PLAYING WITH A CHAIN OR PHYSICAL AND MATHEMATICAL INFORMATICS

Abstract

The article describes an educational laboratory work within the framework of interdisciplinary connec- tions at the intersection of informatics, mathematics and physics: the study of the sagging of a closed chain with different support points. A technology of computer processing of photos and videos of a physical ex- periment with the subsequent processing of the media files on the computer is illustrated. We present the π chain number (1.258... − the optimal ratio of the chain length to the distance between its attachment points) and its relation to the previously unexplored problem of the shape of the sag of a closed chain hanging on two nails. The analytic expression for this constant was first found. A new physical and math- ematical constant is found − the critical angle of the sagging of a closed chain on the “hangers”. The prob- lem of the sagging of a closed chain on a cone is detailed. The applicability of the computer tool “optimiza- tion with constraints” for solving problems of theoretical mechanics is investigated. Three main tools of mathematical computer tools are described: numerical mathematics, symbolic mathematics and graphics. The importance of using units of measurement when solving physical problems on a computer is empha- sized. The problem of publishing mathematical formulas in articles and books is discussed.

Keywords

• Theoretical mechanics
• sagging chain
• chain function
• derivative
• integral
• potential energy
• kinetic energy
• Lagrange-Dirichlet principle
• Newton’s law
• constrained optimization
• system of algebraic equations
• computer graphics
• animation
• Mathcad