V. Limits of Measurement 1. Accuracy and Precision.
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Transcript of V. Limits of Measurement 1. Accuracy and Precision.
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V. Limits of Measurement
1. Accuracy and Precision
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• Accuracy - a measure of how close a measurement is to the true value of the quantity being measured.
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Example: Accuracy• Who is more accurate when
measuring a book that has a true length of 17.0cm?
Susan: 17.0cm, 16.0cm, 18.0cm, 15.0cm
Amy: 15.5cm, 15.0cm, 15.2cm, 15.3cm
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• Precision – a measure of how close a series of measurements are to one another. A measure of how exact a measurement is.
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Example: Precision
Who is more precise when measuring the same 17.0cm book?
Susan: 17.0cm, 16.0cm, 18.0cm, 15.0cm
Amy: 15.5cm, 15.0cm, 15.2cm, 15.3cm
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Example: Evaluate whether the following are precise, accurate or both.
Accurate
Not Precise
Not Accurate
Precise
Accurate
Precise
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2. Significant Figures
• The significant figures in a measurement include all of the digits that are known, plus one last digit that is estimated.
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Exact NumbersAn exact number is obtained when you count
objects or use a defined relationship.
- Counting objects are always exact2 soccer balls4 pizzas
- Exact relationships, predefined values, not measured1 foot = 12 inches1 meter = 100 cm
For instance is 1 foot = 12.000000000001 inches? No 1 ft is EXACTLY 12 inches.
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9
Solution
A. Exact numbers are obtained by
2. counting
3. definition
B. Measured numbers are obtained by
1. using a measuring tool
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2.1 Uncertainty in Measurement 40.16 cm
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2.1. Uncertainty in Measurement
• A measurement always has some degree of uncertainty.
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2.1 Uncertainty in Measurement
• Different people estimate differently.
• Record all certain numbers and one estimated number.
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2.1 Measurement and Significant Figures• Every experimental
measurement has a degree of uncertainty.
• The volume, V, at right is certain in the 10’s place, 10mL<V<20mL
• The 1’s digit is also certain, 17mL<V<18mL
• A best guess is needed for the tenths place.
Chapter Two 13
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14
What is the Length?
1 2 3 4 cm
• We can see the markings between 1.6-1.7cm• We can’t see the markings between the .6-.7• We must guess between .6 & .7• We record 1.67 cm as our measurement• The last digit an 7 was our guess...stop there
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Learning Check
What is the length of the wooden stick?1) 4.5 cm 2) 4.54 cm 3) 4.547 cm
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16
8.00 cm or 3 (2.2/8)?
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Below are two measurements of the mass of the same object. The same quantity is being described at two different levels of precision or certainty.
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2.2 Significant Figures • Significant figures are the meaningful figures in our
measurements and they allow us to generate meaningful conclusions
• Numbers recorded in a measurement are significant. – All the certain numbers plus first estimated number
e.g. 2.85 cm • We need to be able to combine data and still produce
meaningful information• There are rules about combining data that depend on
how many significant figures we start with………
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2.3 Rules for Counting Significant Figures
1. Nonzero integers always count as significant figures.
1457 has 4 significant figures
23.3 has 3 significant figures
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Rules for Counting Significant Figures
2. Zeros
a. Leading zeros - never count0.0025 2 significant figures
b. Captive zeros - always count 1.008 4 significant figures
c. Trailing zeros - count only if the number is written with a decimal point 100 1 significant figure 100. 3 significant figures 120.0 4 significant figures
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Rules for Counting Significant Figures
3. Exact numbers - unlimited significant figures
• Not obtained by measurement Determined by counting:
3 apples Determined by definition:
1 in. = 2.54 cm
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Practice Rule #1 Zeros
45.8736
.000239
.00023900
48000.
48000
3.982106
1.00040
6
3
5
5
2
4
6
• All digits count
• Leading 0’s don’t
• Trailing 0’s do
• 0’s count in decimal form
• 0’s don’t count w/o decimal
• All digits count
• 0’s between digits count as well as trailing in decimal form
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How Many Significant Figures?
1422
65,321
1.004 x 105
200
435.662
50.041
102
102.0
1.02
0.00102
0.10200
1.02 x 104
1.020 x 104
60 minutes in an hour
500 laps in the race
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• One convention about trailing zero
A bar placed over ( or under) the last significant figure; any trailing zeros following this are insignificantExample:500 has 1 s.f. 5500 has 3 s.f. 500. has 3 s.f.
