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Dimensional Analysis and Its Applications

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  • Checking the Dimensional Consistency of Equations
  • Deducing Relation among the Physical Quantities

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Dimensional Analysis and its Applications

Dimensional Analysis:-

  • Only those physical quantities which have same dimensions can be added and subtracted. This is called principle of homogeneity of dimensions.

  • Dimensions can be multiplied and cancelled like normal algebraic methods.

  • In mathematical equations, quantities on both sides must always have same dimensions.

  • Arguments of special functions like trigonometric, logarithmic and ratio of similar physical

  • Quantities are dimensionless.

Equations are uncertain to the extent of dimensionless quantities.
Example Distance = Speed x Time. In Dimension terms, [L] = [LT-1] x [T]
Since, dimensions can be cancelled like algebra, dimension [T] gets cancelled and the equation becomes [L] = [L].

Homogeneity Principle

If the dimensions of left hand side of an equation are equal to the dimensions of right hand side of the equation, then the equation is dimensionally correct. This is known as homogeneity principle.
Mathematically [LHS] = [RHS]

Limitations of Dimensional Analysis
1. Dimensionless quantities cannot be determined by this method. Constant of proportionality cannot be determined by this method. They can be found either by experiment (or) by theory.
2. This method is not applicable to trigonometric, logarithmic and exponential functions.
3. In the case of physical quantities which are dependent upon more than three physical quantities, this method will be difficult.
4. In some cases, the constant of proportionality also possesses dimensions. In such cases, we cannot use this system.
5. If one side of the equation contains addition or subtraction of physical quantities, we cannot use this method to derive the expression.

Applications:-
Dimensional analysis is very important when dealing with physical quantities. In this section, we will learn about some applications of the dimensional analysis.

Fourier laid down the foundations of dimensional analysis. The Dimensional formulas are used to:
1. Verify the correctness of a physical equation.
2. Derive a relationship between physical quantities.
3. Converting the units of a physical quantity from one system to another system.

Checking the Dimensional Consistency
As we know, only similar physical quantities can be added or subtracted, thus two quantities having different dimensions cannot be added together. For example, we cannot add mass and force or electric potential and resistance.

For any given equation, the principle of homogeneity of dimensions is used to check the correctness and consistency of the equation. The dimensions of each component on either side of the sign of equality are checked, and if they are not the same, the equation is considered wrong.

Let us consider the equation given below,
`1/2`mv2 = mgh

The dimensions of the LHS and the RHS are calculated
LHS: [M] [LT-1]2 = [M][L2T-2] = [ML2T-2]
RHS: [M] [LT-2][L] = [ML2T-2] 
As we can see the dimensions of the LHS and the RHS are the same, hence, the equation is consistent.

Deducing the Relation among Physical Quantities
Dimensional analysis is also used to deduce the relation between two or more physical quantities. If we know the degree of dependence of a physical quantity on another, that is the degree to which one quantity changes with the change in another, we can use the principle of consistency of two expressions to find the equation relating these two quantities. This can be understood more easily through the following illustration.

Example: Derive the formula for centripetal force F acting on a particle moving in a uniform circle.
As we know, the centripetal force acting on a particle moving in a uniform circle depends on its mass m, velocity v and the radius r of the circle. Hence, we can write
F = ma vb rc

Writing the dimensions of these quantities
[MLT-2] = Ma[LT-1]bLc
[MLT-2] = MaLc+bT-b
As per the principle of homogeneity, we can write,
a = 1, b + c = 1 and b = 2

Solving the above three equations we get, a = 1, b = 2 and c = -1.
Hence, the centripetal force F can be represented as,
F = Km1v2r-1
F = k`"mv"^2/"r"`

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