A contribution to calculation of the mathematical pendulum

Abstract

In this work, as a continuation of rigorous solutions of the mathematical pendulum theory, calculated dependences were obtained in elementary functions (with construction of plots) for a complete description of the oscillatory motion of the pendulum with determination of its parameters, such as the oscillation period, deviation angles, time of motion, angular velocity and acceleration, and strains in the pendulum rod (maximum, minimum, zero, and gravitational). The results of calculations according to the proposed dependences closely (≪1%) coincide with the exact tabulated data for individual points. The conditions of ascending at which the angular velocity, angular acceleration, and strains in the pendulum rod reach their limiting values equal to $2\sqrt {\frac{g} {l}} ,0.77\frac{g} {l}$ and 5m ₁ g, respectively, are shown. It was revealed that the angular acceleration does not depend on the pendulum oscillation amplitude; the pendulum rod strain equal to the gravitation force of the pendulum R _s = m ₁ g at the time instant $t \approx 1\sqrt {\frac{1} {g}}$ is also independent on the amplitude. The dependences presented in this work can also be invoked for describing oscillations of a physical pendulum, mass on a spring, electric circuit, etc.