8/15/2013
SPH4C
COLLEGE PHYSICS
REVIEW: MATH SKILLS
L Scientific Notation
(P.547)
Scientific Notation
In science we frequently encounter numbers which are difficult to write in
the traditional way - velocity of light, mass of an electron, distance to the
nearest star. Scientific notation, or standard notation, is a technique,
using powers of ten, for concisely writing unusually large or small numbers.
Expression
Common decimal notation
Scientific notation
124.5 million
kilometres
124 500 000 km
1.245 x 108 km
154 thousand
picometres
154 000 pm
1.54 x 105 pm
602 sextillion
molecules
602 000 000 000 000 000
000 000 molecules
6.02 x 1023
molecules
August 15, 2013
4CR - Scientific Notation
1
Scientific Notation
SCIENTIFIC NOTATION
uses powers of ten to write large/small numbers
Expression
Common decimal notation
Scientific notation
124.5 million
kilometres
124 500 000 km
1.245 x 108 km
154 thousand
picometres
154 000 pm
1.54 x 105 pm
602 sextillion
molecules
602 000 000 000 000 000
000 000 molecules
6.02 x 1023
molecules
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Scientific Notation
In scientific notation, the number is expressed by:
1.
writing the correct number of significant digits with one non-zero digit
to the left of the decimal point, and then
2.
multiplying the number by the appropriate power (+ or -) of ten (10).
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4CR - Scientific Notation
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Scientific Notation
For example,
=
=
2 394
2.394 x 1000
2.394 x 10 3
=
=
0.067
6.7 x 0.01
6.7 x 10 -2
NOTE!
Scientific notation also enables us to show the correct number of significant
digits. As such, it may be necessary to use scientific notation in order to
follow the rules for certainty (discussed later).
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4CR - Scientific Notation
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Scientific Notation
PRACTICE
1. Express each of the following in scientific notation.
6.807 x 103
(a) 6 807
5.3 x 10-5
(b) 0.000 053
(c) 39 879 280 000
3.987928 x 1010
8.13 x 10-7
(d) 0.000 000 813
(e) 0.070 40
7.040 x 10-2
(f) 400 000 000 000
4 x 1011
(g) 0.80
8.0 x 10-1
(h) 68
6.8 x 101
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Scientific Notation
PRACTICE
2. Express each of the following in common notation.
70
(a) 7 × 101
5 200
(b) 5.2 × 103
(c) 8.3 × 109
8 300 000 000
0.101
(d) 10.1 × 10-2
(e) 6.386 8 × 103
6 386.8
(f) 4.086 × 10-3
0.004 086
(g) 6.3 × 102
630
(h) 35.0 × 10-3
0.035 0
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4CR - Scientific Notation
6
Scientific Notation With Calculators
On many calculators, scientific notation in entered using a special key,
labelled EXP or EE. This key includes “x 10” from the scientific notation;
you need to enter only the exponent. For example, to enter
7.5 x 10 4
press
7.5 EXP 4
3.6 x 10 -3
press
3.6 EXP +/- 3
NOTE!
Depending on the type of calculator you have, the “+/-” signs may need to
be entered after the relevant number.
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4CR - Scientific Notation
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3
8/15/2013
SPH4C
COLLEGE PHYSICS
REVIEW: MATH SKILLS
L International System of Units (SI)
(P.572)
SI
Over hundreds of years, physicists (and other
scientists) have developed traditional ways (or
rules) of expressing their measurements. If we
can’t trust the measurements, we can put no
faith in reports of scientific research. As such,
the International System of Units (SI) is used
for scientific work throughout the world –
everyone accepts and uses the same rules, and
understands that there are limitations to the
rules.
August 15, 2013
4CR - SI
1
SI
SI RULES
•
In the SI system all physical quantities can
be expressed as some combination of
fundamental units, called base units. (i.e.,
mol, m, kg, EC, s, ...). For example:
1 N = 1 kg@m/s 2
1 J = 1 kg@m
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2/s 2
7 unit for force
7 unit for energy
4CR - SI
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SI
SI RULES
•
The SI convention includes both quantity
and unit symbols. Note that these are
symbols (e.g., 60 km/h) and are not
abbreviations (e.g., 40 mi./hr.).
