Critics of Existent Theory of Mathematical Pendulum Part 2

• General Background: A mass m hanging from a string whose length is L and a pivot point on which this mass is fixed are what a simple pendulum (which was discovered during the 10 th century by Ibn Yusuf) consists of. During the 17 th century, it is developed by some physicist, especially by Galileo. When the mass hanging from the string is released with an initial angle, it starts to move with a periodic motion. The motion can be approximated as a simple harmonic motion if the pendulum swings through a small angle (so sin (ө) can be approximated as ө). The frequency and period for the simple pendulum are the independent of the initial angle of the movement (initial position of the mass to the vertical reference line). In addition to the initial angle of the mass, the period doesn't depend on the mass of the object. However, it is affected by the length of the string which the mass is hanged on and the acceleration of gravity. The most widespread applications of the simple pendulum are for timekeeping, gravimetry (the existence of the variable g in the period equation of simple pendulum-• means that the pendulum frequency is different at different places on Earth), seismology, scholar tuning, and coupled pendula. It is also used for entertainment and religious practice. • Aim: To determine the effects or contribution of the length of the string on the period for the simple pendulum and find out a mathematical relationship between the length and the period. • Hypothesis: Since the length of the string which the mass is hanged on is shortened, the magnitude of the period for the simple pendulum gets increased. Different masses of the object hanging from the string have no effect on the period.

Physics Education, 2013

Geometric demonstration located in v, t diagram explains the nature of the period of a pendulum, i.e. the universal connection of time with velocity (radius) and acceleration. Conceptual nature of the principles proof points to its universal validity. In other words, if it is valid for a circle, it is valid. Through the geometry of free fall we describe physical

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

European Journal of Physics Education, 2013

Solving the position of a simple pendulum at any time is apparently one of the most simple and basic problems to solve -in high school and college physics courses. However, because of this apparent simplicity, teachers and physics texts often assume that the solution is immediate -without pausing to reflect on the problem formulation or verifying that the solution obtained is indeed correct. This process causes conceptual errors to be carried out with them -students and even worse teachers. This paper presents some of the misconceptions found in teachers solving simple pendulum problems, moreover it presents proposals made in the texts, which generate the creation of such misconceptions in both teachers and students, and finally, it presents the proposal of a solution to correct this problem.

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.

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.

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

arXiv: Classical Physics, 2019

In this paper we revisit the construction by which the $SL(2,\mathbb{R})$ symmetry of the Euler equations allows to obtain the simple pendulum from the rigid body. We begin reviewing the original relation found by Holm and Marsden in which, starting from the two Casimir functions of the extended rigid body with Lie algebra $ISO(2)$ and introducing a proper momentum map, it is possible to obtain both the Hamiltonian and equations of motion of the pendulum. Important in this construction is the fact that both Casimirs have the geometry of an elliptic cylinder. By considering the whole $SL(2,\mathbb{R})$ symmetry group, in this contribution we give all possible combinations of the Casimir functions and the corresponding momentum maps that produce the simple pendulum, showing that this system can also appear when the geometry of one of the Casimirs is given by a hyperbolic cylinder and the another one by an elliptic cylinder. As a result we show that from the extended rigid body with Li...

Physics

Since the pioneering works of Newton (1643–1727), mechanics has been constantly reinventing itself: reformulated in particular by Lagrange (1736–1813) then Hamilton (1805–1865), it now offers powerful conceptual and mathematical tools for the exploration of dynamical systems, essentially via the action-angle variables formulation and more generally through the theory of canonical transformations. We propose to the (graduate) reader an overview of these different formulations through the well-known example of Foucault’s pendulum, a device created by Foucault (1819–1868) and first installed in the Panthéon (Paris, France) in 1851 to display the Earth’s rotation. The apparent simplicity of Foucault’s pendulum is indeed an open door to the most contemporary ramifications of classical mechanics. We stress that adopting the formalism of action-angle variables is not necessary to understand the dynamics of Foucault’s pendulum. The latter is simply taken as well-known and simple dynamical s...

