Abstract
In this study we used image processing algorithms to determine pendulum ball motion and present an approach for solving the nonlinear differential equation that governs its movement. This formulae show an excellent agreement with the exact period calculated with the use of elliptical integrals, and they are valid for both small and large amplitudes of oscillation.Also, we reveal some interesting aspects of the study simple pendulum or procedures for determining analytical approximations to the periodic solutions of nonlinear differential equations.
Highlights
We reveal some interesting aspects of the study simple pendulum or procedures for determining analytical approximations to the periodic solutions of nonlinear differential equations
Arun A simple approximate expression is derived for the dependence of the period of a simple pendulum on the amplitude
The simple pendulum is rich in physics implications, and an understanding of its behavior over a more realistic range of phenomena is a worthwhile goal
Summary
Making the assumption of small angle allows the approximation to be made sinθ≈θ Substituting this approximation into eq (1) yields the equation for a harmonic oscillator: Under the initial conditions θ(0) = θ0 and dθ/dt(0) = 0, the solution is ISSN 1916-9639 E-ISSN 1916-9647. Represent a simple harmonic motion where θ0 is the semi-amplitude of the oscillation (that is, the maximum angle between the string of the pendulum and the vertical). The period of the motion, the time for a complete oscillation (outward and return) is which called Christiaan Huygens's law for the period In this case, the period of oscillation depends on the length of the pendulum and the acceleration due to gravity, and is independent of the amplitude θ
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