UNITS AND DIMENSIONS - DIMENSIONAL FORMULA

Dimension:
    Dimensions of a physical quantity are the powers to which the base quantities are raised to represent one unit of that quantity. The dimensions of a physical quantity describe the nature of the factors by which that quantity depends on the base or fundamental quantity.

Dimensional Analysis
    Dimensional analysis is the practice of checking relations between physical quantities by identifying the dimensions of the physical quantities. These dimensions are independent of the numerical multiples and constants and all the quantities in the world can be expressed as a function of the fundamental dimensions.
    The seven fundamental quantities are represented by Capital letters & enclosed in square brackets [ ] to represent its dimensions.
1.    Dimension of Length is described as [L]
2.    Dimension of time is described as [T]
3.    Dimension of mass is described as [M]
4.    Dimension of electric current is described as [A]
5.    Dimension of the amount of quantity can be described as [mol].
6.    Dimension of temperature is [K]
7.    Dimension of luminous intensity is [Cd]
Dimensional Formula:
    An expression showing the powers to which the fundamental units are to be raised to obtain one unit of the derived quantity is called Dimensional formula of that quantity.
    Consider a physical quantity Q which depends on base quantities like length, mass, time, electric current, the amount of substance and temperature, when they are raised to powers a, b, c, d, e, and f. Then dimensions of physical quantity Q can be given as:
[Q] = [L^aM^bT^cA^dmol^eK^f]
    It is mandatory for us to use [ ] in order to write dimension of a physical quantity. In real life, everything is written in terms of dimensions of mass, length and time. Look out few examples given below:
It is mandatory for us to use [ ] in order to write dimension of a physical quantity. In real life, everything is written in terms of dimensions of mass, length and time. Look out few examples given below:
1. The volume of a solid is given as the product of length, breadth and its height. Its dimension is given as:
    Volume = Length × Breadth × Height
    Volume = [L] × [L] × [L] (as length, breadth and height are lengths)
    Volume = [L]³
    As volume is independent on mass and time, the powers of time and mass will be zero while expressing its dimensions i.e. [M]⁰ and [T]⁰
    The final dimensional formula of volume will be [M]⁰[L]³[T]⁰ = [M⁰L³T⁰]

2. In a similar manner, dimensions of area will be
    Area = Length × Breadth 
    Area = [L] × [L] (as length& breadth are lengths)
    Area = [L]²
    =>[M⁰L²T]⁰
3. Speed of an object is distance covered by it in specific time and is given as:
    Speed = Distance/Time
    Dimension of Distance = [L]
    Dimension of Time = [T]
    Dimension of Speed = [L]/[T]
    [Speed] = [L]1[T]^-1 = [L¹T^-1] = [M⁰L¹T^-1]
    Velocity also having same dimensional formula as speed.
4.  Acceleration of a body is defined as rate of change of velocity with respect to time, its dimensions are given as:
    Acceleration = Velocity / Time
    Dimension of velocity = [LT^-1]
    Dimension of time = [T]
    Dimension of acceleration will be = [LT^-1]/[T]
    [Acceleration] = [L¹T^-2] = [M⁰LT^-2]
5. Density of a body is defined as mass per unit volume, and its dimension are given as:
    Density = Mass / Volume
    Dimension of mass = [M]
    Dimension of volume = [L³]
    Dimension of density will be = [M] / [L³]
    [Density] = [ML^-3] or [ML^-3T⁰]
6. Force applied on a body is the product of acceleration and mass of the body
    Force = Mass × Acceleration
    Dimension of Mass = [M]
    Dimension of Acceleration = [LT^-2]
    Dimension of Force will be = [M] × [LT^-2]
    [Force] = [MLT^-2]
Dimensional formula for some other physical quantities:
Charge, (q) = [AT]
Specific heat, (s) = [L²T²K^-1]
Gas constant, [R] = [ML²T^-2K^-1 mol^-1]
Rules for writing dimensions of a physical quantity
We follow certain rules while expressing a physical quantity in terms of dimensions, they are as follows

• Dimensions are always enclosed in [ ] brackets
• If the body is independent of any fundamental quantity, we take its power to be 0
• When the dimensions are simplified we put all the fundamental quantities with their respective power in single [  ] brackets, for ex: as in velocity we write [L][T]^-1 as [LT^-1].
• We always try to get derived quantities in terms of fundamental quantities while writing a dimension.
• Laws of exponents are used while writing dimensional formula of physical quantity so basic requirement is a must thing
• If the dimension is written as it is we take its power to be 1, which is an understood thing
• Plane angle and Solid angle are dimensionless quantity, that is they are independent of fundamental quantities

Dimensional Formula for different physical quantities:
    It consumes a lot of time while deriving dimensions of quantities. So, in order to save time, we learn some basic dimensions of certain quantities like velocity, acceleration, and other related derived quantities. 
    For Example, suppose you’re asked to find dimensions of Force and you remember dimension of acceleration is [LT^-2], you can easily state that the dimension of force as [MLT^-2] as force is the product of mass and acceleration of a body.

                                                                                           IMPORTANT DIMENSIONAL FORMULAE

Dimensional Constants:
    The physical quantities which have dimensions and have a fixed value are called as dimensional constants.
Ex: Gravitational Constant (G), Planck’s Constant (h), Universal gas constant (R), Velocity of light in vacuum (c) etc.,
Dimensional variables:
    Dimensional variables are those physical quantities which have dimensions and do not have fixed value.
Ex:  Velocity, acceleration, force, work, power... etc
Dimensionless constants:
    Dimensionless quantities are those which do not have dimensions but have a fixed value.
(a)Dimensionless quantities without units. 
    Ex: Pure numbers, pi(p) etc.,
 (b)Dimensionless quantities with units.
    Ex: Angular displacement - radian, Joule’s constant-joule/calorie, etc.,
Dimensionless variables: 
    Dimensionless variables are those physical quantities which do not have dimensions and do not have fixed value.
Ex: Specific gravity, refractive index, Coefficient of friction, Poisson’s Ratio etc.,
Advantages of Dimensions:

•     Advantages of describing a physical quantity in terms of its dimensions are as follows:
•     Describing dimensions help in understanding the relation between physical quantities and its dependence on base or fundamental quantities, that is how physical quantity rely on mass, time, length, temperature etc.
•     Dimensions are used in dimensional analysis, where we use them to convert from one system to another.
•     Dimensions are used in predicting unknown formulae by just studying how a certain physical quantity depends on base quantities and up to which extent.
•     It makes measurement and study of physical quantities easier.
•     We are able to identify or observe a quantity just because of its dimensions.
•     Dimensions define physical quantities & their existence.

Limitations of Dimensions:

• Besides being a useful quantity, there are many limitations of dimensions, which are as follows:
• Dimensions can’t be used for trigonometric and exponential functions.
• Dimensions never define exact form of a relation.
• We can’t find values of certain constants in physical relations with the help of dimensions.
• A dimensionally correct equation may not be the correct equation always.