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- Round off 52.394 to 1,2,3,4 significant figures
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2.4 Scientific notationWrite number in form:
Standard decimal notation Scientific notation
2 2×100
300 3×102
4,321.768 4.321768×103
−53,000 −5.300×104
6,720,00,000 6.72000×109
0.2 2×10−1
0.000 000 007 51 7.51×10−9
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Chapter Two 28
Two examples of converting standard notation to scientific notation are shown below.
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Chapter Two 29
Two examples of converting scientific notation back to standard notation are shown below.
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• Scientific notation is helpful for indicating how many significant figures are present in a number that has zeros at the end but to the left of a decimal point.
• The distance from the Earth to the Sun is 150,000,000 km. Written in standard notation this number could have anywhere from 2 to 9 significant figures.
• Scientific notation can indicate how many digits are significant. Writing 150,000,000 as 1.5 x 108 indicates 2 and writing it as 1.500 x 108 indicates 4.
• Scientific notation can make doing arithmetic easier.
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How many sig figs?
1 10302.00
100.00 970
0.001 0.00250
10302 1.0302x104
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2.5 Rules for Multiplication and
Division
• I measure the sides of a rectangle, using a ruler to the nearest 0.1cm, as 4.5cm and 9.3cm
• What does a calculator tell me the area is?• What is the range of areas that my measurements might
indicate (consider the range of lengths that my original measurements might cover)?
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Rules for Multiplication and Division
• The number of significant figures in the result is the same as in the measurement with the smallest number of significant figures.
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2.6 Rules for Addition and Subtraction
• The number of significant figures in the result is the same as in the measurement with the smallest number of decimal places.
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2.7 Rules for Combined Units
• Multiplication / Division– When you Multiply or Divide measurements you must carry out
the same operation with the units as you do with the numbers
50 cm x 150 cm = 7500 cm2
20 m / 5 s = 4 m/s or 4 ms-1
16m / 4m = 4
• Addition / Subtraction– When you Add or Subtract measurements they must be in the
same units and the units remain the same
50 cm + 150 cm = 200 cm
20 m/s – 15 m/s = 5 m/s
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32.27 1.54 = 49.6958
3.68 .07925 = 46.4353312
1.750 .0342000 = 0.05985
3.2650106 4.858 = 1.586137 107
6.0221023 1.66110-24 = 1.000000
49.7
46.4
.05985
1.586 107
1.000
Calculate the following:
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.56 + .153 = .713
82000 + 5.32 = 82005.32
10.0 - 9.8742 = .12580
10 – 9.8742 = .12580
.71
82000
.1
0
Look for the last important digit
Calculate the following:
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Mixed Order of Operation
8.52 + 4.1586 18.73 + 153.2 =
(8.52 + 4.1586) (18.73 + 153.2) =
239.6
2180.
= 8.52 + 77.89 + 153.2 = 239.61 =
= 12.68 171.9 = 2179.692 =
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Calculate the following. Give your answer to the correct number of significant figures and use the correct units
11.7 km x 15.02 km =
12 mm x 34 mm x 9.445 mm =
14.05 m / 7 s =
108 kg / 550 m3 =
23.2 L + 14 L =
55.3 s + 11.799 s =
16.37 cm – 4.2 cm =
350.55 km – 234.348 km =
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practice1.Calculate Volume of sphere with ,55.0 mr
33
33
70.06969.0
)55.0(3
4
3
4
mm
rV
2. Perimeter of the big circle
mm
mrP
5.34557.3
)55.0(22
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Try the following
7.895 + 3.4=
(8.71 x 0.0301)/0.056 = =
A= =
13m
4.91m2
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IV Dimension Analysis – some simple rules
1.In : The product unit is the product of the individual unit of each of those variables. (Ditto for ratios.)
2. : Different terms can only added together in a sum if each term in the sum has the same unit type. (Ditto for subtraction.)
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Example 1
- impossible: 40m + 20m/s or 12.5 s - 20m2
- Can Do: 50.0m + 20.55m=70.6mand 40m/s +11m/s =51m/s
- Can Do, but need to convert into same unit:
40m + 11cm = 40m + 11cm = 40.11m
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Example 2
The above expression yields:
1.5 m 3.0 kg ?
a)4.5 m kgb)4.5 g kmc)A or Bd)Impossible to evaluate (dimensionally invalid)