Wilberforce Pendulum Research Articles

Wilberforce-like Larmor Magnetic Moment and Spin Precession.

In a Wilberforce pendulum, two mechanical oscillators are coupled: one pertains to the longitudinal (tension) motion and the other to the rotational (twisting) motion. It is shown that the longitudinal magnetic moment of circular currents, and similarly the magnetic moment of a spin-chain, can exhibit a Wilberforce-like vibration. The longitudinal oscillation is related to the Langevin diamagnetism, while the twisting motion is superimposed on the magnetic moment and spin precession. The calculations show that the coupling term is nonlinear in this (longitudinal) vibrating and (magnetic moment) precession system. By increasing the strength of the coupling we arrive at a spectrum, where further vibrational modes can be associated with the rotation of the precession. This means that the extent of the change in coherence can be demonstrated. Since the coupling strength can be different due to local effects, this can be an important factor from the point of view of signal propagation and in preserving signal shapes. The amount specifying the dissipation is introduced to express the degree of deviation. A relationship exists between the parameter characteristic of the coupling strength and how its quantity influences decoherence and dissipation.

Coupled oscillations of the Wilberforce pendulum unveiled by smartphones

The Wilberforce pendulum illustrates important properties of coupled oscillators including normal modes and beat phenomena. When helical spring is attached to a mass to create the Wilberforce pendulum, the longitudinal and torsional oscillations are coupled. A Wilberforce can be constructed simply from a standard laboratory spring, and a smartphone's accelerometer and gyroscope can be used to monitor the oscillations. We show that the resulting time-series data match theoretical predictions, and we share the procedures for observing both normal modes and beats.

Initial release condition dependent decoupling motion of a Wilberforce pendulum in the case of near resonance

The vertical and angular oscillation of a Wilberforce pendulum was found to be coupled in the case of resonance. Many studies have discussed the effects of the physical properties of a Wilberforce pendulum on coupled motion. In this article, we mainly study the influence of initial release conditions on the coupled motion of a Wilberforce pendulum in the near resonance case. The experimental results clearly exhibit that the coupled motion phenomenon depends on the initial release condition. Theoretical calculation further demonstrates that the coupled strength between two normal modes is determined by the ratio of initial angular to vertical displacement in the case of near resonance. The experimental results also verify the calculated variation of decoupling coefficient with this displacement ratio. Our study indicates that even in the case of resonance, the motion of a Wilberforce pendulum is probably decoupled at a given initial release condition. Complete decoupled motion is found to appear only when the damping coefficients in vertical and angular motion generate equal energy dissipation rates. The corresponding decoupling phenomenon in the case of resonance, occurs when the initial release energy of vertical and angular motion is equal.

Wilberforce pendulum: modelling linearly damped coupled oscillations of a spring-mass system

The Wilberforce pendulum is a coupled spring-mass system, where a mass with adjustable moment of inertia is suspended from a helical spring. Energy is converted between the translational and torsional modes, and this energy conversion is most clearly observed at resonance, which occurs when the damped natural frequencies of the two oscillation modes are equal. A theoretical model—with energy losses due to viscous damping proportional to velocity accounted for—was formulated using the Lagrangian formalism to predict the pendulum mass’ trajectory. Theoretical predictions were compared with experimental data, showing good agreement. Fourier analysis of both theoretical predictions and experimental data further corroborate the validity of our quantitative model. The dependence of oscillation features like beat frequency and maximum conversion amplitude on relevant parameters such as the initial vertical displacement, initial angular displacement and moment of inertia was also investigated and experimentally verified.

Dynamic simulation of the Wilberforce pendulum using constrained spatial nonlinear beam finite elements

AbstractMore than 100 years ago, Lionel Robert Wilberforce did investigations On the Vibrations of a Loaded Spiral Spring [1]. The spring was clamped at its upper side and on the other side, perpendicular to the spring axis, a steel cylinder was attached. Four screws with adjustable nuts were symmetrically attached around the cylinder in order to change its moment of inertia (Fig. 1). In this paper the Wilberforce pendulum is modeled by a rigid body attached to a constrained spatial nonlinear Timoshenko beam, discretized with B‐spline shape functions. As shown by a numerical experiment, the presented model is capable of reproducing the characteristic pendulum motion.

