bluuebunny/crossref_metadata_2025_split huggingface dataset:
Spring suspensions for accurate pendulums
Authors: Matthys Robert James 
Published: January 2010

Abstract
This chapter deals with the design and performance of various types of spring suspensions for an accurate pendulum clock. It also covers the three basic oscillation modes of a pendulum, the interactions between them, and the effects they have on the clock's timing accuracy. These oscillation modes affect the design of the suspension. To simplify the discussion, left-right pendulum motion is used to refer to the normal left-right motion of the pendulum bob, as viewed from in front of the clock case. Front-to-back pendulum motion refers to the front-to-back motion of the pendulum bob, again as viewed from in front of the clock case. The pendulum rod is made of invar. The lengths given for suspension springs are the free flexure lengths, and do not include the clamped portions at the ends of the springs. The springs in this chapter are made of thin flat stock, with their ends thickened afterward by clamping or soldering ‘chops’ (thicker end pieces) onto the spring's two ends.
James’ suspension spring equations
Authors: Matthys Robert James 
Published: January 2010

Abstract
In 1983, K. James published two useful but rather complicated equations whose purpose was to help design the suspension spring. The equations show the effect that different spring lengths, widths, and thicknesses will have on the pendulum. The two equations are quite helpful, as the suspension spring is without doubt the most complicated part of a pendulum, despite its apparent physical simplicity. The first equation calculates the maximum stress in the spring, which occurs at the spring's top end at the maximum angle of swing. The second calculates how much the pendulum will speed up due to the inherent torque exerted by the suspension spring on the pendulum rod. In this chapter, the second equation is used to show that the suspension spring exerts a temperature effect on the pendulum's timing that is roughly as big as the thermal expansion of the pendulum rod. Anywhere from 16% to 84% of a pendulum's total temperature sensitivity is due to the suspension spring, with the actual amount depending on the spring's dimensions, modulus of elasticity, and suspended weight.
Solid one-piece suspension springs
Authors: Matthys Robert James 
Published: January 2010

Abstract
This chapter discusses how to make solid one-piece type of pendulum suspension springs. There are three types of pendulum suspension springs: the mechanically clamped type, the soldered assembly type, and the solid one-piece type. The mechanically clamped type is the easiest to make, but its clock rate is variable and dependent on the clamping forces and clamping surfaces. A 10-microinch change in the length of a 1-second beat pendulum amounts to an error of 3.7 seconds per year. To meet the microinch length tolerance needed for an accurate pendulum clock, a mechanically bolted assembly must have microinch tolerances on the clamped surfaces, that is, an optical finish. Mechanically machined surfaces are not good enough, and introduce uncertainty as to where the spring ends and the end clamp actually begins. The uncertainty causes the pendulum's length and its timing to vary with temperature and the suspension spring's clamping pressure.
Pendulum rod materials
Authors: Matthys Robert James 
Published: January 2010

Abstract
Over time, several different materials have been used for the pendulum rod such as steel, wood, and invar. The best material is quartz because of its proven stability and low thermal expansion. Steel is used for the pendulum rod in simple ordinary clocks because it is cheap and has relatively low thermal expansion. Wood is sometimes recommended because of its low linear thermal expansion coefficient along the grain, but it is an inherently unstable material. It warps, splits, and exhibits a high mechanical creep under load. Worst of all, wood expands and contracts considerably with relative humidity. Invar is a mixture of 36% nickel and 63% iron. It is magnetic and rusts in a humid environment. A new material of interest for the pendulum rod is carbon fibre, but it may not work too well as a pendulum rod as the epoxy absorbs moisture, changing the rod's length and weight.
Fasteners for quartz pendulum rods
Authors: Matthys Robert James 
Published: January 2010

Abstract
This chapter describes five different ways of fastening things to a quartz pendulum rod. To connect to a metal rod, a hole is drilled in it or a thread is cut on it. But how do you fasten something to quartz? Quartz is like glass it is brittle and breaks easily. Five types of fasteners are described here, with some pros and cons on each. The five fasteners are cemented sleeve, clamp ring, solder joint, dowel pin, and split sleeve. Any of the fasteners can be used at either the top or bottom of a quartz rod, for connecting to a suspension spring, bob, or rating nut. In most cases, the fastener material should be invar because of its low thermal expansion coefficient.
Correcting the pendulum’s air pressure error
Authors: Matthys Robert James 
Published: January 2010