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4CR - SI
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SI
SI RULES
•
When converting units the method most
commonly used is multiplying by conversion
factors (equalities), which are memorized or
referenced (e.g., 1 m = 100 cm, 1 h = 60
min = 3600 s).
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4CR - SI
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SI
SI RULES
•
It is also important to pay close attention to
the units, which are converted by
multiplying by a conversion factor (e.g., 1
m/s = 3.6 km/h).
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SI
USEFUL CONVERSION FACTORS!
H
G
H
M
1000
1000
)
H
k
)
1000
)
H
H
m/s
3.6
H
base 100 c
u
)
n
i
t
H
km/h
yr
)
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d
24
H
:
1000
)
1000
)
H
365
)
H
m
10
H
hr
)
60
0
)
H
min
60
)
sec
)
4CR - SI
6
SI
PRACTICE
1. Use the chart to convert each of the
following measurements to their
base unit.
5.7 x 109 W
(a) 5.7 GW
72 x 10-2 m
(b) 72 cm
6 x 10-6 C
(c) 6 µC
0.50 x 106 J
(d) 0.50 MJ
6.8 x 10-3 L
(e) 6.8 mL
548 x 10-9 m
(f) 548 ηm
0.75 x 103 g
(g) 0.75 kg
Power
Prefix
109
giga
Symbol
G
106
mega
M
103
kilo
k
100
-----
-----
10-2
centi
c
10-3
milli
m
10-6
micro
:
10-9
nano
0
NOTE!
This is only a partial list - refer to P.661 for a complete list.
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SI
PRACTICE
2. An athlete completed a 5-km
race in 19.5 min. Convert this
time into hours.
19.5 min x
=
0.325 hours
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3. A train is travelling at 95 km/h.
Convert 95 km/h into metres
per second (m/s).
1 hour
95 km/h x
60 min
=
4CR - SI
1000 m
1 km
x
1 hour
3600 s
26.4 m/s
8
3
8/15/2013
SPH4C
COLLEGE PHYSICS
REVIEW: MATH SKILLS
L Uncertainty in Measurements
(P. 546)
Uncertainty in Measurements
There are two types of quantities used in
science: exact values and measurements. Exact
values include defined quantities (1 m = 100
cm) and counted values (5 beakers or 10 trials).
Measurements, however, are not exact
because there is always some uncertainty or
error associated with every measurement. As
such, there is an international agreement about
the correct way to record measurements.
August 15, 2013
4CR - Uncertainty in Measurements
1
Significant Digits
The certainty of any measurement is communicated by the number of
significant digits in the measurement. In a measured or calculated value,
significant digits are the digits that are known reliably, or for certain,
and include the last digit that is estimated or uncertain. As such, there are
a set of rules that can be used to determine whether or not a digit is
significant (refer to P.650 of your text).
SIGNIFICANT DIGITS
digits that are certain plus one estimated digit
indicates the certainty of a measurement
rules L P.650
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Significant Digits
WHEN DIGITS ARE SIGNIFICANT ✔
1. All non-zero digits (i.e., 1-9) are significant.
For example:
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259.69
61.2
has five significant digits
has three significant digits
4CR - Uncertainty in Measurements
3
Significant Digits
WHEN DIGITS ARE SIGNIFICANT ✔
2. Any zeros between two non-zero digits are significant.
For example:
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606
6006
has three significant digits
has four significant digits
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4
Significant Digits
WHEN DIGITS ARE SIGNIFICANT ✔
3. Any zeros to the right of both the decimal point and a non-zero digit
are significant.
For example:
August 15, 2013
7.100
7.10
has four significant digits
has three significant digits
4CR - Uncertainty in Measurements
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Significant Digits
WHEN DIGITS ARE SIGNIFICANT ✔
4. All digits (zero or non-zero) used in scientific notation are significant.
For example:
August 15, 2013
3.4 x 10 3
3.400 x 10 3
has two significant digits
has four significant digits
4CR - Uncertainty in Measurements
6
Significant Digits
WHEN DIGITS ARE SIGNIFICANT ✔
5. All counted and defined values have an infinite number of significant
digits.