International Letters of Chemistry, Physics and Astronomy, 2013

In the paper, the adequate theory of oscillator is presented, being a sort of prelude to verification of the classical theory of mechanics. The developed theory is based on a properly understood notion of energy, quantum value changes of its determined measures (potentials), as well as of such changes types of sites of full energetic states which presents the essence of the true principle of the energy conservation. In the first part of the paper the principle of energy conservation was considered. Then the energetic aspects of the oscillator motion, with an exemplary real system motion was presented. The third part was a development of kinetics of a body in the harmonic motion and verification of the adequate theory of the oscillator. At the end, this Part 4 is devoted to the determination of the gravity acceleration by means of the mathematical pendulum to confirm the previously presented findings.

Many first year university students in the Science related fields have a problem in identifying and applying the fundamental mathematical concepts that they have learnt from as far back as Grade 10 in solving some of the physics related problems. The majority of students, mostly from public schools, cannot seem to relate the two fields of study. Previous studies have shown that majority of first year students lack the integrated approach to different scientific disciplines in solving some specific problems and in experimental analysis. This paper presents the findings of an investigation on the ability of the first year students to use the mathematical concept of the straight line equation and graph, in analysing physics properties, with particular reference to a Simple Pendulum motion experiment. The method employed was experimental. Students were given a simple pendulum experiment to determine the gravitational acceleration of the pendulum by graphical analysis. Two graphical approaches were employed, with both expected to yield the same results. Participants were required to analyse the motion of the pendulum, specifically the variation of the length of the pendulum with the period of oscillation. The feedback showed a far reaching implications relating to first years’ abilities to mathematically analyse a physics concept. These findings serve as a basis for a need to improve the teaching of science and mathematics in the schools, especially the practical approach to teaching and learning science.

2017

We assert that, from a pragmatic point of view, mathematicians treat mathematical objects as if they were real. If a theory is consistent, theorems are discovered (sometimes with analyses not necessarily different from those applied in sciences) and proofs are invented; modern technology cannot exist without accepting the law of excluded middle; a constructive proof may provide new ideas or methods but, from a mathematical point of view, a non-constructive proof is as sound as a constructive one. Accordingly, no mathematician, pure or applied, gets by without the axiom of choice; on the other hand, although different theorems and objects may appear depending on the acceptance or not of the continuum hypothesis, no important theorem applicable to the real world exists – at least until now – which depends on accepting or not this hypothesis. Mathematical objects built by applied mathematicians are often as useful as physical objects, even those objects created via computer-assisted or...

1999

Qualitative analysis of a pendulum with a periodically varying length is conducted. It is proved that there are two periodic solutions having a prescribed amplitude A( and a period ¹ which is an even multiple k of the excitation period. Stability analysis is carried out for the principal parametric oscillations (k"2). In this connection it is shown that such a pendulum cannot serve as a mathematical model of swing as it is generally considered.

European Journal of Physics, 2005

We describe a 8085 microprocessor interface developed to make reliable time period measurements. The time period of each oscillation of a simple pendulum was measured using this interface. The variation of the time period with increasing oscillation was studied for the simple harmonic motion (SHM) and for large angle initial displacements (non-SHM). The results underlines the importance of the precautions which the students are asked to take while performing the pendulum experiment.

Beyond Babel: Religion and Linguistic Pluralism (ed. by Andrea Vestrucci), Sophia studies in cross-cultural philosophy of traditions and cultures, vol. 43, 2023

Sometimes an oscillation takes place between two incompatible approaches to an experienced situation: from one to another, then back and then again, and again. The oscillation is not an additional ingredient but an essential aspect of the situation. Both approaches are needed but having assumed one of them the participant is led to the realization of the need of the other. A process of this kind occurs in interfaith dialogue: we oscillate between considering the other religion from an objective standpoint and perceiving it from the perspective of my own religion. Some kind of oscillation is present in other situations. Several examples are given to show that the process can be seen as familiar even though it has hardly been identified as a separate phenomenon to be analyzed. According to the present author, it is related to but possibly distinct from complementarity as it is known in physics. A preliminary attempt is made to formulate the logic of the oscillation process, which can be called sequential paraclassical logic.