Abstract
This chapter describes how to calculate and remove the air pressure error from a clock's time error versus time chart. Depending on geographic location and the density of the pendulum materials, air pressure variations can cause roughly 2-18 seconds of error in a clock in a year's time interval. This is because the pendulum ‘floats’ in a sea of air, and variations in the air pressure make the pendulum slow down or speed up. In this chapter, the pressure error itself is described first, followed by an actual clock data run to show how the pressure error is calculated and then removed from a clock's time error versus time curve. Correcting for the pressure error corrects the clock's time to a constant average pressure at the clock site. The error is the product of the pressure difference times time. More specifically, it is the integral of the pressure difference with respect to time.
Effect of the clock case walls on a pendulum
Authors: Matthys Robert James 
Published: January 2010

Abstract
This chapter describes an experiment designed to determine the effect of the walls of a pendulum clock case on a pendulum. The basic experimental approach was to measure the pendulum's clock rate and drive force, with the walls set at different distances from the pendulum. All measurements were made at atmospheric pressure. The walls affect the pendulum via the air between the walls and the pendulum. The bob is the biggest part of a pendulum, so its size and possibly its shape are important in determining the walls' effect on the pendulum. The results show that the walls made the pendulum run slower and increased the drive force. The walls' slowdown effect on the pendulum is independent of the angle of swing. Relative humidity causes changes in clock rate that are rather large for an accurate clock.
Sinusoidal drive of a pendulum
Authors: Matthys Robert James 
Published: January 2010

Abstract
A sinusoidal drive is believed to be ideal for a pendulum. In 1960, an experiment was carried out demonstrating a semi-sinusoidal pendulum drive consisting of a sinusoid with its peaks clipped off. Here, a pendulum-mounted coil moving in a fixed magnetic field was used to generate a sinusoidal velocity signal. The peaks of this sinusoidal signal were then clipped off at a fixed amplitude, using a parallel reversed set of two diodes and two batteries. The peak-clipped sine wave was then fed to a second pendulum-mounted coil moving in another fixed magnetic field. The magnetic force generated the peak-clipped sine wave current in the second coil drove the pendulum and kept it running. In 1993, a very simple semi-sinusoidal drive circuit that had only one op-amp and one coil-and-magnet structure instead of two was reported. The drive coil both senses and drives the pendulum. The pendulum's drive force is also a peak-clipped sine wave. In this chapter, four electronic circuits for sinusoidally or semi-sinusoidally driving a pendulum are analysed, and their pros and cons discussed.
Kinetic Theory of Granular Gases
Authors: Brilliantov Nikolai V. , Pöschel Thorsten 
Published: January 2010

Abstract
Kinetic Theory of Granular Gases provides an introduction to the rapidly developing theory of dissipative gas dynamics — a theory which has mainly evolved over the last decade. The book is aimed at readers from the advanced undergraduate level upwards and leads on to the present state of research. Throughout, special emphasis is put on a microscopically consistent description of pairwise particle collisions which leads to an impact-velocity-dependent coefficient of restitution. The description of the many-particle system, based on the Boltzmann equation, starts with the derivation of the velocity distribution function, followed by the investigation of self-diffusion and Brownian motion. Using hydrodynamical methods, transport processes and self-organized structure formation are studied. An appendix gives a brief introduction to event-driven molecular dynamics. A second appendix describes a novel mathematical technique for derivation of kinetic properties, which allows for the application of computer algebra. The text is self-contained, requiring no mathematical or physical knowledge beyond that of standard physics undergraduate level. The material is adequate for a one-semester course and contains chapter summaries as well as exercises with detailed solutions. The molecular dynamics and computer-algebra programs can be downloaded from a companion web page.
Pseudo-Liouville and Binary Collision Operators in Dissipative Gas Dynamics
Authors: Brilliantov Nikolai V. , Pöschel Thorsten 
Published: January 2010

