DIMENSIONAL ANALYSIS: TECHNIQUES, PRINCIPLE AND EXERCISES - PHYSICAL - 2024

The dimensional analysis is a tool widely used in various branches of science and engineering to better understand the phenomena involving the presence of different physical quantities. The quantities have dimensions and from these the different units of measurement are derived.

The origin of the concept of dimension is found in the French mathematician Joseph Fourier, who was the one who coined it. Fourier also understood that in order for two equations to be comparable, they must be homogeneous with respect to their dimensions. In other words, meters cannot be added to kilograms.

Thus, dimensional analysis is responsible for studying the magnitudes, dimensions and homogeneity of physical equations. For this reason, it is frequently used to check relationships and calculations, or to construct hypotheses on complicated questions that can later be tested experimentally.

In this way, dimensional analysis is a perfect tool to detect errors in calculations by checking the congruence or inconsistency of the units used in them, putting special focus on the units of the final results.

In addition, dimensional analysis is used to design systematic experiments. It allows to reduce the number of necessary experiments, as well as to facilitate the interpretation of the obtained results.

One of the fundamental bases of dimensional analysis is that it is possible to represent any physical quantity as a product of the powers of a smaller quantity, known as fundamental quantities from which the others are derived.

Fundamental quantities and dimensional formula

In physics, fundamental quantities are considered to be those that allow the others to be expressed as a function of these. By convention, the following have been chosen: length (L), time (T), mass (M), intensity of electric current (I), temperature (θ), luminous intensity (J) and amount of substance (N).

On the contrary, the rest are considered derived quantities. Some of these are: area, volume, density, speed, acceleration, among others.

A dimensional formula is defined as the mathematical equality that presents the relationship between a derived quantity and the fundamental ones.

Dimensional analysis techniques

There are various techniques or methods of dimensional analysis. Two of the most important are the following:

Rayleigh method

Rayleigh, who along with Fourier was one of the forerunners of dimensional analysis, developed a direct and very simple method that allows us to obtain dimensionless elements. In this method the following steps are followed:

1- The potential character function of the dependent variable is defined.

2- Each variable is changed by its corresponding dimensions.

3- The equations of homogeneity condition are established.

4- The np unknowns are set.

5- The exponents that have been calculated and fixed in the potential equation are substituted.

6- The groups of variables are moved to define the dimensionless numbers.

Buckingham method

This method is based on Buckingham's theorem or pi theorem, which states the following:

If there is a homogeneous dimensional relationship between a number “n” of physical or variable quantities where “p” different fundamental dimensions are included, there is also a dimensionally homogeneous relationship between n – p, independent dimensionless groups.

Dimensional homogeneity principle

The Fourier principle, also known as the principle of dimensional homogeneity, affects the proper structuring of the expressions that link physical quantities algebraically.

It is a principle that has mathematical consistency and states that the only option is to subtract or add physical quantities that are of the same nature. Therefore, it is not possible to add a mass with a length, nor a time with a surface, etc.

Similarly, the principle states that for the physical equations to be dimensionally correct, the total of the terms of the members of the two sides of the equality must have the same dimension. This principle makes it possible to guarantee the coherence of the physical equations.

Similarity principle

The principle of similarity is an extension of the dimensional homogeneity character of physical equations. It is stated as follows:

Physical laws remain unchanged when faced with changes in the dimensions (size) of a physical event in the same system of units, whether it is changes of a real or imaginary nature.

The clearest application of the principle of similarity occurs in the analysis of the physical properties of a model made on a smaller scale, to later use the results in the object in real size.

This practice is essential in fields such as the design and manufacture of airplanes and ships and in large hydraulic works.

Applications

The many applications of dimensional analysis include those listed below.

- Locate possible errors in the operations carried out

- Solve problems whose resolution presents some insurmountable mathematical difficulty.

- Design and analyze small-scale models.

- Make observations about how possible modifications influence a model.

Also, dimensional analysis is used quite frequently in the study of fluid mechanics.

The relevance of dimensional analysis in fluid mechanics is due to how difficult it is to establish equations in certain flows as well as the difficulty to solve them, so it is impossible to achieve empirical relationships. For this reason, it is necessary to resort to the experimental method.

Solved exercises

First exercise

Find the dimensional equation for velocity and acceleration.

Solution

Since v = s / t, it is true that: = L / T = L ∙ T ^-1

Similarly:

a = v / t

= L / T ² = L ∙ T ^-2

Second exercise

Determine the dimensional equation for momentum.

Solution

Since the momentum is the product of mass and velocity, it is true that p = m ∙ v

So:

= M ∙ L / T = M ∙ L ∙ T ^-2

References

1. Dimensional analysis (nd). On Wikipedia. Retrieved on May 19, 2018, from es.wikipedia.org.
2. Dimensional analysis (nd). On Wikipedia. Retrieved on May 19, 2018, from en.wikipedia.org.
3. Langhaar, HL (1951), Dimensional Analysis and Theory of Models, Wiley.
4. Fidalgo Sánchez, José Antonio (2005). Physics and chemistry. Everest
5. David C. Cassidy, Gerald James Holton, Floyd James Rutherford (2002). Understanding physics. Birkhäuser.