Dimensional Analysis

Using dimensional analysis, any object’s size and shape can be quantified. In simple terms, dimensional analysis helps study the nature of different objects in mathematical forms. It includes several geometric properties along with angles and lengths such as straightness, flatness, etc. The concept behind dimensional analysis is that the quantities that have similar dimensions can be subtracted. Also, two physical quantities with the same dimensions are equal. 

What is Dimensional Analysis? 

Dimensional analysis can be described as the study of physical quantities along with units of measurement and dimensions. It plays a significant role in keeping the units unaltered, making it easier to perform mathematical calculations smoothly. A thorough understanding of units and dimensions is necessary to solve several mathematical problems. 

There are many different types of quantities to measure in the physical world. From the size of an atom to the space between the planets, the universe is vast. Converting them from one unit to another becomes important as well. Unit analysis, often known as dimensional analysis, is the process of converting one unit to another. Another type of proportional reasoning is unit analysis. It involves multiplying a measurement by a known proportion to produce a result with a different unit. In general, it is a method for finding another unit by multiplying or dividing a value by a particular ratio.

Dimensions and dimensional formula

A single unit must be used to express each quantity. To do so, all of the quantity’s fundamental units are multiplied by a specified number of times. Dimensions are the names given to these abilities.

An expression is made up of all the fundamental units that have been increased to a specific level of power. The dimensional formula for that quantity is this equation.

For example, v = [L1T-1] represents velocity. The dimensions are 1 and -1, and the dimensional formula is [L1T-1].

Dimensional Analysis and Unit Conversion 

Also referred to as Unit Factor Method or Factor Label Method, it uses conversion factor to attain similar units. Let us understand it with an example – 

We already know that 1000 metres make 1 km,

It means, 2 km is equal to 2 x 1000 = 2000 metres 

Here, the conversion factor will be 1000. 

Types of Units

1. Fundamental units: Fundamental quantities are those that are not derived from any other quantity. Fundamental units are the units used to measure these quantities. e.g. Length, mass, time, electric current, temperature, luminous intensity, and the quantity of substance are all fundamental quantities. Metre, kilogram, ampere, kelvin, candela, and mole are all terms used to describe the temperature of a body. Fundamental units are units that measure these quantities.
2. Derived units: Derivative quantities are those that are derived from other quantities. Derived units are the units that are used to measure these quantities. For example, derived quantities include force, acceleration, pressure, energy, and power.

Principle of Homogeneity of Dimensional Analysis

According to the principle of homogeneity of dimensional analysis, the dimensions of every term of the dimensional equation on both sides should be equal. This principle plays a significant role in converting units from one form to another. 

For example – 

1000 metres make 1 km,

It means, 4km is equal to 4 x 1000 = 4000 metres 

Here, the conversion factor will be 1000.

Applications of Dimensional Analysis

Dimensional Analysis is significantly applied in real-life physics and is a significant measurement aspect. Here is the list of applications of dimensional analysis- 

• For checking dimensional consistency 
• For driving the relationship of the physical quantities in the physical phenomena 
• For changing units from one system to another system

Limitations of Dimensional Analysis

Apart from benefits and applications, dimensional analysis has some limitations as well. These include –

• Dimensional Analysis does not provide any information regarding the dimensional constant
• The formula that has the trigonometric function, exponential function, logarithmic function, and so on cannot be derived
• Dimensional analysis does not provide any information regarding physical quantity to check whether it is vector or scalar

What is Measurement in Physics? 

Measurement is one of the most important, therefore fundamental concepts in Physics, without which there might be no value of theories, experiments, or other tests recorded. Without the ability to measure, it would be nearly impossible for a scientist to determine results. 

Measurement includes several tools to describe how many or how much of something is present. Primarily, scientists are dependent on the metric system, which was first developed in the 1790s in France. Its units can are as follows – 

1. Time – Second (s)
2. Length – Metre (m)
3. Temperature – Kelvin (K)
4. Electric current – Ampere (A)
5. Mass – Kilogram (kg)
6. Luminous Intensity – Candela (cd)
7. Amount of a substance – Mole (mol)

Conclusion 

With this, we end our study material on dimensional analysis. In this Introduction to dimensional analysis, we studied that the study of the physical quantities and units of measurement and dimensions is referred to as the dimensional analysis. It plays a significant role in keeping the units unaltered, making it easier to perform mathematical calculations easily and smoothly. Dimensional analysis is hugely applied in real-life physics and is a significant aspect of measurement.