Unit Systems
Various systems of units have been used the field of engineering. The systems used differ between countries. For example, the United Kingdoms previously used the gravitational system of feet, pounds, and seconds (FPS) whilst most other European countries used the metric absolute system. However, from 1960, a modern version of the metric system called the International System of Unit (SI) has been adopted by most countries worldwide.
Basic Units
The SI measurement system involves seven basic units that has been fixed arbitrarily, Table 1. From these basic units, a number of derived units are formed. These are mere combinations of the basic units.
Table 1.1. Basic Units | ||
Quantity | Unit | Symbol |
Length | meter | m |
Mass | kilogram | kg |
Time | second | s |
Electrical Current | Ampere | A |
Amount Of Substance | mol | mol |
Temperature | kelvin | K |
Light intensity | candella | cd |
Note that the basic unit for mass is kilogram, not gram. The kilogram is the only unit defined using a prefix. Prefixes are used for other units only when the magnitudes of the quantities being measured are too large or too small compared to the quantity used in practical situations. The kilogram is used as the basic measurement because the unit gram is too small for use in daily situations. The types of prefixes used are given in Table 1.2.
Table 1.2. Types Of Prefixes | ||
Power | Prefix | Symbol |
10^{12} | tera | T |
10^{9} | giga | G |
10^{6} | mega | M |
10^{3} | kilo | K |
10^{-2} | centi | C |
10^{-3} | milli | M |
10^{-6} | micro | Îœ |
10^{-9} | nano | N |
10^{-12} | piko | P |
In basic Engineering Mechanics, only three basic units are used, namely meter, kilogram, and second. Other units used are units derived from those basic units.
Derived Units
As stated earlier, the SI measurement system has only seven basic units of measurement.From the basic units, other units are derived. These units are known as derived units. There are derives units that have been given specific names. For example, the unit kg.m.s^{-2} is named newton (N) and the unit N.m^{-2} is called pascal (Pa).
A derived unit is formed based on the definition of the quantity it measures. For example, the accelerationa is defined as the rate of change in velocity v with respect to time t, whilst the velocity is defined as the rate of change of position s with respect to time t. Hence
v = ds/dt; a = dv/dt
The units of v are the unit of displacement (meter) divided by the unit of time (seconds), i.e. m/s or m.s^{-1}. Furthermore, the unit for a is obtained as unit of v divided by the unit of time, i.e.
unit of a = (m/s)/s = m.s^{-2}
Besides,the derived units are also obtained from basic equations that defined the quantities measured by the respective units. For example, the unit for force is obtained from the definition of Newtonâ€™s second law
F = ma
whereF is the force acting on a body of mass m and causes it to move with acceleration a. The unit of measurement for a force is the same as that of mass multiplied by the unit of measurement for acceleration. In the SI system, the unit for force is
Unit of F = kg(m/s^{2}) = kg.m/s^{2} = newton (N)
In this book, the derived units are written using the dot to separate the basic units that it is formed of. This will avoid the confusion that may arise because of prefixes. For example, millinewton is written mN whilst meter-Newton is written m.n and ms is millisecond whilst m.s is meter-second.
Earlier on, it was stated that there are derived units that have been given specific names. Such units which are commonly used in engineering mechanics are given in Table 1.3.
Table 1.3. Derived Units | |||
Quantity | Derived Unit | Name | Symbol |
Force | kg.m/s^{2} | newton | N |
Work/Energy | N.m | joule | J |
Power | J.s^{-1} | watt | W |
Homogeneity Of Units
In any equation that describes a physical process, every term of the equation must be expressed in the same unit of measurement. Without homogeneity of dimensions, those terms cannot be handled simultaneously when replaced by numerical values. For example, the expression for potential energy function is given as
V = (1/2)kx^{2} + mgh
where (1/2)kx^{2 }is the elastic potential energy obtained based on the stiffness k of an elastic member and the deformation x it undergoes whilst mgh is the gravitational potential energy obtained based on the weight W=mg of a body and its displacement h from the measurement datum.^{(3)}
The quantity V is measured by using the unit for energy N.m or joule (J). Both the terms on the right of the equal sign in the above equation must have the unit joule. Say, L is the unit of measurement for length, M is for mass, and T for time in any system of measurement. By using the fact that newton itself is a derived unit, i.e. the mass unit M multiplied by the unit for acceleration, and the unit for acceleration itself is the derived unit L/T2, i.e. unit for distance L divided by the square of the unit for time T, the equation produces the unit expression
(ML/T^{2}).L = [(ML/T^{2}).(1/L)]L^{2} + M(L/T^{2})L
ML^{2}/T^{2} = ML^{2}/T^{2} + ML^{2}/T^{2}
The homogeneity of units is obtained since every term has the same unit, ML^{2}/T^{2}.