Physical Quantities

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Physical Quantities
A physical property that can be measured. The
measurement indicates a numeric value followed by
a unit. How much does a crate of oranges
weigh? 48 Without the units, the answer
could mean anything! 48 pounds The pound is a
unit of weight, and indicates unambiguously the
magnitude of the physical measurement. What
exactly is a pound? It is a unit of mass or unit
of weight (force exerted in a gravitational
field). A pound is equal to 7000 grains.
Originally a grain was the weight of a grain seed
from the middle of an ear of barley.
source Wikipedia
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Physical Quantities
Dimension SI Unit symbol English Unit symbol
Metric Unit symbol length (distance) meter
m foot ft
meter m mass kilogram kg pound lb
gram g temperature Kelvin
K Fahrenheit F
Celsius C time second s second s
second s charge coulomb C n/a
n/a
A multitude of other physical quantities can be
defined using these basic dimensions. Ex
velocity length/time and has units of m/s
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Unit prefixes
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conversion between prefixes of the same unit type
is easy How many millimeters are in one
kilometer? 1 mm 10-3 m 1 km 103 m to make
the conversion

1
103 m
1 km


1 km
1
103 m
conversion factors
1
10-3 m
1 mm


1
1 mm
10-3 m
( )
1 mm
1 km (103 m) x
10-3 m
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conversion between prefixes of the same unit type
is easy How many millimeters are in one
kilometer? 1 mm 10-3 m 1 km 103 m to make
the conversion

1
103 m
1 km


1 km
1
103 m
conversion factors
1
10-3 m
1 mm


1
1 mm
10-3 m
( )
( )
1 mm
103 mm

1 km (103 m) x

106 mm
10-3 m
10-3
1
106 mm
1 km

new conversion factor

1 km
1
106 mm
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Measurement and Significant Figures
How much does a tennis ball weigh??
Answer 000000 g 00000 g 0000 g 100 g
54 g 54.1 g 54.07 g 54.074 g
54.0741 g 54.07413 g 54.074127 g
(etc.)
How are these measurements obtained? What do
they mean? Which measurement is useful? How
much did the measurement cost?
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Earth 5,974,210,448,207,921,111,325,700,000 g
Tennis ball 000,000 g

