Unit and measurement

A unit is defined as a convention to define an amount of physical property in a specific system of units. There are several systems of units with different conventions to express these units. There are units defined for every physical quantity and that physical quantity is expressed in terms of that specific unit. For example, in International System of units we have kilogram for mass, meter for length, seconds/minutes/hours for time, etc. Without a unit a physical property cannot be distinguished or described.

For writing, units are expressed in terms of alphabets, referred to as unit symbols. Like meter is written ‘m’, second as ‘s’, grams as ’g’, etc.

To expresses larger quantities of units in terms of power of tens, certain prefixes are used.

Examples: kilogram for 1000 grams, centimeters for 1/100 of a meter meters, milliseconds for 1/1000 of a second, etc.

International system of units:

Initially there were several systems of units in practice such as the MKS system which used meter for length, kilogram for mass and second for time, CGS system which used centimeter for length, gram for mass and second for time, British Engineering system which used foot for length, slugs for mass and second for time. The diversity in the use of system of units by different fields caused problems in coordination between the units and hence there was a need of a system of units to be used internationally. Therefore, the MKS system was promoted to be called as The International System of Units abbreviated as the SI system. The Si system has seven fundamental physical quantities given below:

Derived units:

Derived units are composed of the fundamental units and are originated from the products and ratios of the fundamental units. Derived units belong to physical quantities that are derived from basic physical properties. Examples are:

Velocity – a physical quantity derived from fundamental quantities of length and time

                velocity=length * time= ms

So the units of velocity is ms-1

Acceleration – a physical quantity derived from velocity and time

    acceleration=velocitytime=ms2

The units of acceleration is ms-2

Force – a physical quantity derived from mass and acceleration

${\rm{force}} = {\rm{mass}}* {\rm{acceleration}} = {\rm{kg\

: m\: }}{{\rm{s}}^{ – 1}}{\rm{\: or\: N}}$

The units of force are kg.m.s-1 or N (Newton)

Work – a physical quantity derived from force and length

                                work=force∗length=NmorJ

The units of work are Nm or J (Joules)

Principle of homogeneity of dimensions:

The principle of homogeneity of dimensions is used to verify the physical laws involving physical quantities. For a relation to be valid and correct the dimensional form of the physical law must be balanced, i.e. the dimensions at both side of the equation must be same. Every physical law that is valid obeys the principle of homogeneity of dimensions.

For example consider following equations:

Equations of motion:

                                           vf=vi+at
          LT−1=LT−1+LT−2T
                     LT−1=LT−1+LT−1


                       s=vit+12at2
   L=LT−1T+12LT−2T2
                                                          L=L+12L


                                       2as=v2f−v2i

2LT−2L=(LT−1)2−(LT−1)2
2L2T−2=L2T−2−L2T−2

Mass – Energy equivalence:

                                                                 E=m.c2
                                  J=kg.(ms−1)2
                    ML2T−2=ML2T−2

All the equations that are given above are dimensionally correct i.e. the dimensions are same at both sides of the equation. It is to be noted that all the constants are ignored in dimensional form and also dimensions are never to be canceled by subtraction, just like we have two similar terms with opposite signs in last equation of motion.