurements give credibility to the interpretation
of:
•Theories
•Laws
•Principles
•Hypothesis
This credibility is directly related to the accuracy of
the measurements
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Estimation (uncertainty)
• All measurements have some degree of
estimation.
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It would be difficult to measure with
an certainty beyond a millimeter
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The ruler has a limited amount of certainty.
Thinner lines could increase the amount of
certainty .
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All measuring devices have a certain
amount of uncertainty
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The uncertainty of a measurement is
determined by the
a. precision of the measurement
and
b. accuracy of the measured value
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Precision verses Accuracy
• Precision in its strictest sense refers to the exactness
to which the measuring instrument has been
manufactured.
• If the same measurement is repeated multiple times with the
same instrument, will the measurement be the same each time?
(repeatability)
• smaller units would make the instruments more precise (exact_
• Accuracy is how close the measurement is to the
true value
• Influenced by :
1. person making the measurements
2. precision of the instrument
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Measurement of Uncertainty
• Estimated uncertainty is written with a ± sign;
for example:
• Percent uncertainty is the ratio of the
uncertainty to the measured value, multiplied
by 100:
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Significant Figures
• The number of significant figures is the number of
reliably known digits in a number. It is usually possible
to tell the number of significant figures by the way the
number is written:
• 23.21 cm has 4 significant figures
• 0.062 cm has 2 significant figures (the initial zeroes
don’t count)
• 80 km is ambiguous – it could have 1 or 2 significant
figures. If it has 3, it should be written 80.0 km. If it has
2 it should be written in scientific notation 8.0 x 101 km
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Significant Figures
• When multiplying or dividing numbers, the
result has as many significant figures as the
number used in the calculation with the
fewest significant figures.
• Example: 11.3 cm x 6.8 cm = 76.84 cm2
• 11.3 cm / 77cm = 0.1467
77 cm2
0.15
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Adding and Subtracting Significant Figures
• When adding or subtracting quantities, leave the same number of
decimal places (rounded) in the answer as there are in the quantity
with the least number of decimal places.
• Examples
1)
2)
23.1
157
0.546
-5.5
1.45
25.096
151.5
152
25.1
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1-4 Measurement and Uncertainty; Significant
Figures
Calculators will not give you the right
number of significant figures; they usually
give too many but sometimes give too few
(especially if there are trailing zeroes after a
decimal point).
The top calculator shows the result of 2.0 /
3.0.
The bottom calculator shows the result of
2.5 x 3.2.
Scientific Notation
• Scientific notation is the expression of a
number in the “power of 10”
36,900
3.69 x 104
 allows a number to expressed in significant digits in the
coefficient
 eliminates the need to write multiple zeros
Know how to add, subtract, multiply, and divide numbers
expressed in scientific notation
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Order of Magnitude Estimation
• Method of making an approximate value for a
measurement
 the number is rounded to one (1) significant figure
 and its power of 10
3,675 m ----- 4 x 103 m
5,000 m ----- 5 x 103 m
added together 9 x 103 m
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Order of Magnitude Estimation
• Reasons for
1) Rapid estimation
2) Accurate calculation is not worth the time
3) Quick check of an accurate calculation to check
for large errors
a)
Check the accuracy of the exponent
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Order of Magnitude Example
• Find the volume (V) of a lake
o Lake has
o Average depth of 65 m
o Surface area of 52,500 m2
Volume = area x depth = (7 x 101m) x (5 x 104m2) = 3.5 x 106m3
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