For example:
August 15, 2013
16 students
B = 3.1415...
has 4 significant digits
has 4 significant digits
4CR - Uncertainty in Measurements
7
Significant Digits
WHEN DIGITS ARE NOT SIGNIFICANT ✘
1. If a decimal point is present, zeros to the left of other digits (i.e.,
leading zeros) are not significant – they are placeholders.
For example:
August 15, 2013
0.22
0.000 22
has two significant digits
has two significant digits
4CR - Uncertainty in Measurements
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Significant Digits
WHEN DIGITS ARE NOT SIGNIFICANT ✘
2. If a decimal point is not present, zeros to the right of the last non-zero
digit (i.e., trailing zeros) are not significant – they are placeholders.
For example:
98 000 000
25 000
has two significant digits
has two significant digits
NOTE!
In most cases, the values you will be working with in this course will have
two or three significant digits.
August 15, 2013
4CR - Uncertainty in Measurements
9
Significant Digits
PRACTICE
1. How many significant digits are there in each of the following
measured quantities?
(a) 353 g
3
(b) 9.663 L
4
(c) 76 600 000 g
3
(d) 30.405 ml
5
(e) 0.3 MW
1
(f) 0.000 067 s
2
(g) 10.00 m
4
(h) 47.2 m
3
(i) 2.7 x 105 s
2
(j) 3.400 x 10-2 m
4
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4CR - Uncertainty in Measurements
10
Significant Digits
PRACTICE
2. Express the following measured quantities in scientific notation with
the correct number of significant digits.
(a) 865.7 cm
(4)
8.657 x 102 cm
(b) 35 000 s
(2)
3.5 x 104 s
(c) 0.05 kg
(1)
5 x 10-2 kg
(d) 40.070 nm
(5)
4.0070 x 101 nm
(e) 0.000 060 ns
(2)
6.0 x 10-5 ns
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Precision
Measurements depend on the precision of the measuring instruments used,
that is, the amount of information that the instruments can provide. For
example, 2.861 cm is more precise than 2.86 cm because the three decimal
places in 2.861 makes it precise to the nearest one-thousandth of a
centimetre, while the two decimal places in 2.86 makes it precise only to
the nearest one-hundredth of a centimetre. Precision is indicated by the
number of decimal places in a measured or calculated value.
PRECISION
indicated by the number of decimal places in the number
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4CR - Uncertainty in Measurements
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Precision
RULES FOR PRECISION
1. All measured quantities are expressed as precisely as possible. All
digits shown are significant with any error or uncertainty in the last
digit.
For example, in the measurement 87.64 cm the uncertainty is with the
digit 4.
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4CR - Uncertainty in Measurements
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Precision
RULES FOR PRECISION
2. The precision of a measuring instrument depends on its degree of
fineness and the size of the unit being used.
For example, a ruler calibrated in millimetres (ruler #2) is more precise
than a ruler calibrated in centimetres (ruler #1) because the ruler
calibrated in millimetres has more graduations.
#1
#2
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Precision
RULES FOR PRECISION
3. Any measurement that falls between the smallest divisions on the
measuring instrument is an estimate. We should always try to read
any instrument by estimating tenths of the smallest division.
For example, with ruler #1 we would estimate to the nearest tenth of a
centimetre (i.e. 3.2 cm); with ruler #2 we would estimate to the
nearest tenth of a millimeter (i.e. 3.24 cm).
#1
#2
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Precision
RULES FOR PRECISION
4. The estimated digit is always shown when recording the measurement.
For example, the 7 in the measurement 6.7 cm would be the estimated
digit.
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4CR - Uncertainty in Measurements
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Precision
RULES FOR PRECISION
5. Should the object fall right on a division mark, the estimated digit
would be 0.
For example, if we use a ruler calibrated in centimetres to measure a
length that falls exactly on the 5 cm mark, the correct reading is 5.0
cm, not 5 cm.
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Precision
PRACTICE
3. Use the two centimetre rulers to measure and record the length of the
pen graphic.
