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Dimensional Analysis - Meaning, Examples, FAQs

Dimensional Analysis - Meaning, Examples, FAQs

Edited By Vishal kumar | Updated on Nov 12, 2024 06:03 PM IST

Dimensional analysis is a method used to check the correctness of the equation and simplify complex equations. This article covers what is dimensional analysis, the application of dimensional analysis, uses of dimensional analysis, what is principle of homogeneity of dimensions, limitations of dimensional analysis

This Story also Contains
  1. What Is Principle of Homogeneity Of Dimensions:
  2. What is Meant by Dimension?
  3. What is Dimensional Analysis Class 11?
  4. Application of Dimensional Analysis:
  5. Uses Of Dimensional Analysis Class 11
  6. Examples of Dimensional Analysis:
  7. Limitations Of Dimensional Analysis
  8. Problem Solving Strategy of Dimensional Analysis
Dimensional Analysis - Meaning, Examples, FAQs
Dimensional Analysis - Meaning, Examples, FAQs

What Is Principle of Homogeneity Of Dimensions:

The principle of homogeneity of dimensions says that “ In any physical mathematical equation the dimensions of each term appearing in the equation are the same on each side of that equation”. This is called the principle of homogeneity.

What is Meant by Dimension?

In physics, any physical quantity can be expressed in terms of fundamental units, and the representation of a physical quantity in terms of fundamental units is called the dimension of the physical quantity.

Following are the symbols for fundamental units used in Dimensional Analysis class 11.

Name of the UnitSymbol
Length[L]
Time[T]
Mass[M]
Electric current[A]
Temperature[K]
Amount of substance[mol]
Intensity of light[cd]

What is Dimensional Analysis Class 11?

When we represent each physical quantity of a mathematical equation in its dimensional form then analysis of dimensions to determine whether a given equation is correct or not dimensionally is known as dimensional analysis.

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Application of Dimensional Analysis:

Application of dimensional analysis in various fields are:

  1. Dimensional Analysis is applicable in the physics field for uses like unit conversion, and checking the correctness of equations.
  2. In thermodynamics for deriving dimensionless numbers like Reynold's number.
  3. In environmental science by helping researchers develop equations to predict natural occurrences like cyclones.

Uses Of Dimensional Analysis Class 11

The most widely uses of dimensional analysis are mentioned as:

  1. The validity of a physical equation can be checked using dimensional analysis.
  2. Dimensional analysis is used to determine the dimensions of any unknown variable’s dimension in a given physical equation.
  3. Units can be converted from one system to another using dimensional analysis.
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Examples of Dimensional Analysis:

When we analyze the physical equation by using their dimensions such as Distance, velocity, and Time relation.

We know that the Dimension of physical quantity Velocity is [LT1] while the Dimensions of quantity Time is [T] and from the relation, we know that Distance=Velocity×time so we can find the dimension of quantity distance by multiplying the dimensions of Velocity and Time and we get, Dimension of Distance as LT1 T=[L]. Hence, this is a simple example of dimensional analysis showing a valid physical equation can be checked by using dimensional analysis.

Limitations Of Dimensional Analysis

Some of the limitations of dimensional analysis are:

  1. Cannot determine dimensionless quantities or constants
  2. It depends on known variables only during the analysis.
  3. Not able to determine the addition and multiplication of terms with the same dimensions

Problem Solving Strategy of Dimensional Analysis

1. Check the correctness of the equation f=mv2r

We will use the dimensional analysis and principle of homogeneity which can be used If the dimension of quantity ‘f’ represents force and the dimension of quantity mv2r where m represents mass, v represents velocity and r represents radius are the same then the given equation will be correct dimensionally.

The force has a dimension of mass×acceleration so, the dimension of ‘f’ can be written as M[LT2] or Dimension of force f=[MLT2]

Now, the dimension of radius which is simply the distance will be the Dimension of r=[L], and for mass Dimension of mass m=[M], and velocity Dimension of velocity v=[LT1] Now, on putting these dimensions on the right-handed part of the equation which is mv2r we get, [M][LT1]2[L] on solving we get, [MLT2]

Hence, the dimension of quantity f is the Dimension of force f=[MLT2] and the dimension of quantity mv2r is, [MLT2] which are the same, so from the principle of homogeneity this physical equation has dimensions the same on both sides so, this is a correct equation.

2. Check the correctness of the equation v2u2=2aS

We will use the dimensional analysis and principle of homogeneity which can be used If the dimension of quantity v2u2 where v, u represents velocity and has a dimension of [LT1] and the dimension of quantity 2aS where a represents acceleration and has a dimension of [LT2] and S represent distance which has a dimension of [L] is the same, then the equation will be correct dimensionally. Now, using these dimensions let us find the dimension of quantity v2u2 as [LT1]2=[L2 T2] since u and v are both velocities so their difference is also a velocity.

Now, let us find the dimension of quantity 2aS as 2 is a dimensionless constant, and the multiplication of a and S will have the dimension of LT2 L=[L2 T2] so, we see that both parts have the same dimension of [L2 T2] so, according to the principle of homogeneity, both parts have the same dimension which shows, the given equation is correct.

3. Dimensional analysis of S=ut+0.5at2

We will do the Dimensional analysis of S=ut+12at2 by using the principle of homogeneity to check the correctness of the following equation S=ut+12at2 where S represents distance having a dimension of [L] which is on the left side of the equation. Now coming to the right side, we have u which is velocity having the dimension of [LT1] and t is time having the dimension of [T] so the net dimension of the product of velocity u and time t will be LT1 T=[L] and similarly another part of right side equation is 0.5at2 where a is acceleration having the dimension of [LT2] and the dimension of the square of t is [T2] so the net dimension of term 0.5at2 will be [L]. Hence, the net right-sided equation has a dimension of L+L=[L] since the addition of two dimensions is the same dimension. Hence, the left side and right side of the equation have the same dimension of [L] so, by the principle of homogeneity, the equation S=ut+0.5at2 is correct.

Frequently Asked Questions (FAQs)

1. Which of the following is dimensional constant?

(A) Force F

(B) Velocity V

(C) Planck Constant h

(D) Mass M

A dimensional constant is a constant quantity in physics that has a constant numerical value and has proper dimensions. Force is not a constant quantity as it depends upon mass and velocity is also not a constant quantity as it depends upon distance and time similarly mass can have any numerical value but Plank constant h has a fixed value of h=6.62×10-34Js. and Planck constant has also a fixed dimension of [ML2T-1] Hence, (C) Planck Constant h is a dimensional constant.

2. What is dimensional analysis?

In physics, Dimensional analysis is a method of checking the validity and as well as finding the dimensions of any physical term in a mathematical equation consisting of physical parameters using the principle of homogeneity and dimensions of physical quantities is known as Dimensional analysis.

3. State the principle of homogeneity of Dimensions.

The principle of homogeneity of dimensions states that, If we have a physical mathematical equation then the dimensions of each term in the left side of the equation and dimension of each term in the right side of the equation will always be the same. This principle is known as the principle of homogeneity in dimensions.

4. What is meant by Dimension of physical quantity in physics?

In physics, every physical quantity can be expressed in terms of the fundamental units needed to represent it completely; the representation of a physical quantity with the fundamental units is known as the dimension of that physical quantity.

5. Write two uses of Dimensional analysis.

Two most important uses of dimensional analysis is mentioned as:

  1. Dimensional analysis uses the principle of homogeneity which enables us to check the correctness of any physical mathematical equation.

  2. Dimensional analysis method is also used to convert one system of units to another system of unit by comparing the dimensions of a physical quantity in each units system respectively.

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