Falling chains

The one-dimensional fall of a folded chain with one end suspended from a rigid support and a chain falling from a resting heap on a table is studied. Because their Lagrangians contain no explicit time dependence, the falling chains are conservative systems. Their equations of motion are shown to contain a term that enforces energy conservation when masses are transferred between subchains. We show that Cayley's 1857 energy nonconserving solution for a chain falling from a resting heap is incorrect because it neglects the energy gained when a transferred link leaves a subchain. The maximum chain tension measured by Calkin and March for the falling folded chain is given a simple if rough interpretation. Other aspects of this falling folded chain are briefly discussed.Comment: 9 pages, 1 figure; the Abstract has been shortened, three paragraphs have been re-written for greater clarity, and textual improvements have been made throughout the paper; to be published by the Am. J. Physic

Singularity-Free Breit Equation from Constraint Two-Body Dirac Equations

We examine the relation between two approaches to the quantum relativistic two-body problem: (1) the Breit equation, and (2) the two-body Dirac equations derived from constraint dynamics. The Breit equation is known to be pathological when singularities appear at finite separations $r$ in the reduced set of coupled equations for attractive potentials even when the potentials themselves are not singular there. They then give rise to unphysical bound states and resonances. In contrast, the two-body Dirac equations of constraint dynamics do not have these pathologies in many nonperturbative treatments. To understand these marked differences, we first express these contraint equations in a hyperbolic form. These coupled equations are then re-cast into two equivalent equations: (1) a covariant Breit-like equation with potentials that are exponential functions of certain ``generator'' functions, and (2) a covariant orthogonality constraint on the relative momentum. This reduction enables us to show in a transparent way that finite-$r$ singularities do not appear as long as the the exponential structure is not tampered with and the exponential generators of the interaction are themselves nonsingular for finite $r$. These Dirac or Breit equations, free of the structural singularities which plague the usual Breit equation, can then be used safely under all circumstances, encompassing numerous applications in the fields of particle, nuclear, and atomic physics which involve highly relativistic and strong binding configurations.Comment: 38 pages (REVTeX), (in press in International Journal of Modern Physics

Electroproduction of the d* dibaryon

The unpolarized cross section for the electroproduction of the isoscalar $J^\pi = 3^+$ di-delta dibaryon $d^*$ is calculated for deuteron target using a simple picture of elastic electron-baryon scattering from the $\Delta \Delta (^7D_1)$ and the $NN (^3S_1)$ components of the deuteron. The calculated differential cross section at the electron lab energy of 1 GeV has the value of about 0.24 (0.05) nb/sr at the lab angle of 10$^\circ$ (30$^\circ$) for the Bonn B potential when the dibaryon mass is taken to be 2.1 GeV. The cross section decreases rapidly with increasing dibaryon mass. A large calculated width of 40 MeV for $d^*(\Delta\Delta ^7S_3)$ combined with a small experimental upper bound of 0.08 MeV for the $d^*$ decay width appears to have excluded any low-mass $d^*$ model containing a significant admixture of the $\Delta\Delta (^7S_3)$ configuration.Comment: 11 journal-style pages, 8 figure

Color mixing in high-energy hadron collisions

The color mixing of mesons propagating in a nucleus is studied with the help of a color-octet Pomeron partner present in the two-gluon model of the Pomeron. For a simple model with four meson-nucleon channels, color mixings are found to be absent for pointlike mesons and very small for small mesons. These results seem to validate the absorption model with two independent color components used in recent analyses of the nuclear absorption of $J/\psi$ mesons produced in nuclear reactions.Comment: 3 journal-style page

Three-body decay of the d* dibaryon

Under certain circumstances, a three-body decay width can be approximated by an integral involving a product of two off-shell two-body decay widths. This ``angle-average'' approximation is used to calculate the $\pi NN$ decay width of the $d^*(J^\pi=3^+, T=0)$ dibaryon in a simple $\Delta^2$ model for the most important Feynman diagrams describing pion emissions with baryon-baryon recoil and meson retardation. The decay width is found to be about 0.006 (0.07, 0.5) MeV at the $d^*$ mass of 2065 (2100, 2150) MeV for input dynamics derived from the Full Bonn potential. The smallness of this width is qualitatively understood as the result of the three-body decay being ``third forbidden''. The concept of $\ell$ forbiddenness and the threshold behavior of a three-body decay are further studied in connection with the $\pi NN$ decay of the dibaryon $d'(J^\pi=0^-, T=0 or 2)$ where the idea of unfavorness has to be introduced. The implications of these results are briefly discussed.Comment: 15 pages, RevTeX, two-column journal style, six figure

A mobile game (safe city) designed to promote children's safety knowledge and behaviors: protocol for a randomized controlled trial

Background: Children have high levels of curiosity and eagerness to explore. This makes them more vulnerable to danger and hazards, and they thus have a higher risk of injury. Safety education such as teaching safety rules and tips is vital to prevent children from injuries. Although game-based approaches have the potential to capture children’s attention and sustain their interest in learning, whether these new instructional approaches are more effective than traditional approaches in delivering safety messages to children remains uncertain. Objective: The aim of this study is to test the effectiveness of a game-based intervention in promoting safety knowledge and behaviors among Hong Kong school children in Grades 4-6. It will also examine the potential effect of the game-based intervention on these children’s functioning and psychosocial difficulties. Methods: This study comprises the development of a city-based role-playing game Safe City, where players are immersed as safety inspectors to prevent dangerous situations and promote safety behavior in a virtual city environment. The usability and acceptability tests will be conducted with children in Grades 4-6 who will trial the gameplay on a mobile phone. Adjustments will be made based on their feedback. A 4-week randomized controlled trial with children studying in Grades 4-6 in Hong Kong elementary schools will be conducted to assess the effectiveness of the Safe City game–based intervention. In this trial, 504 children will play Safe City, and 504 children will receive traditional instructional materials (electronic and printed safety information). The evaluation will be conducted using both child self-report and parent proxy-report data. Specifically, child safety knowledge and behaviors will be assessed by a questionnaire involving items on knowledge and behaviors, respectively, for home safety, road safety, and sport-related safety; child functioning will be assessed by PedsQL Generic Core Scales; and psychosocial difficulties will be assessed by the Strength and Difficulties Questionnaire. These questionnaires will be administered at 3 time points: before, 1 month, and 3 months after the intervention. Game usage statistics will also be reviewed. Results: This project was funded in September 2019. The design and development of the Safe City game are currently under way. Recruitment and data collection will begin from September 2020 and will continue up to March 1, 2021. Full analysis will be conducted after the end of the data collection period. Conclusions: If the Safe City game is found to be an effective tool to deliver safety education, it could be used to promote safety in children in the community and upgraded to incorporate more health-related topics to support education and empowerment for the larger public. Trial Registration: ClinicalTrials.gov NCT04096196; https://clinicaltrials.gov/ct2/show/NCT04096196 International Registered Report Identifier (IRRID): PRR1-10.2196/1775