General Physics (PHYS101)
Length, time and mass
Dimensional analysis
Lecture 02
www.cmt.ua.ac.be/golib/PHYS101
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International System (SI) of Units
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•
•
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Length
Time
Mass
Basic SI Units
meter
seconds
kilogram
m
s
kg
These are the only units necessary to describe any quantity.
• [Volume] = m3 cubic meter
• [Density] = kg/m3 kilogram per cubic meter
• [Speed] = m/s meter per second
• [Acceleration] = m/s2 meter per second squared
• [Force]: N (Newton) = kg m/s2
• [Frequency]: Hz (Hertz) = s-1
• [Pressure]: Pa (Pascal) = N/m2
• [Energy]: J (Joule) = N m
• [Power]: W (Watt) = J/s
Conversion of Units: Chain-link method
Example 1: Express 3 min in seconds?
1min = 60 s
Conversion Factor?
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Conversion of Units
Example 3: Express 200 km/h in miles/s?
1 km = 0.6 miles
1h = 60 min = 60 x 60 s = 3600 s
200 km/h= 200 x 0.6 miles/3600 s = 0.03 miles/s
Example 4: Express 200 km/h in m/s?
1 km = 1000 m
1 h = 3600 s
200 km/h= 200 x 1000 m/3600 s = 55.56 m/s
km/h --> m/s :3.6
m/s --> km/h x3.6
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Scientific notations
Scientific notation
Expanded form
1 x 100
1
1 x 101
10
1 x 102
100
1 x 103
1000
1x
106
1 000 000
1x
10-1
1/10 or 0.1
1 x 10-3
1/1000 or 0.001
1 x 10-6
0. 000 001
1.23 = 1.23 x 100
0.25 = 2.5 x 10-1
0.0007925 = 7.925 x 10-4
The following prefixes indicate
multiples of a unit.
Multiplier
Prefix
Symbol
1012
tera
T
109
giga
G
106
mega
M
103
kilo
k
10-3
milli
m
10-6
micro

10-9
nano
n
10-12
pico
p
10-15
femto
f
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Rounding
Speed of light: c=299 792 458 m/s
c=2.99 792 458 x 108 m/s
• Overestimation: digits 5 to 9 can be dropped from the
decimal place during the rounding, however, one
should be added to the digit in front of it.
• Underestimation: the following digits can just be
dropped from the decimal place: 0, 1, 2, 3, an 4.
Example 1. Round c to a nearest 1000th. c=2.998 x 108 m/s.
Example 2. Round c to a nearest 10th.
c=3.0 x 108 m/s.
Example 3. Round 273.587 to a nearest integer. 274
Example 4. Round 273.587 to 2 significant figures. 270
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Uncertainties in measurements
• The accuracy of the measurements are determined by significant figures.
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Rules for Significant Figures
1. All nonzero figures are significant
359 87678 1245 987889
2. All zeros between nonzeros are significant
205 1003 508009 800009002
3. Zeros at the end are significant if there is a decimal
point before them
4.200 1003.5600 30.003000
4. All other zeros are non-significant
30000 0.0000344
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Rules for Significant Figures
Not significant
Not significant
Significant
zero at the beginning
zero used only to locate
the decimal point
all zeros between nonzero
numbers
0.004004500
Significant
Significant
all nonzeros
zeros at the end of
integers
a number to the right
of the decimal point
Just take care of zeros
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Operations with Significant Figures
• When adding or subtracting, round the results to the
smallest number of decimal places of any term in the sum
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Operations with Significant Figures
• When multiplying or dividing, round the result to the same
accuracy as the least accurate measurements (i.e. the
smallest number of the significant figures)
Example: Calculate the surface area of a plate with
dimensions 4.5 cm by 7.32 cm.
A=4.5 cm x 7.32 cm=32.94 cm2.
A=33 cm2.
1.0:9 = 0.11111… = 0.1
1.0:9.0 = 0.11111… = 0.11
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Definition of kilogram
Length
[L]
m
Time
[T]
s
Mass
[M]
kg
• Unit for mass is defined in terms of kilogram, based on a specific
Pt-Ir cylinder kept at the International Bureau of Standards.
Why is it hidden under two
glass domes?
• Another definition is based on the mass of carbon atom.
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Definition of second
• Defined in terms of the oscillation of radiation from a cesium
atom (9 192 631 700 times frequency of light emitted).
• US time standard NIST-F1: accurate to 1 second in 80 million years.
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Definition of meter
• Meter is the distance travelled by light in a vacuum
during a given time (1/299 792 458 s).
• The speed of light is:
c=299 792 458 m/s
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Dimensional analysis: Equation analysis
• Complicated formulas can be checked for consistency by looking
at the units (dimensions) to make sure that both sides of the
equation match.
Example 1: Is the equation correct?
Example 2: Show that the expression
dimensionally correct, where and
represent velocities,
is acceleration, and is a time interval.
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Dimensional analysis: Equation analysis
Example 3: Is the equation correct?
[𝑣] =
𝐿
𝑇
𝑣 = 𝑠𝑡
[st] = 𝐿𝑇
𝐿
≠ 𝐿𝑇
𝑇
Equation is not correct!
Example 4: The volume of a sphere:
V=
4
𝜋𝑅2
3
[V] = 𝐿3
[R] = 𝐿2
L3≠ 𝐿2
Equation is not correct!
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Dimensional analysis: Predictions
Example. Use dimensional analysis to determine how the
time of a falling apple t depends on the height h.
h
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Summary
• Scientific notations
• Order of magnitude: 10x (x=1,2,3 ..)
• Rounding
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Summary
• Uncertainties in the measurements
• Significant figures
• It is important to control the number of digits or significant
figures in the measurements.
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Summary
• Dimensional analysis is a technique to check the correctness
of an equation
• Dimensional analysis can be useful in making predictions.
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