UNITS AND DIMENSIONS

By international agreement a small number of physical quantities such as length, time etc. are chosen and assigned standards. These quantities are called ‘base quantities’ and their units as ‘base units’. All other physical quantities are expressed in terms of these ‘base quantities’. The units of these dependent quantities are called ‘derived units’.

The standard for a unit should have the following characteristics.

(a) It should be well defined.
(b) It should be invariable (should not change with time)
(c) It should be convenient to use
(d) It should be easily accessible

The 14th general conference on weights and measures (in France) picked seven quantities as base quantities, thereby forming the International System of Units abbreviated as SI (System de International) system.

Base quantities and their units

The seven base quantities and their units are

Base quantity             Unit            Symbol

Length                     Metre                   M

Mass                         Kilogram            Kg

Time                          Second             Sec

Electric current            Ampere             A

Temperature                Kelvin              K

Luminous intensity       Candela           Cd

Amount of substance    Mole               Mole

DERIVED UNITS

derived units  are those units which can be expressed in terms of base units. For example, speed is defined to be the ratio of
distance to time.

Unit of Speed = (unit of distance (length))/(unit of time)  = m/s = ms-1  (Read as metre per sec.)

SOME DERIVED SI UNITS AND THEIR SYMBOLS

Quantity        Unit        Symbol        Expressed  in base units

Force          newton         N                     Kg-m/sec2

Work           joules          J                      Kg-m2 /sec2

Power         watt            W                     Kg-m2 /sec3

Pressure     pascal          Pa                     Kg m-1/S2

IMPORTANT

The following conventions are adopted while writing a unit.

(1) Even if a unit is named after a person the unit is not written IN capital letters. i.e. we write joules not Joules.

(2) For a unit named after a person the symbol is a capital letter e.g. for joules we write ‘J’ and the rest of them
are in lowercase letters e.g. seconds is written as ‘s’.

(3) The symbols of units do not have plural form i.e. 70 m not 70 ms or 10 N not 10Ns.

(5) Punctuation marks are not written after the unit
e.g. 1 litre = 1000 cc not 1000 c.c.

It has to be borne in mind that SI system of units is not the only system of units that is followed all over the
world. There are some countries (though they are very few in number) which use different system of units. For
example: the FPS (Foot Pound Second) system or the CGS (Centimeter Gram Second) system.

DIMENSIONS

The unit of any derived quantity depends upon one or more fundamental units. This dependence can be expressed with
the help of dimensions of that derived quantity. In other words, the dimensions of a physical quantity shows how its
unit is related to the fundamental units.

To express dimensions, each fundamental unit is represented by a capital letter. Thus the unit of length is denoted
by L, unit of mass by M. Unit of time by T, unit of electric current by I, unit of temperature by K and unit of
luminous intensity by C.

Remember that speed will always remain distance covered per unit of time, whatever is the system of units, so the
complex quantity speed can be expressed in terms of length L and time T. Now, we say that dimensional formula of
speed is LT-1. We can relate the physical quantities to each other (usually we express complex quantities in terms
of base quantities) by a system of dimensions.

Dimension of a physical quantity are the powers to which the fundamental quantities must be raised to represent the
given physical quantity.

Example:

Density of a substance is defined to be the mass contained in unit volume of the substance.

Hence, [density] = ([mass])/([volume]) = M/L3 = ML-3

So, the dimensions of density are 1 in mass, -3 in length and 0 in time.

Hence the dimensional formula of density is written as

[ρ]= ML-3T0

It is to be noted that constants such as ½ π, or trigonometric functions such as “sin wt” have no units or
dimensions because they are numbers, ratios which are also numbers.

Units and Dimensions are important from IIT JEE perspective. Objective questions are framed on this section. AIEEE
definitely has 1-2 questions every year directly on these topics. Sometimes both IIT JEE and AIEEE do not ask
questions on units and dimensions directly but they change units and involve indirect application. So it’s very
important to master these concepts at early stage as this forms the basis of your preparation for IIT JEE and AIEEE
Physics.