Author: Nurmatov O.R., Yulchiyev I.I., Axmadjonov M.F., Xidirov D.Sh., Nasirov M.X.

Annotation: Vibrations explain the physical properties of many substances in nature. It is therefore important to study the laws of vibration. Many entrants and students have memorized the results of the laws of vibration of a mathematical pendulum, one of the simplest vibration systems, and are interested in how these results came about. However, these solutions cannot be found with explanations in a school math course. The article shows how these results can be obtained by mathematical methods.

Keywords: Mathematical pendulum, free oscillations, forced oscillations, harmonic oscillations, enharmonic oscillations, oscillation phase, oscillation amplitude, tangential acceleration, inertia force, elastic force, exponential function, differential equation.