Derived units
We can define all the derived units 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

Express in base units

Force

newton

N

Kgm/sec^{2}

Work

joules

J

Kgm^{2}/sec^{2}

Power

watt

W

Kgm^{2}/sec^{3}

Pressure

pascal

Pa

Kg m^{1}/S^{2}

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 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.
(4) Not more than one solid’s is used i.e. all units of numerator written together before the ‘/’ sign and all in the denominator written after that.
i.e. It is 1 ms^{2} or 1 m/s^{2} not 1m/s/s.
(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 show 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/L^{3} = 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^{3}T^{0}
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 12 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.
At askIITians we provide you free study material on units and dimensions so that you get all the professional help needed to get through IIT JEE and AIEEE easily. AskIITians also provides live online IIT JEE preparation and coaching where you can attend our live online classes from your home!