Theory of Harmonic Oscillations: A Gross Error in Physics

Abstract

The critical analysis of the foundations of the standard theory of harmonic oscillations is proposed. The unity of formal logic and rational dialectics is methodological basis of  the analysis. The analysis leads to the conclusion that this theory represents gross error.  The substantiation (validation) of this statement is the following main results. I. In the  case of the material point suspended on the elastic spring, the linear differential  equation of harmonic oscillations is the equation (condition) of balance of Newton’s  force (Newton’s second law) and “Hooke’s force” (“Hooke’s law” as pseudolaw). This  equation contains the following gross methodological errors: (a) the differential  equation of motion of the material point does not satisfy the dialectical principle of the  unity of the qualitative and quantitative determinacy of physical quantities (i.e.,  Newton’s force and Hooke’s force). In other words, the left and right sides of the  differential equation (i.e., the equation of balance of the forces) have no identical  qualitative determinacy: the left side of the the equation of balance of the forces  represents Newton’s force, and the right side of the the equation of balance of the forces  represents the “Hooke’s force” (as pseudolaw); (b) the sum of Newton’s force and the  “Hooke force” (as pseudolaw) in the the equation of balance of the forces is equal to  zero. This means that the sum of the numerical values of Newton’s force and “Hooke’s  force” (as pseudolaw) is equal to zero. Consequently, the numerical values of Newton’s  force and “Hooke’s force” (as pseudolaw) are equal to zero in the region of neutral real  numbers. This means that the equation of balance of the forces is incorrect; (c) “Hooke’s  force” (as pseudolaw) in the equation of balance of the forces represents the product of  the spring constant (coefficient of stiffness of the spring) and the coordinate of the  material point. In this case, “Hooke’s force” (as pseudolaw) does not represent Hooke’s  law. “Hooke’s force” (as pseudolaw) contradicts to Hooke’s law because the coordinate  of rhe material point does not determine the spring constant (coefficient of stiffness of  the spring). “Hooke’s force” (as pseudolaw) has the dimension of Newton’s force. But,  as the practice of measurement of Hooke’s force with the help of a dynamometer  shows, the dynamometer readings are real neutral numbers with the dimension  “kilogram-force”; (d) the mathematical operation of division of the equation of balance  of the forces by the mass of the material point leads to the linear equation of balance of 

the accelerations of the material point. In this case, the mathematical operation gives  rise to the term “frequency”: (spring stiffness coefficient)-to-(mass) ratio is “squared  frequency”. But the spring stiffness coefficient is the constant that does not define the  concept of frequency. Therefore, the quantity of the acceleration of the material point  does not define the concept of the frequency of periodic motion; (e) the solution of the  linear differential equation of balance of the accelerations of the material point has  imaginary roots. This leads to the following contradiction: the coordinate of the  material point is both an exponential function and a trigonometric function. II. In the  case of oscillations of the mathematical pendulum, the linear differential equation of  harmonic oscillations of the material point suspended on the inextensible thread  represents a mathematical description of the angular displacement of the inextensible  thread in the Cartesian coordinate system. This equation is a mathematical consequence  of the standard differential equation of the rotational motion dynamics and contains  the following gross methodological errors: (a) the differential equation of motion of the  material point suspended on the inextensible thread does not satisfy the dialectical  principle of the unity of the qualitative and quantitative determinacy of physical  quantities (i.e., the physical quantity of rate of change of the angular momentum  (moment of momentum) and the physical quantity of moment of the acting force). This  equation expresses the condition of balance of the rate of change of the angular  momentum (moment of momentum) and the moment of the acting force. Gross error is  that the left and right sides of the balance equation have no identical qualitative  determinacy: the left side of the balance equation is the rate of change of the angular  momentum (moment of momentum), and the right side of the balance equation is the  moment of the acting force; (b) the sum of the rate of change of the angular momentum  and the moment of the acting force is equal to zero in the balance equation. This means  that the sum of the numerical values of the rate of change of the angular momentum  and the moment of the acting force is equal to zero. Consequently, the numerical values  of the rate of change of the angular momentum and the moment of the acting force are  equal to zero in the region of neutral real numbers. This means that the balance  equation is incorrect; (c) the mathematical operation of division of the equation of  balance of the rate of change of the angular momentum and the moment of the acting  force by the mass of the material point and the square of the thread length results in the  equation of balance of the angular accelerations. In this case, the mathematical  operation results in the term “frequency” (“squared frequency”). But the quantity of  the angular acceleration does not determine the frequency of the periodic motion; (d)  the linear differential equation of balance of the angular accelerations is analogous to  the linear differential equation of balance of the accelerations of the material point  suspended on the spring. Therefore, the solution of the linear differential equation of  balance of the angular accelerations has imaginary roots and leads to the following  contradiction: the angle of displacement of the pendulum from the equilibrium position  is both an exponential function and a trigonometric function.