(a) Child’s ruler
~ 4.9 cm
(b) Ordinary ruler
~ 4.92 cm
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Precision
PRACTICE
4. An object is being measured with a ruler calibrated in millimetres. One
end of the object is at the zero mark of the ruler. The other end lines
up exactly with the 5.2 cm mark. What reading should be recorded for
the length of the object? Why?
5.20 cm should be recorded since the object falls right on division.
Since the ruler is calibrated in millimetres, we need to estimate to the
nearest tenth of the smallest division
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4CR - Uncertainty in Measurements
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Precision
PRACTICE
5. Which of the following values of a measured quantity is most precise?
(a) 4.81 mm, 0.81 mm, 48.1 mm, 0.081 mm
(b) 2.54 cm, 12.64 cm, 126 cm, 0.5400 cm, 0.304 cm
(a) 0.081 mm – has 3 decimal places
(b) 0.5400 cm – has 4 decimal places
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Uncertainty in Measurements
PRACTICE
6. Copy and complete the following table.
# of Sig.Dig.
Measurement
Measurement
Precision
now
needed
rounded
in sci.not.
a 63.479 km (example)
3
5
3
63.5
6.35 × 101
b 46 597.2 cm
1
6
2
47 000
4.7 x 104
c 0.5803 L
4
4
1
0.6
6 x 10-1
d 325 kg
0
3
2
320
3.2 x 102
e 0.067 80 mm
5
4
3
0.0678
6.78 x 10-2
f
3
6
4
485.0
4.850 x 102
485.000 kW
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8
Accuracy & Significant Figures
Accuracy relates to how close a measurement agrees with the accepted
value. This is an indication of the quality of the measuring instrument and
the technique of the user. The difference between an observed value (or the
average of the observed values) and the accepted value is called the error.
The degree of accuracy of any instrument depends on two things: the
precision of the measuring instrument and the skill and care of the user.
The precision of a measuring instrument depends on the size of the unit
being used. It depends on the place value of the last digit obtained from a
measurement or calculation. For example, 2.861 is more precise than
581.86.
Certainty is determined by how many certain digits are obtained by the
measuring instrument.
Any measurement that falls between the smallest division on the measuring
instrument is an estimate and is therefore uncertain.
Example: Determine the length of each line with the appropriate number of
significant figures.
a)
b)
Significant figures record all digits that are certain, plus one uncertain digit.
To determine the number of significant figures in a measurement, count the
number of digits. All digits in a given measurement are significant except
for the leading zeros.
Example: State the number of significant figures for each of the following
measurements.
a) 32.58g 
b) 6.07cm 
c) 0.0025mL 
d) 0.180L 
e) 4.148 x 103g 
f) 2.00 x 10-1m 
CERTAINTY RULE FOR MULTIPLYING AND DIVIDING:
***The answer has the same number of significant figures as the
measurement with the fewest significant figures***
Example: State the answer with the appropriate number of significant
figures.
a)
21.8

3.9
b)
(9.80)(27.06) 
PRECISION RULE FOR ADDING AND SUBTRACTING:
***The answer has the same number of decimal places as the measured
value with the fewest decimal places***
Example:
State the answer with the appropriate number of significant
figures.
a)
104.2km + 11km + 0.67km =
b)
5.5m + 0.597m + 0.1262m =
EXACT NUMBERS:
When you directly count the number of something, this is an exact value.
Objects that have set values, such as 100cm/m or 60s/min, are defined
values. Exact and defined values are said to contain an infinite number of
significant figures and therefore don’t affect the rules of certainty or
precision.
Example: State the answer with the appropriate number of significant
figures.
a)
1
(.25)(16.2) 2 
2
b)
4
 (2.3) 3 
3
CONVERTING UNITS
The method most commonly used is multiplying by conversion factors,
which are either memorized or referenced.
Example: Perform the following conversions, stating your final answer with
the appropriate number of significant figures
a) 28.6 minutes to hours.
b) 18 minutes to seconds
c) 16 km/h to m/s.
8/15/2013
SPH4C
COLLEGE PHYSICS
REVIEW: MATH SKILLS
L Calculations Using Measurements
(P.546-547)
Rounding
If measurements are approximate, the
calculations based on them must also be
approximate. Scientists agree that calculated
answers should be rounded so they do not give
a misleading idea of how precise the original
measurements were. Use these rules when
making calculations and rounding answers to
calculations.
