1
MEASUREMENTS
Measurements are needed to explore relationships between quantities.
Relationships can be qualitative.
The top of campanile is higher than the library.
Illiteracy contributes to poverty.
Daddy is taller than Mary Grace.
Daddy is taller than Sarah.
Note: These qualitative relationships limit knowledge about Mary Grace and Sarah.
Relationships are more powerful when they are quantitative.
Daddy is 75 inches.
Mary Grace is 64 inches.
Sarah is 52 inches.
Note: These quantitative relationships yield much more information.
All measurements need units.
Temperature in Edmonton is 20.
20 what???
20 C (degrees Celsius) (about 68 F)
Gasoline in Edmonton is 1.20.
1.20 what???
1.20 Canadian dollars per liter (about $4.32/gal)
**Measurements without units are meaningless.**
ACCURACY AND PRECISION
Precision – how close measurement are to each other
Accuracy – how close measurements are to actual value (on average)
Precision  Accuracy
Ave.
18.5
18.6
18.6
18.3
18.5
Ave.
20.5
18.7
21.6
18.8
19.9
Ave.
19.9
20.1
20.1
20.0
20.0
True value is 20.0
precise but
not accurate
accurate but
not precise
systematic error
random error
accurate and
precise
no error
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SIGNIFICANT FIGURES
- how precise is measurement?
- uncertainty of measurement is  1 of last of significant figures
Rules for Determining Significant Figures
1. All nonzero digits are significant.
23.2 m (3 SF); 5 L
(1 SF); 3347.52 g (6 SF)
note: 23.2 m means 23.2 m  0.1 m
5 L means 5 L  1 L (maybe 4 L, maybe 6 L)
3347.52 g means 3347.52 g  0.01 g
2. Zeroes between nonzero digits are significant.
42.02 cm (4 SF); 5.002 kg (4 SF); 20104 mL (5 SF)
3. Zeroes at the beginning of first non-zero digit are never significant.
0.007 m (1 SF); 0.00204 L (3 SF); 0.123 g (3 SF)
4. Zeroes at the end of a number
a) and after decimal point are significant.
231.00 km (5 SF); 2.010 cm3 (4 SF); 80.0100 mg (6 SF)
b) and has no decimal point are not significant.
7300 nm (2 SF); 80900 GL (3 SF); 240000 kg (2 SF)
5. Integers and other exact numbers have infinite number of significant figures, i.e., there
is no uncertainty in their measurements.
One can measure 45 or 46 students
45.007 students makes no sense.
3
CALCULATING WITH SIGNIFICANT FIGURES
1. In multiplication, the number of SF of an answer is the lowest number of SF for the
measurements used
9.52 3 SF
x14
.
2 SF
13.328
13
(2 SF)
7518
.
 8.44719  8.4
8.9
55824
.
2.8514
 0.062257641  0.06226
4580
.
x66.2
5 SF
3 SF
36955488
.

3700  3.70 x 103 (3 SF)
2. For the operation of addition, we only report answer with the uncertainty of the coarsest
measurement.
281
.
5.32
 1431
.
176.52
 176.5
1.37
3.28
7.35
12.00
 12.00
1176
.
12.19
 134.8
148166
.
 148.2
1100
128
 4.7
1232.7
 1200
SCIENTIFIC NOTATION
Convenient to express very large or very small numbers.
Multiply significant figures by a power of ten.
- number of power of ten is found by counting number of places that decimal is moved
until only one digit is to the left of the decimal point
- moving decimal to the left means positive powers of ten
12.3 = 1.23 x 101
14,100. = 1.41 x 104
7.501 x 103 = 7501
- moving decimal to the right means negative powers of ten
0.0123 = 1.23 x 10-2
0.00004 = 4 x 10-5
7.8 x 10-3 = 0.0078
Note scientific notation allows unambigious way to state significant figures of number.
Consider the measurement of 300 miles to the nearest mile.
The number reported needs to reflect the uncertainty of measurement: 300  1 miles
The problem is that reporting the measurement as 300 miles implies 300  100 miles.
Ambiguity is removed by using scientific notation: 3.00 x 102 miles.
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Multiplying in scientific notation
- multiply significant portions of numbers
- add exponents of powers of ten
- make adjustment to power of ten if necessary
Example: 1.23 x 104  4.2 x 103 = (1.23 x 4.2)  10(4+3) = 5.2 x 107
Example: 3.234 x 105  5.03 x 10-2 = (3.234  5.03) x 10(5-2) = 16.3 x 103 = 1.63 x 104
Dividing in scientific notation
- divide significant portions of numbers
- subtract exponents of powers of ten
- make adjustment to power of ten if necessary
9.43  10 2
9.43
 10 2  6  112
.  10 4
6 
8.432
8.432  10
17
.  103 17
.
 10 3 7  0.77  104  7.7  105
7 
2.2  10
2.2
UNITS OF MEASUREMENT
SI Units – metric system
Quantity
SI Units
Mass
kilogram
Length
meter
Time
second
Amount
mole
Temperature Kelvin
Volume
liter
Energy
Joule
Pressure
Pascal
SI Prefixes
Prefix
teragigamegakilohectodecadecicentimillimicronanopicofemto-
Abbr.
T
G
M
k
h
dk
d
c
m

