1.3 More About the Metric System
• Metric units come in two slightly different flavours:
– mks: metre* (m), kilogram (kg), second (s)… “human scale”
– cgs: centimetre (cm), gram (g), second (s)… still used in some
branches of physics
• Unless explicitly stated otherwise, we will use metric MKS
units in this course, but you should have some practice in
converting among different systems.
* the Canadian spelling is “metre”, whereas the American spelling – used in the text – is
“meter”. To confuse matters, any apparatus that is used for measurement purposes is
called a “meter”, as in “speedometer” or “parking meter”.
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Slide 1
1.3 More About the Metric System
The following table contains prefixes used to describe very large or
very small values in SI units.
Note that the symbols are case-sensitive. For example, 1 ms = 10-3
seconds, while 1 Ms = 106 seconds.
μ is the Greek letter “mu”
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Slide 2
1.3 More About the Metric System
Even though “kilo” is a prefix (meaning 103), it is the kilogram – not
the gram – that is the SI mks base unit, since 1 gram is too small a
quantity for everyday measurement.
This may result in some confusion when determining the magnitude
of quantities that are not expressed in SI base units. We’ll come back
to this point later on.
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Slide 3
1.3 More About the Metric System
You are undoubtedly familiar with many of these prefixes…
• Your car’s odometer measures km, while the speedometer
measures km/h (kilometers per hour…not fully metric!)
• Your driver’s license list your height in cm
• Measuring cups use units of mL (millilitres)
• Digital cameras quote a number of megapixels
• Computer processor speeds are measured in GHz
• Nanotechnology involves the creation of structures with feature
sizes in the nm range.
Caution! computer memory appears to use SI prefixes…
• Memory is measured in MB or GB (megabytes / gigabytes)
However, these are based on the closest base-2 representation. That
is, 1 GB is not 109 bytes, it’s 1 073 741 824 bytes (230 bytes) – the
reason is that computers format data in binary form.
PC141 Intersession 2013
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Slide 4
Problem #1: SI Prefixes
WBL LP 1.9
Which of the following metric prefixes is the smallest?
PC141 Intersession 2013
A
micro-
B
centi-
C
nano-
D
milli-
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Slide 5
1.3 More About the Metric System
It is assumed that students in PC141 are comfortable with scientific
notation, which is used to represent very large or very small numbers.
If not, here is a quick summary (there is also a review in Appendix I of
the text).
• In scientific notation, we place one digit before the decimal place and one or more
digits after the decimal place; this forms the significand. The exponent tells us
how many powers of 10 to multiply by the significand.
• For example:
3 560 000 000 m = 3.56 x 109 m and 0.000 000 492 s = 4.92 x 10-7 s
• It is good practice to avoid combining scientific notation with SI prefixes. While it
is correct that 3.56 x 109 m = 3.56 x 106 km, the simultaneous use of both
notations can be confusing.
• In many programming languages, scientific notation is indicated with the syntax
“3.56e+9” (or simply “3.56e9”) or “4.92e-7”. However, MasteringPhysics does
not follow this convention. Instead, type “3.56*10^9” or “4.92*10^-7”. If you
previously used WebAssign for online assignments, this will be new to you.
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Slide 6
1.3 More About the Metric System
The following slides (not from the text) list approximate values
for a range of lengths, masses, and times.
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Slide 7
Length
Some approximate lengths
Measurement
Length (m)
Radius of visible universe
4 x 1026
Distance to Andromeda galaxy
2 x 1022
Distance to Proxima Centauri
4 x 1016
Distance travelled by light in 1 second, in vacuum
3 x 108
Radius of Earth
6 x 106
Height of Mt. Everest
9 x 103
Length of football field
1 x 102
Height of a human
2 x 100
Thickness of paper
1 x 10-4
Wavelength of yellow light
5 x 10-7
Length of typical virus
1 x 10-8
Radius of hydrogen atom
5 x 10-11
Radius of proton
1 x 10-15
Planck length
2 x 10-35
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Slide 8
Mass
Some approximate masses
Measurement
Mass (kg)
The known universe
1 x 1053
Milky way galaxy
2 x 1041
Sun
2 x 1030
Earth
6 x 1024
Moon
7 x 1022
Ocean liner
7 x 107
Elephant
5 x 103
Average human
8 x 101
Grape
3 x 10-3
Speck of dust
7 x 10-10
Proton
2 x 10-27
Electron
9 x 10-31
Neutrino
1 x 10-38
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Slide 9
Time
Some approximate times
Measurement
Time (s)
Predicted lifetime of the proton
3 x 1040
Age of the universe
5 x 1017
Age of the pyramid of Cheops
1 x 1011
Human life expectancy
2 x 109
One year
3 x 107
One day
9 x 104
Time for light to travel from earth to moon
1 x 100
One cycle of middle C
4 x 10-3
One cycle of FM radio wave
1 x 10-8
Shortest man-made light pulse
8 x 10-17
Planck time
1 x 10-43
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Slide 10
1.3 More About the Metric System
The standard SI unit of volume is the cubic metre (m3), the
volume of a cube that is 1 metre to a side. This turns out to be
inconveniently large for many applications, so it is common to
use the litre (L) instead. This is the volume of a cube 10 cm to a
side.
