We quantify the size and shape of things using Dimensional Analysis. It helps us study the nature of objects mathematically. It involves lengths and angles as well as geometrical properties such as flatness and straightness. The basic concept of dimension is that we can add and subtract only quantities with the same dimensions. Similarly, two physical quantities can be equal only if they have the same dimensions.
Dimensional Analysis Explained
The study of the relationship between physical quantities with the help of dimensions and units of measurement is termed dimensional analysis. Dimensional analysis is essential because it keeps the units the same, helping us perform mathematical calculations smoothly.
Unit Conversion and Dimensional Analysis
Dimensional analysis is also called Factor Label Method or Unit Factor Method because we use conversion factors to get the same units. To help you understand the stated better, let’s say you want to know how many metres make 3 km?
We know that 1000 metres make 1 km,
Therefore,
3 km = 3 × 1000 metres = 3000 metres
Here, the conversion factor is 1000 metres.
Using Dimensional Analysis to Check the Correctness of Physical Equation
Let’s say that you don’t remember whether
- time = speed/distance, or
- time = distance/speed
We can check this by making sure the dimensions on each side of the equations match.
Reducing both the equations to its fundamental units on each side of the equation, we get
- \(\begin{array}{l}[T]=\frac{[L][T]^{-1}}{L}=[T]^{-1}\,\,(Wrong) \end{array} \)
- \(\begin{array}{l}[T]=\frac{[L]}{[L][T]^{-1}}=[T]\,\,(Right) \end{array} \)
However, it should be kept in mind that dimensional analysis cannot help you determine any dimensionless constants in the equation.
Read more like this:
| Dimensions of Electric Charge | Dimensions of Potential Difference |
| Dimensions of Resistance | Dimensions of Kinematic Viscosity |
| Dimensions of Joule | Dimensions of Pressure Gradient |
Homogeneity Principle of Dimensional Analysis
Principle of Homogeneity states that dimensions of each of the terms of a dimensional equation on both sides should be the same. This principle is helpful because it helps us convert the units from one form to another. To better understand the principle, let us consider the following example:
Example 1: Check the correctness of physical equation s = ut + ½ at2. In the equation, s is the displacement, u is the initial velocity, v is the final velocity, a is the acceleration and t is the time in which change occurs.
Solution:
We know that L.H.S = s and R.H.S = ut + 1/2at2
The dimensional formula for the L.H.S can be written as s = [L1M0T0] ………..(1)
We know that R.H.S is ut + ½ at2 , simplifying we can write R.H.S as [u][t] + [a] [t]2
[L1M0T-1][L0M0T-1] +[L1M0T-2][L0M0T0]=[L1M0T0]………..(2)
From (1) and (2), we have [L.H.S] = [R.H.S]
Hence, by the principle of homogeneity, the given equation is dimensionally correct.
Applications of Dimensional Analysis
Dimensional analysis is a fundamental aspect of measurement and is applied in real-life physics. We make use of dimensional analysis for three prominent reasons:
- To check the consistency of a dimensional equation
- To derive the relation between physical quantities in physical phenomena
- To change units from one system to another
Limitations of Dimensional Analysis
Some limitations of dimensional analysis are:
- It doesn’t give information about the dimensional constant.
- The formula containing trigonometric function, exponential functions, logarithmic function, etc. cannot be derived.
- It gives no information about whether a physical quantity is a scalar or vector.
Additional Solved Problems
1. Check the correctness of the physical equation v2 = u2 + 2as2.
Solution:
The computations made on the L.H.S and R.H.S are as follows:
L.H.S: v2 = [v2] = [ L1M0T–1]2 = [ L1M0T–2] ……………(1)
R.H.S: u2 + 2as2
Hence, [R.H.S] = [u]2 + 2[a][s]2
From (1) and (2), we have [L.H.S] ≠ [R.H.S]
Hence, by the principle of homogeneity, the equation is not dimensionally correct.
2. Evaluate the homogeneity of the equation when the rate flow of a liquid has a coefficient of viscosity η through a capillary tube of length ‘l’ and radius ‘a’ under pressure head ‘p’ given as
From (1) and (2), we have [L.H.S] = [R.H.S]
Hence, by the principle of homogeneity, the given equation is homogenous.
Frequently Asked Questions – FAQs
What is dimensional analysis?
State true or false: Dimensional analysis can not be used to find dimensionless constants.
State the principle of homogeneity of dimensions
Why do we use dimensional analysis?
- To check the consistency of a dimensional equation
- To derive the relation between physical quantities in physical phenomena
- To change units from one system to another.
What are the limitations of dimensional analysis?
- It doesn’t give information about the dimensional constant.
- The formula containing trigonometric function, exponential functions, logarithmic function, etc. cannot be derived.
- It gives no information about whether a physical quantity is a scalar or vector.
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Can the SI unit of a physical quantity be obtained knowing its dimensional formula? Explain with an example.
Yes! The unit of a physical quantity can be obtained by its dimensional formula.
As an example, let us consider force.
By Newton’s second of law of motion, we know that force is given by the following equation:
F=ma
The above equation can be rewritten as
F=m(v/t)
Rewriting the above equation using basic dimensions, we get
F=(m(d/t))/t
F=(md)/(t^2)=M^1L^1T^(-2)
Substituting the measuring unit for these basic dimensions, we get
(kg.m)/(s^2)
The above unit is nothing but the Newton, the SI unit of force. Newton is equal to the force that would give a mass of one kilogram an acceleration of one metre per second per second
This is how knowing the dimensional formula helps us find the unit of the physical quantity.
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