Theoretical and experimental studies of the Wilberforce pendulum

This work focuses on exploring the phenomenon of Wilberforce pendulum, a research problem for the International Young Physicists’ Tournament in 2021. A Wilberforce pendulum is a system which has a mass hanging vertically from the bottom of a long helical string. The mass is allowed to undergo both rotational motion about and translational motion along its vertical axis with the spring. Under appropriate initial conditions, the pendulum can exhibit coupled oscillations, where the mass switches between vertical oscillations (where it bobs up and down) and azimuthal oscillations (periodic rotations about the vertical axis). This paper aims to explore this phenomenon in three stages: firstly, an analysis on the mechanics of helical springs for an intuitive explanation for the onset of coupled oscillations; secondly, a comprehensive Lagrangian formulation which models the ‘beating’ motion path of the pendulum and its lightly damped oscillations; thirdly, an experimental verification of our model and the conditions required for clear coupling between the two modes of oscillations. The findings presented in this paper reveal some interesting features of the Wilberforce pendulum and help us gain some understanding of its physical properties. This phenomenon can also be regarded as an extremely insightful problem for introductory college physics courses focussed on the subject of oscillations and resonance, as its construction is easy and inexpensive, and it is a clear demonstration of coupled oscillatory motion.

Closed stable orbits in a strongly coupled resonant Wilberforce pendulum

We prove the existence of closed stable orbits in a strongly coupled Wilberforce pendulum, for the case of a 1:2 resonance, by using techniques of singular geometric reduction combined with the more classical averaging method of Moser.

PhysLab: a 3D virtual physics laboratory of simulated experiments for advanced physics learning

We introduce a virtual physics laboratory, ‘PhysLab’, created using 3D video game technology suitable for advanced level physics courses in secondary schools. This comprises 32 simulated experiments covering a range of physics topics, selected in collaboration with practicing school teachers. PhysLab is made available at no cost for the physics education community as an installable application, ready to run, while giving the instructor some control over how information is displayed, and suitable ranges of experimental parameters. While we focus on the theory, design and pedagogical aspects of PhysLab, we provide some critical reflections on the use of simulations in physics teaching in general, and especially how these could be most effectively used in the physics classroom. The experiments are classified according to their nature: supporting theory; ‘What if’-scenarios, like playing badminton on the Moon; or as hypothetical situations, such as what happens if you drop your home town into a hole through the centre of the Earth. It is not our intention to advocate the replacement of real practical work; we suggest various ways in which simulations can be integrated into the classroom, including instruction and real practical work. We also discuss the details of several PhysLab simulations: the Wilberforce Pendulum; oscillations of a mass on a rubber band, which involves non-linearity; the Drude theory of electrical conduction; the Cyclotron, and a ‘Journey to the Centre of the Earth’.

Cross-camera tracking and frequency analysis of a cheap Slinky Wilberforce pendulum

A Wilberforce pendulum is a mechanical oscillator often used as an example for the effect of coupling. After establishing a theoretical model of this oscillator and building a prototype, tracking algorithms have been developed to get the position of the oscillator. These tools enable us to highlight the existence of normal modes and determine the constants of the system. We also test different initial conditions to project the state of the pendulum on the normal modes. At the end, we discuss the match between our model and the experimental results. This comparison leads to possible improvements of the model and of the measurements.

Experimental verification of the adiabatic transfer in Wilberforce pendulum normal modes

A pendulum bob has been constructed to experimentally probe the adiabatic transfer in the normal modes of a Wilberforce pendulum. The rotational inertia of the pendulum bob can be changed during its motion in order to follow the gradual transition from a pure translational to a pure rotational oscillation. The reported extension of the conventional use of the Wilberforce pendulum helps understand and demonstrate the concept of normal modes, coupled oscillators, beating, and avoided crossing in undergraduate university classes.

Electronic system for the complex measurement of a Wilberforce pendulum

The authors present a novel application of a micro-electro-mechanical measurement system to the description of basic physical phenomena in a model Wilberforce pendulum. The composition of the kit includes a tripod with a mounted spring with freely hanging bob, a module GY-521 on the MPU 6050 coupled with an Arduino Uno, which in conjunction with a PC acts as measuring set. The system allows one to observe the swing of the pendulum in real time. Obtained data stays in good agreement with both theoretical predictions and previous works. The aim of this article is to introduce the study of a Wilberforce pendulum to the canon of physical laboratory exercises due to its interesting properties and multifaceted method of measurement.