In chemistry (and cooking!), we often use millilitres (mL). One
mL is the volume of a cube 1 cm to a side.
Since 1 mL is one cubic centimetre, it is
often abbreviated as “1 cc”, particularly
in the medical fields.
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Slide 11
1.4 Unit Analysis
• Unit analysis (or “dimensional analysis”) is an easy way to spot
errors in your work.
• The two sides of any (correct) equation must be identical in
magnitude and dimensions. In other words, it makes sense to write
“2+1=3”, but not “2 metres + 1 metre = 3 seconds”.
• Before submitting any answers, make sure that you perform a quick
unit analysis to see if you’ve made a mistake.
• Also keep in mind that quantities with different dimensions can not
be added or subtracted (“1 metre + 2 seconds = ???”), but they can
be multiplied or divided (“4 meters / 2 seconds = 2 metres per
second”).
• Quantities with the same dimensions but different units CAN be
added or subtracted, after converting to the same units (“1 inch + 1
cm = 2.54 cm + 1 cm = 3.54 cm”). Unit conversion is coming up
next…
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Slide 12
Problem #2: Unit Analysis
Consider a pendulum of length L and mass m. The gravitational
acceleration is g (units of L / T2). Which of the following formulas could
be correct for the period P of the pendulum’s oscillations?
2
A
𝑃 = 2𝜋 𝑔𝐿
B
𝑃 = 2𝜋𝑚𝐿/𝑔
C
𝑃 = 2𝜋 𝐿/𝑔
D
𝑃 = 2𝜋𝐿𝑚
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Slide 13
1.4 Unit Analysis
Many quantities of interest in PC141 can not be described using just
one of the seven base units. However, they can be described using
products or quotients of the base units. These are called derived
units.
For example:
• Speed has units of L / T
(length divided by time; i.e. metres per second)
• Density has units of M / L3
(mass divided by length cubed)
• Force has units of ML / T2
(mass multiplied by length divided by time squared)
• Energy has units of ML2/ T2
(mass multiplied by length squared divided by time squared)
These units will arise naturally from the physics that defines them.
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Slide 14
1.4 Unit Analysis
Here are a few examples of derived SI units that we will
encounter in PC141:
Quantity
Unit Name
SI Unit Symbol
Force
Newton
N
Expression in terms of SI
base units
kg∙m/s2
Energy
Joule
J
kg∙m2/s2
Power
Watt
W
kg∙m2/s3
Note that parameter symbols are italicized, whereas unit
symbols are not. For example, “I walked a distance d = 1 km”.
Furthermore, unit symbols named after people are capitalized.
So are the unit names, although our textbook doesn’t seem to
follow that convention.
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Slide 15
1.5 Unit Conversions
While we can’t magically convert one dimension to another (i.e.
metres to seconds), we are often required to convert between
units having the same dimension (i.e. miles to km, or hours to
seconds).
This is accomplished using Google conversion factors. These are
ratios between units that have a magnitude of 1. Multiplying any
value by a conversion factor will change its units, but not its
magnitude.
The units can be cancelled like any other algebraic quantity.
Example: What is 4 cm, expressed in inches?
Conversion factor: 1 inch = 2.54 cm
4 cm ×
PC141 Intersession 2013
1 inch
2.54 cm
=1.57 inches
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Slide 16
1.5 Unit Conversions
Often, more than one conversion factor must be used. These can
be cascaded in a process termed chain-link conversion.
Regardless of the number of conversion factors necessary, always
remember to choose conversion factors such that you can cancel
units in the numerator and denominator to result in the desired
units.
Example: What is 100 km/h, expressed in m/s?
Solution:
km
1000 m 1 hour 1 min
100
×
×
×
= 27.8 m/s
hour
1 km
60 min 60 s
PC141 Intersession 2013
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Slide 17
1.5 Unit Conversions
Extra care must be taken when converting powers of units (i.e.
square feet to square metres). For instance, 1 m = 100 cm, but it
is incorrect to assume that 1 m2 = 100 cm2. Instead, the
conversion factor also needs to be raised to the same power.
Example: How many cm3 are in 2 m3?
𝟑
Solution:
𝟏𝟎𝟎 cm
𝟏𝟎𝟎𝟑 cm𝟑
𝟐m𝟑 ×
𝟏m
= 𝟐 m𝟑 ×
𝟏 𝟑 m𝟑
= 𝟐 × 𝟏𝟎𝟔 cm𝟑
There are two million cubic centimetres in 2 cubic metres!
