1.2 System of Units
A single system is efficient, convenient, economical, and makes sense. The US system units is difficult
for the United States to give up because a lot of expert technical knowledge and intuition gained over
generations is tied to this particular unit system.
The world will use a single system once the US decides it wants too give up the US system. The
intent has been there for decades but the US units are still used vigorously. Living 2 miles from the
grocery store is better imagined than 3.21 kilometers for persons living in the US.
There are two popular “Systems of Units” in practice. The first one is the “System International” or SI
System of Units – used across the world where the unit for weight is Newtons. The second is the US
or the English System – mostly used in USA – where the unit for weight is pound (or pound force to
be strict). Any material for sale in the USA will display the specifications in both system of units.
Figure 1.2.1 Widespread use of US Units
US, Burma, Liberia are three countries that have not adopted the SI as the sole system of units
(from Wikipedia)
1.2.1 SI - or System International
The International system of units is used globally for commercial purposes as well as in science and
engineering. It appeared in 1960 and some of the standards are maintained at the International
Bureau of Weights and Measures at Severn, near Paris, France. It is also sometimes referred to as
the metric system.
The SI is a more complete system.
In fact the US/English system have no units for many of the physical quantities that have been defined
or discovered recently.
The SI is a system of units of measurement defined through seven (7) BASE UNITS
Table 1.2.1 SI Base Units
Name
Unit Symbol
Quantity
Symbol
metre/meter
kilogram
second
ampere
kelvin
candela
m
kg
s
A
K
cd
length
mass
time
electric current
thermodynamic temperature
luminous intensity
l
m
t
I
T
Iv
mole
mol
amount of substance
n
SI: Base and Derived Units
The meter stick is an instrument that measures one meter between marked ends.
Figure 1.2.2 The meter stick
In SI units the measured length is 1 meter. The information on length can be expressed in several
alternate units in the same unit system.
In the same SI units it is also :
● 100 centimeters
● 1000 millimeters
● 106 microns
● 109 nanometers
It is important to use the appropriate unit defined through context and practice. For example the
nanometer is useful in discussing the wavelength of light. Microns may be appropriate when
discussing products in the semiconductor industry. Millimeters may be appropriate for discussing
blood vessels. The BASIC unit of length is the meter (alternately metre)
In addition to BASIC units many physical quantity can be expressed through DERIVED units.
Derived units imply a combination of basic units. For example
The density of water is 1000 kilogram/meter3 [kg/m3]. This is a derived unit. Every physical quantity
can be associated with a corresponding basic or derived unit in the SI system of units.
Consider another example of the linear elastic spring from physics. We will apply a force (F) to stretch
it through a distance (s). Not only is the force equal to spring constant (k) times the deflection (s), the
units for the force must equal to the product of the units for the spring constant and the deflection in
base units.
F=ks
k = F/s
or
or
[N] = [k] [s]
[k] = [N/m]
Figure 1.2.3 A linear spring - derived units
Units used in Mechanics
Table 1.2.2 SI Units in Mechanics
Quantity
Unit
Unit Symbol
length
mass
time
acceleration
meter
kilogram
second
meter per second
squared
radian
square meter
kilogram per cubic meter
Joule
Newton
Newton meter
m
kg
s
m/s2
angle
area
density
energy
force
moment of a force
(torque)
pressure
stress
volume (solid)
volume (liquid)
work
rad
m2
kg/m3
J
N
Nm
Newton per square meter N/m2
Newton per square meter N/m2
cubic meter
m3
liter
10-3 m3
Joule
J
Alternate Symbol
1 Joule = 1Nm
1 N: 1 kg-m/s2
Pa: Pascal
Pa: Pascal
L
1 Joule = 1Nm
There are two corollaries that we will respect:
1. Any law or equation we develop must have the same unit on both side of the equal sign
2. Every term in the equation must also have the same units
1.2.2 US System of Units
The US customary unit system has some overlap with the British Imperial system. It was defined in
modern form in 1959. In 1988 Congress passed a bill for confirming adoption of the SI system but left
the timing indefinite. So the adoption is in transition. We will refer to this system as the US system.
We can recognize base units and derived units like we did for SI units. We also need a way to
translate information between the two systems. However the US system does not have units available
translate information between the two systems. However the US system does not have units available
for a lot of physical quantities that have been important recently.
Unlike the SI system where units for the same quantity , for example length, can vary by orders of 10
(millimeter, centimeter, meter, kilometer), the units for the same quantity in the US have no
connection. The units for length in the US system of units can be expressed in several ways. This is
displayed in Table 1.2.3.
Table 1.2.3 Various US units for length (source Wikipedia)
Unit
1 inch (in)
1 foot (ft)
1 yard (yd)
1 mile (mi)
Survey
1 link (li)
1 (survey) foot (ft)
1 rod (rd)
1 chain (ch)
1 furlong (fur)
1 survey (or statute) mile
(mi)
1 league (lea)
Nautical
1 fathom (ftm)
1 cable (cb)
1 nautical mile (NM or nmi)
Alternate Unit
SI Equivalent
12 in
3 ft
1760 yd
2.54 cm
0.3048 m
0.9144 m
1.609344 km
33⁄ ft or 7.92
50
1200⁄
3937 m
25 li or 16.5 ft
4 rd
10 ch
8 fur
0.2011684 m
0.3048006 m
5.029210 m
20.11684 m
201.1684 m
1.609347 km
3 mi
4.828042 km
2 yd
120 ftm or 1.091 fur
8.439 cb or 1.151 mi
1.8288 m
219.456 m
1.852 km
in
Similar tables can be established for area, volume, mass, cooking measures etc.