Predator-prey model for the self-organization of stochastic oscillators in dual populations.

A predator-prey model of dual populations with stochastic oscillators is presented. A linear cross-coupling between the two populations is introduced following the coupling between the motions of a Wilberforce pendulum in two dimensions: one in the longitudinal and the other in torsional plain. Within each population a Kuramoto-type competition between the phases is assumed. Thus, the synchronization state of the whole system is controlled by these two types of competitions. The results of the numerical simulations show that by adding the linear cross-coupling interactions predator-prey oscillations between the two populations appear, which results in self-regulation of the system by a transfer of synchrony between the two populations. The model represents several important features of the dynamical interplay between the drift wave and zonal flow turbulence in magnetically confined plasmas, and a novel interpretation of the coupled dynamics of drift wave-zonal flow turbulence using synchronization of stochastic oscillator is discussed.

On the periodic orbits of the perturbed Wilberforce pendulum

The aim of the present work is to study the periodic structure of the Wilberforce pendulum under the effects of certain dependent perturbations. We use as a mathematical tool the averaging theory of dynamical systems by providing a system of two nonlinear equations whose simple zeros are linked to periodic solutions of the perturbed system.

The Slinky Wilberforce pendulum: A simple coupled oscillator

The Wilberforce pendulum is an effective classroom demonstration of coupled oscillations and the beat-like behavior that arises in weakly coupled tuned oscillators. We describe a simple and inexpensive version constructed from a Slinky spring toy and a soup can.

Pêndulo de Wilberforce: uma proposta de montagem para ambientes educativos informais e laboratórios didáticos

The Wilberforce Pendulum is a well-known experiment used to demonstrate coupled oscillations commonly restricted to longitudinal and rotational modes of a mass-spring system. It is interesting for the study of the principle of conservation of energy. For this purpose, there is a lot said on how to discuss this at Basic Levels and University Education. For those involved in informal educational environments, such as Museums of Science and Technology (MCT), this paper presents a proposal for setting up this experiment by inserting an innovative automation that allows the visitors themselves to perform a demonstration from a playful perspective that these places wish for. Teaching laboratories and people interested in these topics in general can also benefit from this development.

Using a digital video camera to examine coupledoscillations

In our previous paper (Debowska E, Jakubowicz S and Mazur Z 1999Eur. J. Phys. 20 89-95), thanks to the use of anultrasound distance sensor, experimental verification of thesolution of Lagrange equations for longitudinal oscillations ofthe Wilberforce pendulum was shown. In this paper the sensor anda digital video camera were used to monitor and measure thechanges of both the pendulum's coordinates (verticaldisplacement and angle of rotation) simultaneously. Theexperiments were performed with the aid of the integratedsoftware package COACH 5. Fourier analysis in Microsoft Excel 97 was used to find normal modes in each case of the measured oscillations. Comparison of theresults with those presented in our previous paper (as givenabove) leads to the conclusion that a digital video camera is apowerful tool for measuring coupled oscillations of aWilberforce pendulum. The most important conclusion is that avideo camera is able to do something more than merely registerinteresting physical phenomena - it can be used to performmeasurements of physical quantities at an advanced level.

Computer visualization of the beating of a Wilberforce pendulum

In this paper experimental verification of the important features of the motion of the Wilberforce pendulum is shown. The vertical displacement of the pendulum has been `watched' by computer in real time. The experiments were carried out with different combinations of initial conditions. Fourier analysis of the experimental results revealed the existence of two frequencies corresponding to two normal modes. When their amplitudes are equal the modes combine to produce `beats'. In this case the frequencies of longitudinal and torsional vibrations of the pendulum are equal. This is called the resonance case. Our experiment allows one to observe not only the beats of the two normal modes but also each of the modes separately.

Apparatus for teaching physics: Wilberforce pendulum, demonstration size

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Wilberforce pendulum oscillations and normal modes

This article summarizes our calculation of the normal modes and the normal coordinates for a commercially available Wilberforce pendulum. A procedure is presented by which the normal coordinates may be produced experimentally, so that the frequencies of the normal modes can be obtained both theoretically and experimentally. Where possible, results have been compared with those from previous papers. Finally, PC basic programs have been written in which the behavior of the Wilberforce pendulum has been theoretically reproduced.

L.R. Wilberforce and the Wilberforce Pendulum

L.R. Wilberforce and the Wilberforce Pendulum