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Slide 18
Problem #3: Volume
WBL EX 1.7
What is the volume, in litres, of a cube 20 cm on a side?
Solution: In class
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Slide 19
1.6 Significant Figures
An important distinction must be made between exact numbers and
measured numbers.
Exact numbers are numbers with no uncertainty or error. For
example, if r is the radius of a circle and d is its diameter, we can relate
r and d by the equation r = d / 2. Here, “2” is an exact number.
Measured numbers have some degree of uncertainty in their last
digit.
When calculations are performed on measured numbers, the
uncertainty propagates through to the result of the calculations. As a
result, many of the digits that your calculator shows are meaningless!
The number of significant figures in your answer can not exceed the
number of significant figures in any of the exact numbers contained
in the equation. And in fact, MasteringPhysics may consider your
answers incorrect if you use the wrong number of SigFigs.
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Slide 20
1.6 Significant Figures
Example:
A car travels with constant speed s, and covers a distance of
d = 123 m in a time t = 7.89 s.
The speed is s = d/t = (123 m)/(7.89 s) = 15.5893536 m/s,
according to my calculator. However, it is meaningless to use 9
significant figures to represent s when d and t only use 3
significant digits. It would be correct to say that s = 15.6 m/s.
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Slide 21
1.6 Significant Figures
What constitutes a significant figure?
• Leading zeros are not significant (0.0043 has 2 sigfigs, the 4
and the 3)
• Zeros in the middle of a number are significant (2305 has 4
sigfigs)
• Trailing zeros after a decimal point are significant (3.540 has 4
sigfigs)
• Trailing zeros at the end of a whole number may or may not be
significant – it is ambiguous as to whether a measurement of
“300 metres” has 1, 2, or 3 sigfigs. This can be clarified by
using scientific notation: 3 x 102 metres, 3.0 x 102 metres, and
3.00 x 102 metres have 1, 2, and 3 sigfigs, respectively.
Otherwise, assume that all trailing zeros that you encounter
in PC141 are significant.
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Slide 22
1.6 Significant Figures
Rounding
When a calculated answer contains insignificant digits, they
should be rounded off. If the leftmost of the digits to be
discarded is greater than 5, then the last remaining digit is
rounded up; otherwise, it is retained as is. For example, if we
want to keep 3 significant digits, then 11.3516 rounds UP to 11.4,
while 11.3279 rounds DOWN to 11.3.
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Slide 23
Problem #4: Significant Figures
WBL LP 1.15
Which of the following has the greatest number of
significant figures?
PC141 Intersession 2013
A
103.07
B
124.5
C
0.09916
D
5.408 x 105
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Slide 24
1.7 Problem Solving
The flowchart at left outlines a
useful problem-solving strategy.
It can be used for most types of
physics problems.
We will frequently refer to it
later in the course.
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Slide 25
1.7 Problem Solving
Hints
1. Solve problems algebraically, only inserting numbers at the
very last moment. When you numerical calculations at each
intermediate step, you are accumulating round-off errors.
2. When answering multi-part questions, use the appropriate
number of significant figures in your answers, but carry all
figures to the next part of the question. This also prevents
the accumulation of round-off errors.
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Slide 26
Problem #5: Atoms in Earth
Earth has a mass of 5.98 x 1024 kg. The average mass of the atoms that make up
Earth is 40 u, where 1 u = 1.660 x 10-27 kg (this is the “atomic mass unit”; 1 atom
of carbon-12 has, by definition, a mass of 12 u). How many atoms are in Earth?
Solution: In class
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Slide 27
Problem #6: Grains of Sand
Grains of fine California beach sand are approximately spheres with an average
radius of 50 μm, and are made of silicon dioxide, which has a density of 2600
kg/m3. What mass of sand grains would have a total surface area equal to the
surface area of a cube with edge length 1.00 m?
Solution: In class
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Slide 28
Problem #7: Fuel Efficiency
A tourist purchases a car in England and ships it home to the United States. The
car sticker advertised that the car’s fuel consumption was at the rate of 40 miles
per gallon. However, the tourist didn’t realize that the UK gallon isn’t the same
as the US gallon:
1 UK gallon = 4.546 090 0 litres
1 US gallon = 3.785 411 8 litres
For a trip of 750 miles: (a) how much fuel does the tourist think she requires?
(b) how much fuel does she actually require?
Solution: In class
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Slide 29
Problem #8: Human Blood
WBL EX 1.66
In 1 mm3, human adult blood contains, on average, 7000 white blood cells
(leukocytes) and 250 000 platelets (thrombocytes). If a person has a blood
volume of 5.0 L, estimate the total number of leukocytes and thrombocytes in
the blood.
Solution: In class
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Slide 30