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4CR - Calculations Using Measurements
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Rounding
RULES FOR ROUNDING
1. When the first digit to be dropped is 4 or less, the last digit retained
should not be changed.
For example:
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3.141 326
rounded to 4 digits is
4CR - Calculations Using Measurements
3.141
2
1
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Rounding
RULES FOR ROUNDING
2. When the first digit to be dropped is greater than 5, or if it is a 5
followed by at least one digit other than zero, the last digit retained is
increased by 1 unit.
For example:
2.221 372
4.168 501
August 15, 2013
rounded to five digits is
rounded to four digits is
4CR - Calculations Using Measurements
2.2214
4.169
3
Rounding
RULES FOR ROUNDING
3. When the first digit discarded is five or a five followed by only zeros,
the last digit retained is increased by 1 if it is odd, but not changed if it
is even.
For example:
2.35
2.45
-6.35
rounded to two digits is
rounded to two digits is
rounded to two digits is
2.4
2.4
-6.4
NOTE!
This is sometimes called the even-odd rule.
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4CR - Calculations Using Measurements
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Adding & Subtracting
RULES FOR ADDING & SUBTRACTING
When adding and/or subtracting, the answer has the same number of
decimal places as the measurement with the fewest decimal places.
For example:
=
=
6.6 cm + 18.74 cm + 0.766 cm
26.106 cm
26.1 cm
NOTE!
The answer must be rounded to 26.1 cm because the first measurement
(6.6 cm) limits the precision to a tenth of a centimetre.
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4CR - Calculations Using Measurements
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Multiplying & Dividing
RULES FOR MULTIPLYING & DIVIDING
When multiplying and/or dividing, the answer has the same number of
significant digits as the measurement with the fewest number of significant
digits.
For example:
=
=
77.8 km/h x 0.8967 h
69.76326 km
69.8 km
NOTE!
The certainty of the answer is limited to three significant digits, so the
answer is rounded up to 69.8 km. The same applies to scientific notation.
For example,
(5.5 x 10 4) ) (5.675 x 10 -2) = 9.7 x 10 5
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4CR - Calculations Using Measurements
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Multistep Calculations
RULES FOR MULTISTEP CALCULATIONS
For multistep calculations, round-off errors occur if you use the rounded-off
answer from an earlier calculation in a subsequent calculation. Thus, leave
all digits in your calculator until you have finished all your calculations and
then round the final answer.
For example:
=
=
=
5.21 x 0.45 ) 0.00600
2.3445 ) 0.00600
or
390.75
390 U
=
=
=
2.3 ) 0.00600
383.333333
380 Y
NOTE!
The certainty of the answer is limited to two significant digits, so the
answer is rounded accordingly. In the second example though, rounding
occurred during the calculation which introduced a round-off error.
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4CR - Calculations Using Measurements
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Calculations – Summary
ADDING & SUBTRACTING
fewest decimal places
MULTIPLYING & DIVIDING
fewest number of significant digits
MULTISTEP CALCULATIONS
leave all digits in the calculator until finished and then round
NOTE!
If a combination of addition, subtraction, multiplication and division are
involved, follow the rules for multiplying and dividing.
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4CR - Calculations Using Measurements
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Calculations Using Measurements
PRACTICE
1. Perform the following operations.
(a) 67.8 + 968 + 3.87
(b) 463.66 + 29.2 + 0.17
(c) 68.7 - 23.95
(d) (2.6)(42.2)
(e) (65)(0.041)(325)
(f) (0.0060)(26)(55.1)
(g) 650 ) 4.0
(h) 3.52
(i) (1.62 × 10-3)(7.3 × 10-1)
(j) (5.019 × 10-4)÷(3.1 × 10-7)
August 15, 2013
Round your answers accordingly.
1039.67
= 1040
493.03
= 493.0
44.75
= 44.8
109.72
= 110
866.125
= 870
8.5956
= 8.6
162.5
= 160
12.25
= 12
0.0011826
= 0.0012
1619.0322...
