Dimensional Analysis

Dimensional analysis and its applications is the study of connections between physical variables using their units and dimensions of measurement. We utilise dimensional analysis to translate a unit between several forms. We must maintain the same unit to facilitate mathematical operations in science and mathematics. 

Dimensional Analysis is often referred to as the factor label method or the unit factor method because we utilise a conversion factor to get the same units.

Definition 

The dimensional analysis uses a set of units to determine the form of an equation or, more commonly, to ensure that the result of a computation is correct as a safeguard against many common errors.

The units included are:

These units are also known as SI measurement system base units. Base units are used to measure light intensity and material amount.

These units are also known as SI measurement system base units.

Only a combination of the units above can measure all quantities. Density, for example, is measured in Kg/m3.

We would write this as [M]/[L3] or [M L-3] in dimensional symbols.

Quantities with dimensions, Quantities without dimensions 

Quantities may be classified into four groups based on their dimension. 

1. Dimensional variables

Dimensional variables are physical quantities that have dimensions and have changeable values. For instance, length, velocity, and acceleration are all examples. 

2. Dimensionless variables

Dimensionless variables are physical quantities that lack dimensions yet have changeable values. Specific gravity, strain, and refractive index are all examples. 

3. Dimensional constant

Dimensional constants are physical quantities that have dimensions and have constant values. The gravitational constant, Planck’s constant, and others are examples. 

4. Dimensionless constant

Dimensionless constants are quantities with constant values and no dimensions. Examples include e and numerals. 

Dimensional homogeneity principle 

The notion of dimension homogeneity requires that all words in a physical statement have the same dimensions. In the physical formula v2 = u2 + 2as, for example, the dimensions of v2, u2, and 2as  are identical and equal to [L2T-2].

Factor labelling using dimensional analysis

It is possible to convert units between different measurement systems. Factor label technique, unit factor method, or dimensional analysis are used to describe this approach.

Conversion factors are used to express the connection between different systems’ units, making it possible to move an item from one to another. Based on the fact that the ratio of each basic quantity in one unit with their counterpart in another unit is equal to one, the formula is used.

For instance,

1) How many hours and minutes are in a day?

Solution:

60 minutes is equal to one hour. Three hours equals 180 minutes.

We use conversion factors to ensure that the answer is correct and that conversion factors do not skew the results. The conversion factor is used to change the unit system of each basic quantity, such as mass, length, and time.

Application of dimensional analysis

In real-life physics, dimensional analysis is a crucial part of the measurement. We use dimensional analysis for the following reasons:

• To ensure that a dimensional equation is consistent.
• Determining the relationship between physical quantities in physical phenomena
• To switch from one system to another’s units
• Development of a fluid phenomena equation
• The number of variables necessary in an equation is reduced.

Dimensional Analysis Limitations Explained

• This approach cannot be used to determine dimensionless quantities. This approach cannot be used to find the constant of proportionality. They can be discovered either experimentally or theoretically. 
• This procedure does not work with trigonometric, logarithmic, or exponential functions. 
• This strategy will be difficult to apply to physical quantities that are reliant on more than three physical qualities. 
• In some instances, the constant of proportionality has dimensions as well. In such instances, we are unable to use this system. 
• We cannot use this procedure to obtain an expression if one side of the equation comprises the addition operation of physical quantities.

Conclusion 

Dimensional analysis is an incredible technique for determining the dimensions of physical quantities to verify their relationships. The homogeneity principle is founded on the concept that two quantities of the same dimension may only be added, subtracted, or compared. When the sort of quantities involved is known, the dimensional analysis may be utilised to build credible equations.