ON THE PHYSICAL AND MATHEMATICAL CORRECTNESS OF SOME EQUATIONS OF MECHANICS - Pozhidayev, Sergey (Physical and Mathematical Education, 2024)
Formulation of the problem. In the mathematical description of some phenomena of classical and applied mechanics (after this referred to as mechanics), incomprehensible dimensional phenomena arise. For example, angular velocity can have two different units of measurement, normal acceleration - three, and such physical quantities as moment of force and mechanical work get the same unit newton-meter. From a scientific point of view, such phenomena are contradictions that should be eliminated. After all, Nature is one and uncontroversial; therefore, the description of its phenomena must also be uncontroversial. Each physical quantity (in this or that particular field) should have only one coherent unit, and each unit should characterize only one physical quantity. Therefore, the mathematical apparatus of mechanics needs improvement.Materials and methods. One of the author's previous works established that incomprehensible dimensional phenomena result from the dimensional incorrectness of mathematical ratios due to their laxity. In particular, it was found that the well-known fundamental dependences of elementary geometry, which date back to Archimedes and have been considered impeccable analytical since time immemorial, are mathematically loose empirical and semi-empirical expressions. That prompts us to apply dimensional analysis to check the correctness of mechanics equations. This work also applied the basic principles of analytical mechanics and the law of energy conservation.Results. A review of the nineteen mechanics equations showed that most need to be corrected - mathematically loose or inconsistent with the energy conservation law. The incorrect equations were brought to a mathematically rigorous form and reconciled with the energy conservation law. They lead to an unambiguous system of units of measurement for mechanics, which, in particular, includes five refined units. In this system of units, there are no cases where the exact quantity has different units or different quantities have the same unit.Conclusions. It was established that the existing mathematical apparatus of classical and applied mechanics has significant shortcomings and can be improved.
Keywords: dimensional analysis; wheel speed; normal acceleration; moments of force; moment of inertia; moment of cross-sectional resistance; angular momentum