US: Base and Derived Units
Table 1.2.4 lists some of the base and derived units in the US system. It also shows the corresponding
SI Units and the corresponding conversion factors for equivalence.
Table 1.2.4 US Units in Statics and the corresponding SI Units with conversion factors
Quantity
US Unit
Equivalent SI Unit
length
mass
mass
time
acceleration
angle
area
density
energy
force
moment of a force (torque)
pressure
pressure
Stress
volume (solid)
volume (liquid)
work
1 feet [ft]
1 slug [slug]
1 pound mass [lbm]
1 second [s]
1 [ft/s2]
1 [rad]
1 [ft2]
1 [slug/ft3]
1 [ft-lb]
1 pound force [lb]
1 [lb-ft]
1 [lb/ft2]
1 [lb/in2], [psi]
1 [lb/in2], [psi]
1 [ft3]
1 gallon [gal]
1 [ft-lb]
0.3048 [m]
14.594 [kg]
0.4536 [kg]
1 [s]
0.3048 [m/s2]
1 [rad]
0.0929 [m2]
515.38 [kg/m3]
1.356 [J ]
4.448 [N]
1.356 [Nm]
47.88 [Pa]
6.895 x103 [Pa]
6.895 x103 [Pa]
0.02832 [m3]
3.785 Liter [L]
1.356 [J or Nm]
Conversion between Systems of Units
To work in both system of units you need to how to convert between them. It is a simple arithmetic
process with both the numbers and the units themselves.
For example, in USA, the speed at which you are traveling is usually expressed in miles per hour –
say 25 [mph]. In basic units you will need to express the speed in feet per second [ft/s]. In the SI
system the corresponding base units are meter per second [m/s], while the popular expression for the
speed is kilometer per hour [km/hr].
In order to convert from [mph] to feet per second [ft/s] it we need to know that
1 [mile] is 5280 feet [ft]
1 [hour] is 3600 second [s]
To convert from US units [mph] to SI units [m/s]
To from [m/s] to [km/hr] or [kph] we can use the conversion factor listed in Table 1.2.4
Accuracy
For the conversion of units and other calculations in this book you are likely to use a calculator and
sometimes you will get a long number. Further additional operations with the number you will continue
to obtain these long numbers. For example: cos (25) = 0.906307787 and 2*cos(25) = 1.812615574.
Do we need to know the answer to so many places after the decimals? Let us say you are expressing
your lengths in centimeters.
If you are building desk and chairs you need one place after the decimal for a well built product.
If you are building cars you need two places after the decimal for a good looking car.
If you are building airplanes you need three places after the decimal for a well functioning aircraft.
The number of decimal points is related to the accuracy of your calculations. In engineering problems
the information is usually considered to be accurate to 0.2%. For example if you are using 2500 [N] for
the weight of the pumpkin, a 0.2% error in this estimate is expressed as 2500 ± 5 [N]. Your
calculations are only as good as the accuracy of your information. It does not make sense to report
answers to the ninth place after the decimal. In this eBook we will report our calculations to two places
after the decimal – unless otherwise warranted.
You can choose also to round off your value to the second decimal place. If the number in the third
place after the decimal is greater than or equal to 5 then the second place number is increased by 1,
place after the decimal is greater than or equal to 5 then the second place number is increased by 1,
except if the second place number is 9. You will then have to change the first place number or not
round off. You must be consistent in applying round off. Hence cos (25) = 0.91 and 2*cos(25) = 1.81.
Dimension
How do we know that the units for acceleration in basic SI units is [m/s2]? We can always look it up in
a table (Table 1.2) , in a book ,or obtain it by searching the Internet. A more fundamental idea is that
every physical quantity is associated with a Dimension.
This is very important in the subject of fluid mechanics. Also, this is very different from the concept of
Dimension (D) that is geometrically associated with a problem (like 1 D problem, 2D problem , or 3D
problem) in science and engineering, that progressively includes more information, more equations,
more work, and is more difficult to solve. We will come this path in this book later. A simple way to
understand Dimension that is associated with the units of a physical quantity is to accept that are a set
of basic dimensions – (Table 1.2.5).
Table 1.2.5 Basic Dimensions
Quantity
Symbol
length
mass
time
electric current
thermodynamic temperature
luminous intensity
amount of substance
L
m
t
I
T
Iv
n
Let us consider speed as an example. We know that the average speed is computed by dividing the
distance traveled by the time for the travel.
The dimension for speed is [L/t]. The units for speed can be obtained by substituting appropriate units
for the dimensions in the definition. Units for speed : [m/s]; [km/h]; [ft/s]; [ft/h]; [miles/h]; [in/s]; etc.
Similarly acceleration is
Units for acceleration: [m/s2]; [ft/s2]; [in/s2]; etc.
Example 1.2.1. Using the Newton’s law of gravitation , find (a) the dimension of the universal
gravitational constant (G); (b) Its basic units in the SI system; (c) Its basic unit in the US system.
The Newton’s law of gravitation, where F is the force between the two masses, m1 and m2, and r is
the distance between the centers, can be expressed as:
Assumptions: To proceed we need to know the dimension of the force. We can look up a book or we
can obtain it by using Newton’s second law, F = ma (from physics).
Solution:
Dimension of force, F:
(a) Dimension of G :
(b) [G] = [m3/kg-s2 ] or [m3 kg-1 s-2]
(c) [G] = [ft3/slug-s2] = [ft3 slug-1 s-2]
Note: We work with symbols the same was as we work with numbers.