= 1600
4CR - Calculations Using Measurements
9
Calculations Using Measurements
PRACTICE
2. Solve each of the following. Round your answers accordingly.
(a) Find the perimeter of a rectangular carpet that has a width and
length of 3.56 m and 4.5 m.
(a) 16.1 m
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4CR - Calculations Using Measurements
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Calculations Using Measurements
PRACTICE
2. Solve each of the following. Round your answers accordingly.
(b) Find the area of a rectangle whose sides are 4.5 m and 7.5 m.
(b) 34 m2
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Calculations Using Measurements
PRACTICE
2. Solve each of the following. Round your answers accordingly.
(c) A triangle has a base of 5.75 cm and a height of 12.45 cm.
Calculate the area of the triangle. (Recall A = ½bh)
(c) 35.8 cm2
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4CR - Calculations Using Measurements
12
Calculations Using Measurements
PRACTICE
2. Solve each of the following. Round your answers accordingly.
(d) On the planet Zot distances are measured in zaps and zings. If 3.9
zings equal 7.5 zaps, how many zings are equal to 93.5 zaps?
(d) 49 zings
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4CR - Calculations Using Measurements
13
Calculations Using Measurements
PRACTICE
2. Solve each of the following. Round your answers accordingly.
(e) The Earth has a mass of 5.98 × 1024 kg while Jupiter has a mass of
1.90 × 1027 kg. How many times larger is the mass of Jupiter than
the mass of the Earth?
(e) 318 times
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15/08/2013
SPH4C
COLLEGE PHYSICS
REVIEW: MATH SKILLS
L Error in Measurements
(P.548)
Error in Measurements
Many people believe that all measurements are
reliable (consistent over many trials),
precise (to as many decimal places as
possible), and accurate (representing the
actual value). But there are many things that
can go wrong when measuring. For example:
August 15, 2013
4CR - Error in Measurements
1
Error in Measurements
•
•
•
There may be limitations that make the
instrument
or
its
use
unreliable
(inconsistent).
The investigator may make a mistake or
fail to follow the correct techniques when
reading the measurement to the available
precision (number of decimal places).
The instrument may be faulty or
inaccurate; a similar instrument may give
different readings.
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4CR - Error in Measurements
2
1
15/08/2013
Error in Measurements
PRACTICE
1. What three things can you do during an experiment to help eliminate
errors?
1.
To be sure that you have measured correctly, you should repeat
your measurements at least three times.
2.
If your measurements appear to be reliable, calculate the mean and
use that value.
3.
To be more precise about the accuracy, repeat the measurements
with a different instrument.
August 15, 2013
4CR - Error in Measurements
3
Error in Measurements
PRACTICE
2. There are two types of measurement error. What are they?
random error and systematic error
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4CR - Error in Measurements
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Random Error
Random error results when an estimate is made to obtain the last digit
for any measurement. The size of the random error is determined by the
precision of the measuring instrument. For example, when measuring
length with a measuring tape, it is necessary to estimate between the
marks on the measuring tape. If these marks are 1 cm apart, the random
error will be greater and the precision will be less than if the marks are 1
mm apart. Such errors can be reduced by taking the average of several
readings.
RANDOM ERROR
results when the last digit is estimated
reduced by taking the average of several readings
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4CR - Error in Measurements
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Systematic Error
Systematic error is associated with an inherent problem with the
measuring system, such as the presence of an interfering substance,
incorrect calibration, or room conditions. For example, if a balance is not
zeroed at the beginning, all measurements will have a systematic error;
using a slightly worn metre stick will also introduce error. Such errors are
reduced by adding or subtracting the known error or calibrating the
instrument.
SYSTEMATIC ERROR
due to a problem with the measuring device
reduced by adding/subtracting the error or calibrating the device
August 15, 2013
4CR - Error in Measurements
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Accuracy & Precision
In everyday usage, "accuracy" and "precision" are used interchangeably to
describe how close a measurement is to a true value, but in science it is
important to make a distinction between them. Accuracy refers to how
close a value is to its accepted value. Precision is the place value of the
last measureable digit.
