The Period of a Pendulum

2018

Simulating the simple harmonic motion of pendulum with the Leapfrog integrator based on Hamiltonian canonical equations of first order differential equation wich gives a clear picture how the pendulum behave for different angles and amplitudes. The final proofs of the behaviour of pendulum are the graphics commented at the end.

International Letters of Chemistry, Physics and Astronomy, 2013

In the second part of the paper, the thesis is proved to state that the existent theory describes simply a shadow of the rotating apparent mathematical pendulum. Hence, it appears, even that existent description is not sufficiently adequate. Finally, all defects of the theory, which resulted in so inadequate description of the oscillation motion of the simple mathematical pendulum, have been revealed. The necessity to re-build the existent theory has been indicated in the conclusion. Return to the source is to be the first, essential step on the new path of the cognitive action.

European Journal of Physics, 1999

All kinds of motion of a rigid pendulum (including swinging with arbitrarily large amplitudes and complete revolutions) are investigated both analytically and with the help of computerized simulations based on the educational software package PHYSICS OF OSCILLATIONS developed by the author (see in the web http://www.aip.org/pas). The simulation experiments of the package reveal many interesting peculiarities of this famous physical model and aid greatly an understanding of basic principles of the pendulum motion. The computerized simulations complement the analytical study of the subject in a manner that is mutually reinforcing.

International Letters of Chemistry Physics and Astronomy, 2013

In the second part of the paper, the thesis is proved to state that the existent theory describes simply a shadow of the rotating apparent mathematical pendulum. Hence, it appears, even that existent description is not sufficiently adequate. Finally, all defects of the theory, which resulted in so inadequate description of the oscillation motion of the simple mathematical pendulum, have been revealed. The necessity to re-build the existent theory has been indicated in the conclusion. Return to the source is to be the first, essential step on the new path of the cognitive action.

Science & Education, 2006

In this presentation a number of animations and simulations are utilized to understand and teach some of the pendulum's interpretations related to what we now see as the history of energy conservation ideas. That is, the accent is not on the pendulum as a time meter but as a constrained fall device, a view that Kuhn refers back to Aristotle. The actors of this story are Galileo, Huygens, Daniel Bernoulli, Mach and Feynman (Leibniz's contributions, however important, are not discussed here). The "phenomenon" dealt with is the swinging body. Galileo, focussing on the heights of descent and ascent rather than on trajectories, interprets the swinging body in both ways (time meter and constrained fall), establishes an analogy between pendulums and inclined planes and eventually gets to the free fall law. Huygens expands the analysis to the compound (physical) pendulum and as a by-product of the search for the centre of oscillation (time meter) formulates a version of the vis viva conservation law (constrained fall). Both Galileo and Huygens assume the impossibility of perpetual motion and Mach's history will later outline and clarify the issues. Daniel Bernoulli generalises Huygens results and formulates for the first time the concept of potential and the related independence of the work done from the trajectories (paths) followed: vis viva conservation at specific positions is now linked with the potential. Feynman's modern way of teaching the subject shows striking similarities. Multimedia devices enormously increase the possibility of understanding what is a rather physically complex and historically intriguing problem. Teachers and students are in this way introduced to the beauties of epoch-making scientific research and to its epistemological implications.

This paper looks at the physics behind oscillatory motion and how this can be applied to many different scenarios including using different types of pendulums to explain the phenomena of SHM and DHM (simple and damped harmonic motion respectively). Written as part of my A-Level Physics course (OCR B Advancing Physics) in Autumn 2014 (Nov 2014-Jan 2015)

2009

In this paper we show that there are applications that transform the movement of a pendulum into movements in $\mathbb{R}^3$. This can be done using Euler top system of differential equations. On the constant level surfaces, Euler top system reduces to the equation of a pendulum. Those properties are also considered in the case of system of differential equations with

Science & Education, 2004

Articles about the pendulum in four journals devoted to the teaching of physics and one general science teaching journal (along with other miscellaneous articles from other journals) are listed in three broad categories-types of pendulums, the contexts in which these pendulums are used in physics teaching at secondary or tertiary levels and a miscellaneous category. A brief description of the sub-categories used is provided. * Paul McColl, a physics teacher from Bundoora Secondary College in Victoria and a doctoral student in science education at Monash University, assisted with the collection of the bibliographical material below and I owe a debt of gratitude to him for his contribution to this project.

iaeme

For solving the nonlinear differential equation of the pendulum, here we adopt a method that transforms the nonlinear differential equation into an equivalent linear one and then evaluate the period oscillation. We also apply the energy conservation principle to find the dependence of the time period on the amplitude of oscillation. Also harmonic balance method is applied to find an expression for the period of oscillation. Theoretical curves, simulation results and experimental results are given in support of the findings. Nonlinear method 1 proves that the pendulum oscillation is periodic but it has also very small amount of third harmonic. FFT analysis has been carried out of the data as obtained from the simulation results and it is found that system is almost free from harmonic distortion