Dimensional Analysis

To solve the mathematical problems of physical quantities, it is important to have a brief knowledge of units and dimensions. The basic concept of dimensions is that only those quantities can be added or subtracted which have the same dimension. This concept helps us to derive relationships between physical quantities.

Dimensional analysis is the study of the relation between physical quantities based on their units and dimensions. It is used to convert a unit from one form to another. While solving mathematical problems, it is necessary to keep the units the same to solve the problem easily.

Do you know what is the significance of dimensional analysis? Well! In engineering and science, dimensional analysis describes the relationships between different physical quantities based on their fundamental qualities such as length, mass, time, and electric current, and units of measure like miles v/s kilometres, or pounds v/s kilograms.

In other words, in Physics, we study two types of physical quantities, i.e., fundamental and derived. The seven fundamental units include mass, length, amount of substance, time, luminous intensity, and electric current. However, if we combine two or more fundamental units, we get derived quantities.

For examples, we denote [M] for mass, [L] for length, and so on. Similarly, for speed, which is a derived quantity given by distance/time, we denote it with [M]/[L] or [ML^-1]. This is how we derive the dimensional formula of various quantities. 

The conversion factor used is based on the unit that we desire in the answer. Further, we will derive the dimensional formula of various quantities on this page.

How to Perform Dimensional Analysis?

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Unit Conversion

Dimensional analysis is also called a Unit Factor Method or Factor Label Method because a conversion factor is used to evaluate the units.

For example, suppose we want to know how many meters there are in 4 km.

Normally, we calculate as-

1 km = 1000 meters

4 km = 1000 × 4 = 4000 meters

(Here the conversion factor used is 1000 meters)

Principle of Homogeneity of Dimensional Analysis

This principle depicts that, “the dimensions are the same for every equation that represents physical units. If two sides of an equation don’t have the same dimensions, it cannot represent a physical situation.”

For example, in the equation

[M^aL^bT^c] = MxLyTz

As per this principle, we have

a = x,

b = y, and

c = z

Example of Dimensional Analysis

For using a conversion factor, it is necessary that the values must represent the same quantity. For example, 60 minutes is the same as 1 hour, 1000 meters is the same as 1 kilometre, or 12 months is the same as 1 year.

Let us try to understand it in this way. Imagine you have 15 pens and you multiply that by 1, now you still have the same number of 15 pens. If you want to find out how many packages of the pen are equal to 15 pens, you need the conversion factor.

Now, suppose you have a packaged set of ink pens in which each package contains 15 pens. Let's consider that you have 6 packages. To calculate the total pens, you have to multiply the number of packages by the number of pens in each package. This comes out to be:

15 × 6 = 90 pens

Some other examples of conversion factors that are used in day to day life are:

• A simple example: the time taken by a harmonic oscillator.

• A complex example: the energy of vibrating conduction or wire.

• A third example: demand versus capacity for a disk that is rotating.

Applications of Dimensional Analysis

Dimensional analysis is an important aspect of measurement, and it has many applications in Physics. Dimensional analysis is used mainly because of five reasons, which are:

• To check the correctness of an equation or any other physical relation based on the principle of homogeneity. There should be dimensions on two sides of the equation. The dimensional relation will be correct if the L.H.S and R.H.S of an equation have identical dimensions. If the dimensions on two sides are incorrect, then the relations will also be incorrect.

• Dimensional analysis is used to convert the value of a physical quantity from one system of units to another system of units.

• It is used to represent the nature of physical quantity.

• The expressions of dimensions can be manipulated as algebraic quantities.

• Dimensional analysis is used to derive formulas.

Limitations of Dimensional Analysis

Some major limitations of dimensional analysis are:

• Dimensional analysis doesn't provide information about the dimensional constant.

• Dimensional analysis cannot derive trigonometric, exponential, and logarithmic functions.

• It doesn't give information about the scalar or vector identity of a physical quantity.

Example of Dimensional Formula: Derivation for Kinetic Energy

The dimensional formula of any physical entity is the mathematical expression representing the powers to which the fundamental units (mass M, length L, time T) are to be raised to obtain one unit of a derived quantity. 

Let us now understand the dimensional formula with an example. Now, we know that kinetic energy is one of the fundamental parts of Physics, hence its formula plays a vital role in many fields of Physics. So, let us derive the dimensional formula of kinetic energy. 

The kinetic energy has a dimensional formula of,

[ML²T^-2]

Where,

M = Mass of the object

L = Length of the object

T = Time taken

Derivation

Kinetic energy (K.E) is given by = \[\frac {1} {2}\]  [Mass x Velocity²]---- (I)

The dimensional formula of Mass is =[M¹L⁰T⁰]--- (ii)

We know that,

Velocity = Distance × Time^-1

= L x T^-1 (dimensional formula)

Velocity has a dimensional formula [M⁰L¹T^-1]----- (iii)

On substituting equation (ii) and iii) in the above equation (i) we get,

Kinetic energy (K.E) = \[\frac {1} {2}\]  [Mass x Velocity²]

Or, K.E = [M¹L⁰T⁰] [M⁰L¹T^-1]² = [M^{(0 +1)}L^{(1 + 1)} T^{(-1 + -1)}]

Therefore, on solving, we get the dimensional formula for kinetic as [M⁰L²T^-2].

From this context, we understand that in dimensional analysis a set of units helps us establish the form of an equation and to check that the answer is free of even minute errors. 

Solved Example

1. Find out how many feet are there in 300 centimeters (cm).

Ans. We need to convert cm into feet.

Firstly, we have to convert cm into inches, and then inches into feet, as we can't directly convert cm into feet.

The calculation of two conversion factors is required here:

Then, 300 cm = 300 x \[\frac {1} {30.48}\]    feet

= 9.84 feet