ACCURACY
how close a value is to its accepted value
PRECISION
place value of last measureable digit
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4CR - Error in Measurements
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Accuracy & Precision
For example, the position of the darts in each of the figures are analogous
to measured or calculated results in a laboratory setting. The results in (a)
are precise and accurate, in (b) they are precise but not accurate, and in
(c) they are neither precise nor accurate.
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Percentage Error
No matter how precise a measurement is, it still may not be accurate. The
percentage error is the absolute value of the difference between
experimental and accepted values expressed as a percentage of the
accepted value.
% error =
experimental value − accepted value
accepted value
x 100
NOTE!
The bars (||) in the equation above represent “absolute value”.
means that, mathematically, if a = 3 and b = -3 then |a| = |b| = 3.
August 15, 2013
4CR - Error in Measurements
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Percentage Difference
Sometimes if two values of the same quantity are measured, it is useful to
compare the precision of these values by calculating the percentage
difference between them.
% difference =
measurement 1 − measurement 2
x 100
 measurement 1 + measurement 2 


2


NOTE!
“Magnitude” is a term frequently used by physicists. The magnitude of a
quantity is the same as its absolute value.
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4CR - Error in Measurements
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Error in Measurements
PRACTICE
3. At a certain location the acceleration due to gravity is 9.82 m/s2[down].
Calculate the percentage error of the following experimental values of
“g” at that location.
(a) 8.94 m/s2[down]
(b) 9.95 m/s2[down]
(a) 8.96
(b) 1.32
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4
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Error in Measurements
PRACTICE
4. Calculate the percentage difference between the two experimental
values (8.94 m/s2 and 9.95 m/s2) used in question #3.
10.7
August 15, 2013
4CR - Error in Measurements
12
U Check Your Learning
WIKI (REVIEW)
O.... 4CR - WS1 (Math Skills)
O.... 4CR - QUIZ1 (Math Skills - Part 1)
August 15, 2013
4CR - Error in Measurements
13
5
8/15/2013
SPH4C
COLLEGE PHYSICS
REVIEW: MATH SKILLS
L Measuring & Estimating
(P.549)
Making Precise Measurements
In order to make precise measurements you need to use a device that has
a double scale. A double scale consists of a main scale that is an ordinary
metric scale with centimetres and millimetres and a sliding or vernier
scale.
vernier scale
L
main scale
L
August 15, 2013
4CR - Measuring & Estimating
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Making Precise Measurements
If you look at the diagram carefully you will see there are 10 graduations
on the vernier scale that occupy the same space as 9 graduations on the
main scale. Therefore, only one graduation on the vernier can line up with
a graduation on the main scale.
vernier scale
L
main scale
L
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4CR - Measuring & Estimating
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1
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Making Precise Measurements
A double scale can be placed on various types of instruments. One
common instrument is the vernier caliper. It is used to measure the
outside diameter of a cylinder, the inside diameter of a hollow cylinder, or
the depth of a hole.
August 15, 2013
4CR - Measuring & Estimating
3
Making Precise Measurements
Another instrument with a double scale is the outside micrometer
caliper.
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4CR - Measuring & Estimating
4
Using a Vernier Caliper
Vernier calipers are precision measuring instruments used to make
accurate measurements. The bar and movable jaw are graduated on both
sides, one side for taking outside measurements and the other side for
inside measurements. Vernier calipers are available in metric and in inch
graduations, and some types have both scales. Digital models are also
available.
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2
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Using a Vernier Caliper
NOTE!
When using high-precision instruments, such as the vernier caliper or
outside micrometer caliper, it is necessary to check the zero setting before
taking a reading. If, for example, the instrument is supposed to read 0.000
cm but instead reads 0.002 cm, the error must be taken into consideration
with each reading.
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4CR - Measuring & Estimating
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Using a Vernier Caliper
HOW TO READ A METRIC VERNIER CALIPER
1.
Find the first line (the ZERO line) on the vernier (sliding) scale. Look
on the main (stationary) scale and record the number you just passed
(or are currently on) as #.# cm.
5.0 cm
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4CR - Measuring & Estimating
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Using a Vernier Caliper
HOW TO READ A METRIC VERNIER CALIPER
2.
