On Some Applications of the Method of Dimensional Analysis in the Course of Physics and Mathematics

Abstract

Key words: problem, non-dimensional coefficient, quantity, exponent, dependence In the article, it was attempted to classify and represent the effective applications of the dimensional analysis method in the process of solving problems based on the principle of gradual complexity, which is the scientific and methodological novelty of the work, in order to highlight the role of the method in ensuring the informality and effectiveness of training. Particular attention is paid to the issues of the limits of the method’s applicability. In the first part of the work, the essence of the dimensional analysis method is briefly outlined, and then the simplest problems easily solved by its direct application are considered. The next section represents the cases when the desired physical expressions determined by this method can be uniquely determined with the accuracy of a constant dimensionless factor. As an example, the problem of determining the period of an oscillatory circuit is solved. The cycle of physical problems ends with a discussion of the problem of the Casimir phenomenon, for the solution of which, along with the method of dimensions, additional physical reasoning is used. When getting acquainted with the method of dimensional analysis, an initial erroneous impression may arise that the method is most applicable in physics, and is unlikely to have a possible effective application in mathematics. However, this is not true. No wonder the famous physicist, academician Migdal began his famous book on approximation and modeling of physical phenomena by proving the Pythagorean theorem using the dimensional analysis method. In the last section of the work, some tasks and test tasks of mathematics are considered, in the solution of which the dimensional analysis method is used.