(Fictitious gravity net for measuring
mass) sensitive to units of 105 g
An exceedingly sensitive tool for measuring mass
of planets totally useless for measuring the
mass of a tennis ball!
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Capacity 0 600 g Resolution 0.1 g Cost
240 Capacity 0 210 g Resolution 0.01
g Cost 305 Capacity 0 120 g Resolution
0.001 g Cost 435
Capacity 0 210 g Resolution 0.0001 g Cost
5150
Capacity 0 2000 g Resolution 1 g Cost 109
Capacity 0 1.250 g Resolution 0.0000001
g Cost 15,633
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Significant Figures
Let us make a measurement, maybe the mass of an
apple. We place it onto a balance and it reads
73.5 g. The number 73.5 can be written in many
ways 00000073.500000000000 73.500000000000000000
0000000000073.5000000 00000000000000073.500 00000
0000000000073.50 etc.. All of these are exactly
equal to each other, but these numbers each
indicate different levels of precision. The
balance only yields its data with a level of
precision equal to /- 0.1 g. Therefore, in
the above examples, there are zeros that do not
have a valid physical meaning in our measurement.
Furthermore, some zeros are only place holders,
and are not significant.
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Significant Figures
Significant figures indicate the highest and
lowest resolution digits within a number that
have physical meaning. When the number is
associated with a measurement, the last digit
usually indicates an uncertainty of /- one
unit. Example 94.072 g The lowest resolution
digit is 9 The highest resolution digit is 2
The digits in between are significant The
digits 9, 4, 0, 7 are known with certainty the
digit 2 is known with a certainty of /- 1 This
number contains five significant figures
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Significant Figures
Rule 1. All non-zero digits are
significant. Rule 2. Zeros in the middle of a
number are like any other digit they are always
significant. Rule 3. Zeros at the beginning of
a number are not significant they act only to
locate the decimal point. Rule 4. Zeros at the
end of a number and after the decimal point are
significant. It is assumed that these zeros
would not be shown unless they were
significant. Rule 5. Zeros at the end of a
number and before an implied decimal point may or
may not be significant (we cannot tell). Thus,
23,000 kg may have two, three, four, or five
significant figures. Rule 6. Exact numbers
have an undefined (effectively infinite) number
of significant digits. There are exactly 166
students enrolled in CHEM 106, not 166.1 or 165.8.
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Significant Figures
Examples How many significant figures do the
following measurements have? a) 2730.78 m b)
0.0076 mL c) 3400 kg d) 3400.0 m2
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Arithmetic and Significant Figures
Lets make two measurements in order to calculate
the velocity of a sprinter distance 105
m time 11.3 s What is the sprinters
velocity, with the correct number of significant
figures? velocity distance / time v 105 m /
11.3 s The arithmetic yields a numeric value of
9.29203539823009 m/s This value indicates a
precision of 1 hundred billionth of a m/s, which
is unrealistic given that the values of distance
and time that were measured only contain three
significant figures! Therefore, the velocity
cannot have more significant figures than any of
the original measurements. It can only have
three significant figures. As a result, we round
the arithmetic value v 9.29 m/s
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Arithmetic and Significant Figures
This time we are measuring the mass of aspirin
synthesized in a lab Group 1 synthesized 6.3108
g aspirin Group 2 synthesized 3.2 g aspirin Group
3 synthesized 5.991 g aspirin Group 4 synthesized
0.77 g aspirin How much total aspirin was
synthesized? Report with the correct significant
figures. Notice that group 1 measured their
yield with high precision, while group 2 measured
the mass of aspirin with very low precision.
The data can only be as precise as the least
precise data 6.3108 g 3.2??? g 5.991? g
0.77?? g 16.2718 g The value
is only precise to the tenths decimal place, and
must be rounded 16.2718 g ? 16.3 g
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Arithmetic and Significant Figures
Rule 1. In carrying out a multiplication or
division, the answer cannot have more significant
figures than either of the original
numbers. Rule 2. In carrying out an addition or
subtraction, the answer cannot have more digits
after the decimal point than either of the
original numbers. If a complex operation is
being carried out, first complete the arithmetic,
then round the value to the keep the number of
significant digits determined by the least
precise number. Rounding Rules Rule 1. If the
first digit you remove is 4 or less, drop it and
all following digits. Rule 2. If the first
digit you remove is 5 or greater, round the
number up by adding a 1 to the digit to the left
of the one you drop.
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Arithmetic and Significant Figures

• 4.87 mL 46.0 mL b) 3.4 x 0.023 g
• 19.333 m 7.4 m d) 55 mg 4.671 mg 0.894 mg
• e) 62,911 / 611

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Arithmetic and Significant Figures

• 4.87 mL 46.0 mL b) 3.4 x 0.023 g
• 19.333 m 7.4 m d) 55 mg 4.671 mg 0.894 mg
• e) 62,911 / 611

50.9 mL
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Arithmetic and Significant Figures

• 4.87 mL 46.0 mL b) 3.4 x 0.023 g
• 19.333 m 7.4 m d) 55 mg 4.671 mg 0.894 mg
• e) 62,911 / 611

50.9 mL
0.078 g
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Arithmetic and Significant Figures

• 4.87 mL 46.0 mL b) 3.4 x 0.023 g
• 19.333 m 7.4 m d) 55 mg 4.671 mg 0.894 mg
• e) 62,911 / 611

50.9 mL
0.078 g
11.9 m
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Arithmetic and Significant Figures

• 4.87 mL 46.0 mL b) 3.4 x 0.023 g
• 19.333 m 7.4 m d) 55 mg 4.671 mg 0.894 mg
• e) 62,911 / 611

50.9 mL
0.078 g
11.9 m
51 mg
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Arithmetic and Significant Figures

• 4.87 mL 46.0 mL b) 3.4 x 0.023 g
• 19.333 m 7.4 m d) 55 mg 4.671 mg 0.894 mg
• e) 62,911 / 611

50.9 mL
0.078 g
11.9 m
51 mg
103
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Scientific Notation
A number expressed as the product of a number
between 1 and 10, times 10 raised to a
power. Powers of ten 10-6 0.000001 10-5
0.00001 10-4 0.0001 10-3 0.001 10-2
0.01 10-1 0.1 100 1 101 10 102
100 103 1,000 104 10,000 105
100,000 106 1,000,000 etc

• Examples
• 3.97 x 102
• 0.0000013070 1.3070 x 10-6
• 434,700,000,000,000 4.347 x 1014

Note The number of digits preceding the power
of ten is exactly the number of significant
digits in the value. For this reason, scientific
notation is absolutely unambiguous.
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