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2. Dimension

When a quantity is expressed in terms of the basic quantities, it is written as a product of different powers of the basic quantities. The exponent of a basic quantity that enters into the expression is called the Dimension of the quantity in that base, For Example;

Thus, the dimensions of force are 1 in mass, 1 in length and -2 in times. In this type of calculation the magnitudes are not considered.

The dimensions of a physical quantity are the powers to which the fundamental units of mass, length and time have to be raised to obtain its unit.

Dimensional Formula and Equations:
The expression derived for force F = [MLT-2] is said to its dimensional formula. The dimensional formula of a physical quantity may be defined as the expression which indicates which of the fundamental units of mass; length and time enter into the derived unit of that quantity and with what power.
The expression [F] = [MLT-2] is known as dimensional equation.
Then the equation obtained when a physical quantity is equated with its dimensional formula is known as dimensional equation.

Use of Dimensional Equation
The dimensional equations have got following three uses;
i. To test the correctness of a physical equation.
ii. To derive the relation between different physical quantities involved in a physical phenomenon.
iii. Conversion of one system of units to another.

To Check the Correctness of the Physical Equation
Checking the correctness of a physical equation, the principle of homogenizing of dimensions is used. According to this principle, the dimensions of the fundamental quantities (mass, length and time) are same in each and every terms on either side of the equation.
The dimensions of all the terms in an equation must be identical. One can add or subtract similar physical quantities. A velocity cannot be added to a force or an electric current cannot be subtracted from the temperature. This simple principle is called principle of homogeneity of dimensions in an equation. It is an extremely useful method to check whether an equation is correct or not. If the dimensional of all the terms are not same, the equation must be wrong.
Let us check the equation;

To Derive the Relation between Various Physical Quantities
To derive the relation of a physical quantity, we should use principle of homogeneity.
In the  case of simple pendulum, let us suppose the time period of oscillation (T) depends on the length of the sring (l), mass of the bob (m) and acceleration due to gravity (g). We assume that, the dependence of time period of their quantities as;

We can deduce that the time period of a simple pendulum in independent of its mass and proportional to the square root of the length and inversely proportional to the square root of the acceleration due to gravity.

To convert a Unit of Physical Quantity from One System to  Another
The value of a physical quantity X, is given by;
X = nu
Where, u is unit of measurement
             n is number of time the unit is contained in the physical quantity, i.e, its numeric value.
If u1 and u2 are units of measurement in two systems and n1 and n2 be the numerical values of the physical quantity then;
u1n1 = u2n2

Convert 1 Joule into ergs.
Let, u1 = Unit of work in S.I. system
and, n1 = 1 Joule
Since, W = F× S = [MLT-2][L] = [ML2T-2]