Dimensional Analysis

Dimensional analysis helps in understanding the nature and relationships between objects mathematically. Most of the object’s that we see can be measured. To find the measurements of any object we must know its unit. For example, the card box is 56cm long. Here, the unit cm helps in understanding the measurements and dimensions of the objects. 

But, there are times that we convert one unit to another. The dimensional analysis revolves around this concept of converting units.

I know you have a lot of questions. Don’t worry we are here to help. Keep reading to learn more about dimensional analysis and the homogeneity principle of dimensional analysis.

What is Dimensional Analysis?

In engineering and science, dimensional analysis can be defined as the study of the relationship between different physical quantities that takes the units and dimensions of the objects into consideration in the calculations. It helps us to convert one unit into another unit. It would have been extremely difficult to solve mathematical equations and formulas without the concept of dimensional analysis.

To derive relationships between physical objects their base quantities like time, mass, length, and electric current and units of measure such as miles vs. kilometres are used in the calculation. Rules of algebra are also used in dimensional analysis. 

Types of Units:

To study dimensional analysis, you need to know the different types of units. They are categorised into two types:

Fundamental Units:

The International System of Units has defined 7 fundamental units that are used extensively in all fields. These units cannot be derived from any other unit.

1. Ampere (A) for Electric current
2. Kelvin (K) for temperature
3. Metre (m) for Length
4. Second (s) for Time
5. Kilogram (kg) for Weight
6. Candela (cd) for Luminous intensity
7. Mole (mol) for Amount of substance

Derived Units:

As the name suggests these units are derived from the 7 fundamental units. Energy, pressure, power, acceleration, and force are examples of derived units.

Unit Conversion:

Dimensional analysis is popularly known as Factor Label Method or the Unit Factor Method. This is because the method of the conversion factor is used to get the same units. For example, if I ask you how many millilitres are there in 4 litres of milk. Then your answer would be 4000 millilitres, right? 

Your calculation would be something like this:

1 litre: 1000 millilitres

4 litre: 4000 millilitres

Here the conversion factor is 1000 millilitres.

Homogeneity Principle of Dimensional Analysis:

According to this principle, the dimensions of every term of any dimensional equation should always be the same. If the terms on both sides of the equation have different dimensions then the equation cannot be solved and the relationship between two physical quantities cannot be derived. This principle helps us to convert one unit to another unit. For example, if the unit of time i.e. seconds is present on the left side of the equation then seconds must also be present on the right side of the equation. Comparing hours with kilometres doesn’t make any sense, does it?

Using Dimensional Analysis to Check the Correctness of Physical Equation:

Let’s say you want to check if this equation is v=u+atis correct or not. You can check the correctness of this equation using the dimensional analysis as follows:

v=u+at

Where,

v is final velocity

u is initial velocity

a is acceleration

Hence, this equation is dimensionally correct and the homogeneity principle of dimensional analysis.

Applications of Dimensional Analysis:

Dimensional analysis forms the basics of measurement and has several real-life applications in physics. They are as follows:

• To check the correctness of any equation and to see if it is dimensionally correct or not. The equation would be dimensionally correct if the terms on both sides of the equation have the same units.
• Convert units from one system to another.
• Derive the relationship between two physical quantities.
• To formulate various formulas.

Limitations of Dimensional Analysis:

As with any other concept this too has some limitations:

• Using dimensional analysis you cannot find the dimensional constant.
• Trigonometric, exponential, and logarithmic functions cannot be derived using this concept.
• You cannot find out if the given quantity is a scalar or vector.

Conclusion:

Dimensional analysis is very important in measurements and to solve various mathematical equations. It lays down the core measurement principle using which we can easily derive and compare relationships between two physical objects. This is the reason they have wide applications in engineering and physics. Dimensional analysis is used to check the correctness of any physical equation. The Homogeneity Principle of Dimensional Analysis states that every term of any dimensional equation should always be the same. If the terms on both sides of the equation have different dimensions then the equation cannot be solved.