Find the FIRST pair of lines that match up perfectly. Read the line
number off the vernier (sliding) scale and add this to the
measurement.
5.08 cm
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3
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Using a Vernier Caliper
HOW TO READ A METRIC VERNIER CALIPER
3.
Determine the error – half of the smallest measurement possible. In
our case the smallest measurement possible is 1 mm so the error is 0.5
mm or 0.05 cm. Add this to your measurement.
5.08 " 0.05 cm
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4CR - Measuring & Estimating
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Using a Vernier Caliper
PRACTICE
1. What is the reading (including the error) of the following metric vernier
calipers?
(a)
1
0
2
0.69 " 0.05 cm
0 2 4 6 8 0
(b)
3
4
3.18 " 0.05 cm
0 2 4 6 8 0
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4CR - Measuring & Estimating
10
Using a Vernier Caliper
PRACTICE
1. What is the reading (including the error) of the following metric vernier
calipers?
(c)
0
1
2
0.87 " 0.05 cm
0 2 4 6 8 0
(d)
2
3
4
2.63 " 0.05 cm
0 2 4 6 8 0
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4
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Using a Vernier Caliper
PRACTICE
1. What is the reading (including the error) of the following metric vernier
calipers?
(e)
1
2
1.13 " 0.05 cm
0 2 4 6 8 0
(f)
6
7
8
7.05 " 0.05 cm
0 2 4 6 8 0
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5
8/15/2013
SPH4C
COLLEGE PHYSICS
REVIEW: MATH SKILLS
L Trigonometry
(P.550-551)
Trigonometry
The first application of trigonometry was to solve right-angle triangles.
Trigonometry derives from the fact that for similar triangles, the ratio of
corresponding sides will be equal. For a given angle " in a right triangle,
there are three important ratios: sine, cosine, and tangent. These are
called the primary trigonometric ratios and they can be used to find
the measures of unknown sides and angles in right triangles.
August 15, 2013
4CR - Trigonometry
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Trigonometry
NOTE!
DEG, RAD, and GRAD are different units/modes used for measuring angles.
For this course make sure that your calculator is always in DEG mode.
(Hint: if you are not getting the correct answers for a trigonometry
problem start by checking this!)
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4CR - Trigonometry
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1
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Trigonometry
PRACTICE
1. Determine the value of each ratio rounded to four decimal places.
(a) sin 35°
0.5736
(b) cos 60°
0.5000
(c) tan 45°
1.0000
(d) cos 75°
0.2588
(e) sin 18°
0.3090
August 15, 2013
4CR - Trigonometry
3
Trigonometry
RECALL!
If the value of a trigonometric ratio is known, its corresponding angle can
be found using the inverse of that ratio. For example, if cos 2 = 0.50 then
2 = cos -1 (0.50).
•
for sin -1 use:
2nd
sin
•
for cos -1 use:
2nd
cos
•
for tan -1 use:
2nd
tan
August 15, 2013
4CR - Trigonometry
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Trigonometry
PRACTICE
2. Determine the size of
(a) sin A = 0.5299
(b) cos A = 0.4226
(c) tan A = 4.3315
(d) cos A = 0.5000
(e) sin A = 0.2419
August 15, 2013
∠A rounded to the nearest degree.
32E
65E
77E
60E
14E
4CR - Trigonometry
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2
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Trigonometry
PRACTICE
3. Solve for x rounded to
(a) sin 35° = x/8
(b) cos 70° = x/15
(c) tan 20° = 3/x
(d) sin 85° = 6/x
(e) cos 25° = 5/x
one decimal place.
4.6
5.1
8.2
6.0
5.5
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4CR - Trigonometry
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Trigonometry
PRACTICE
4. Use two different methods to find the value of the unknown(s) in each
triangle. Round your answers to one decimal place.
7 mm
X
2
60 cm
26 cm
35E
Y
10 mm
X = 12.2 mm
Y = 54.1 cm
2 = 64.3 E
August 15, 2013
4CR - Trigonometry
7
U Check Your Learning
WIKI (REVIEW)
O.... 4CR - QUIZ2 (Math Skills - Part 2)
August 15, 2013
4CR - Trigonometry
8
3