n
p
f
Abbr.
kg
m
s
mol
K
L
J
Pa
Factor of ten
1012
109
106
103
102
101
10-1
10-2
10-3
10-6
10-9
10-12
10-15
Other Units
pound, ounce
inch, foot
minute, day
dozen, gross
Cel., Fahr.
ounce, gallon
calorie
atm, mmHg
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Examples:
1 km = 103 m = 1000 m
1 dL = 10-1 L = 0.1 L
1 ns = 10-9 s = 0.000000001 s
1 mmol = 10-3 mol = 0.001 mol
NOTES ON VOLUME
Liter is an SI unit
By definition: 1 L = 1 dm3 = 1 cubic decimeter
Often liter is too large of unit, we use milliliters instead.
1 mL = 1 cm3 = 1 cc
In other words, 1 milliliter = 1 cubic centimeter
NOTES ON MASS AND WEIGHT
Mass is not the same as Weight
Mass  Weight
- mass is quantity of matter
- weight is a force (exerted on earth, moon, etc…)
Force = mass x gravity
Astronaut has same mass on Earth and Moon, but different weights.
CONVERSION OF MEASUREMENT UNITS
- often need to convert measurement from one system of units to another, i. e. 93 cm = ? in.
- need to have equality that relates cm to in
1 in = 2.54 cm
or
1 cm = 0.3937 in
- equality can be written as conversion factor
1in  2.54 cm 
1in
or
2.54 cm
2.54 cm
1in
Many conversion factors relate integers to each other, therefore they have an infinite number of
significant figures.
12 in = 1 ft  12.0000000…in = 1.0000000…ft
FACTOR-LABEL METHOD
- To convert measurement units, one arranges conversion factors to cancel units in
numerator and denominator, similar to canceling factors in fractions.
4 22 2


6 23 3
6
Example: How many inches in 93 centimeters?
Which conversion factor should we use?
1in
2.54 cm
or
2.54 cm
1in
- we want to cancel cm therefore use conversion factor with cm in denominator.
1in  2.54 cm 
1 in
 37 in (2 SF)
2.54 cm
Example: How many inches in 5.76 feet?
Which conversion factor should we use?
93 cm 
- We want to cancel ft,  use factor with ft in denominator.
5.76 ft 
12 in
 69.12 in  69.1 in (3 SF)
ft
Why not 2 SF?
Example: How many pounds is 22.7 kg given that 1 kg = 2.20 lb?
- Use factor-label method to convert with metric system.
- With factor-label method, more than one unit conversion can be done at a time.
Example: How many feet per second is 65 miles per hour?
1 mi = 5280 ft
1 hr = 60 min
1 min = 60 s
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NOTES ON TEMPERATURE UNITS
Three temperature units will be used
Fahrenheit
F
- “normal” American temperature
Celsius
C
- “normal” metric temperature
Kelvin
K
- SI absolute temperature scale
- no negative Kelvin temperature
- 0 K is the coldest one can get
Important Conversion between Kelvin and Celsius
K = C + 273
Temperatures to know
Boiling Point of Water
212 F = 100 C = 373 K
Room Temperature
77 F = 25 C = 298 K
Freezing Point of Water
32 F = 0 C = 273 K