Dimensional Analysis and its Principle, Limitations

The method of determining the dimensions and units of measurement of physical quantities is known as dimensional analysis. 

Dimensional analysis is an excellent method for ensuring that equations are dimensionally correct. We can also utilise dimensional analysis to build credible equations if the quantities are known. It can be used to quantify the size and shape of objects. Today, dimensional analysis has made it feasible to explore the nature of objects mathematically.

Most physical quantities may be expressed in terms of basic dimensions. With the help of fundamental quantities and derived quantities such as mass, length, and time, we will now study dimensional analysis and its applications.

Definition of Dimensional Analysis

Dimensional analysis transforms physical derived quantities to their fundamental quantities to observe them further. It is also used to investigate the relationship between quantity dimensions and units.

Dimensional analysis is a method for comparing relationships between various quantities without mixing their units. Dimensional analysis reduces derived physical properties such as viscosity, density, acceleration, and energy to their three fundamental dimensions of length (L), mass (M), and time (T).

Fundamental and Derived Quantities

The fundamental physical quantities in mechanics are mass, length, and time, and the units of these physical quantities are called fundamental units. These units are known as fundamental since they are all self-contained.

Other physical quantities such as temperature, luminous intensity, electric current, and the amount of substance are also named fundamental physical quantities, and their units are kelvin (K), Candela (Cd), ampere (A), and mole (mol), respectively. The radian for plane angles and the steradian for solid angles are two other fundamental units.

Derived units are units that are not independent of each other and are derived from the primary units. Examples are area, velocity, acceleration, volume, force, and other derived units.

Principle of Homogeneity of Dimensional Analysis

The dimensions of the equations expressing physical circumstances must be the same. This theory is based on the notion that one can only add, subtract, or compare two quantities of the same dimension.

Based on the principle of homogeneity, the dimensions of each dimensional equation on both sides should be the same. 

When the variables x, a, b, and c, all have the exact dimensions, an equation of the form x=a+b+c+…. is dimensionally correct. This technique can be used for algebraic equations and differential and integral equations. However, it should not be assumed that an empirical equation is always dimensionally homogeneous.

Unit Conversion as Dimensional Analysis

It is possible to convert units from one system to another. The factor label method, unit factor method, or dimensional analysis are used.

A unit can be converted from one system to another, using a conversion factor to define the relationship between units. It assumes that the ratio of any fundamental quantity in one unit to its equivalent quantity in another unit is one. The conversion factor transforms each fundamental quantity, such as mass, length, and time, into another preferred unit system.

Some examples of conversion factors in everyday life:

• A pendulum’s rhythmic oscillation.
• A vibrating object’s energy.
• For a revolving disc, demand vs capacity.

Limitations of Dimensional Analysis

Although dimensional analysis is highly beneficial, it has some drawbacks, 

• As multiple physical quantities have the same dimensions, a dimensional formula does not determine the uniqueness of physical quantities. For example, if a physical quantity’s dimensional formula is [ML²T^-2], the quantity might be work, energy, or torque.

• The dimensional methods cannot be used to determine a numerical constant K with no dimensions such as 1 or 2π.

• We are unable to obtain any information regarding dimensional constants.

• We cannot derive complex functions like trigonometric, exponential, or logarithmic functions using dimensional analysis.

• We do not know whether the physical quantity is a matrix or a numeric quantity.

• Dimensional analysis can only determine whether or not a physical relationship is dimensionally correct. It has no way of knowing whether or not the relationship is correct.

• It cannot identify all variables that influence a specific physical quantity.

Application of Dimensional Analysis

Dimensional analysis is a fundamental aspect of measurement applied to solve real physics problems. The following are five reasons why dimensional analysis is used:

• We can check the accuracy of an equation or any other physical relationship using the homogeneity principle. If the dimensions of the left and right sides are the same, the relationship will be correct.

• It is used to convert physical units from one to the other.

• It is also used to highlight the physical quantity’s nature.

• The algebraic quantities can be used to denote the dimensional expressions.

• Deriving formulas is the essential use of dimensional analysis.

Conclusion

Dimensional Analysis is the technique of converting physical derived quantities to their fundamental quantities to observe them further. Length (L), Mass (M), and Time (T) are the three fundamental dimensions (T). We can check the accuracy of an equation or any other physical relationship using the principle of homogeneity. The unit factor technique, or factor label method, is another name for dimensional analysis. Dimensional formulae cannot determine whether a physical quantity is a matrix or a numeric quantity. Some complicated functions, such as trigonometric, exponential, and logarithmic, are not derivable.