Abstract
For hard spheres the interaction potential diverges, that is, an infinite force arises when two particles are brought into contact. This causes serious problems when the standard tools of kinetic theory are applied to hard-sphere fluids. The pseudo-Liouville operator was developed to avoid this problem. This chapter illustrates its application to dissipative gases.
Brownian Motion in Granular Gases
Authors: Brilliantov Nikolai V. , Pöschel Thorsten 
Published: January 2010

Abstract
This chapter analyzes Brownian motion in granular gases. Topics discusses include Boltzmann equation for the velocity distribution function of Brownian particles, Fokker–Planck equation for Brownian particles, velocity distribution function for Brownian particles, and diffusion of Brownian particles.
Kinetic Coefficients For Granular Gases Of Simplified Particles (ε = CONST)
Authors: Brilliantov Nikolai V. , Pöschel Thorsten 
Published: January 2010

Abstract
This chapter generalizes the Chapman–Enskog approach for the case of granular gases whose particles collide inelastically. Under the assumption of constant coefficient of restitution, the kinetic coefficients and velocity distribution function are derived.
Foundations
Authors: BARNETT STEPHEN M. , RADMORE PAUL M. 
Published: January 2010

AbstractThis chapter presents the basic elements of quantum theory, including the Dirac bra and ket notation for quantum states, Hermitian operators and their eigenstates, commutators and the Schrödinger and Heisenberg equations of motion. Simple model atoms are described as an effective spin and their properties related to the familiar Pauli spin-half operators. The density matrix is introduced to describe incoherent processes. The electric-dipole coupling between and atom and the electromagnetic field is treated leading to coupled equations for the probability amplitudes. Finally, quantized field modes are introduced and their elementary properties described leading to the fundamental Jaynes–Cummings model of a single two-state atom interacting with a quantized field mode.
Operators and States
Authors: BARNETT STEPHEN M. , RADMORE PAUL M. 
Published: January 2010

AbstractThis chapter is concerned with providing the necessary rules governing the manipulation of the operators and the properties of the states to enable us top model and describe some physical systems. Atom and field operators for spin, angular momentum, the harmonic oscillator or single mode field and for continuous fields are introduced. Techniques are derived for treating functions of operators and ordering theorems for manipulating these. Important states of the electromagnetic field and their properties are presented. These include the number states, thermal states, coherent states and squeezed states. The coherent and squeezed states are generated by the actions of the Glauber displacement operator and the squeezing operator. Angular momentum coherent states are described and applied to the coherent evolution of a two-state atom and to the action of a beam-splitter.
Quantum Statistics of Fields
Authors: BARNETT STEPHEN M. , RADMORE PAUL M. 
Published: January 2010

AbstractThis chapter considers methods for describing the quantum statistics of a single mode of the electromagnetic field. Some of these methods may be extended to multimode fields, and this is discussed in the last section of the chapter. The moment generating function is developed for studying the photon number statistics of a single field mode. The quantum properties of optical phase are described using the optical phase operator. The characteristic functions and quasi-probability distributions provide a complete statistical description of the field. These rely on the properties the coherent states and the Glauber displacement operator.
Analytical Mechanics for Relativity and Quantum Mechanics
Authors: Johns Oliver 
Published: January 2010

Abstract
This book provides an innovative and mathematically sound treatment of the foundations of analytical mechanics and the relation of classical mechanics to relativity and quantum theory. A distinguishing feature of the book is its integration of special relativity into teaching of classical mechanics. After a thorough review of the traditional theory, the book introduces extended Lagrangian and Hamiltonian methods that treat time as a transformable coordinate rather than the fixed parameter of Newtonian physics. Advanced topics such as covariant Langrangians and Hamiltonians, canonical transformations, and Hamilton-Jacobi methods are simplified by the use of this extended theory. And the definition of canonical transformation no longer excludes the Lorenz transformation of special relativity. This is also a book for those who study analytical mechanics to prepare for a critical exploration of quantum mechanics. Comparisons to quantum mechanics appear throughout the text. The extended Hamiltonian theory with time as a coordinate is compared to Dirac’s formalism of primary phase space constraints. The chapter on relativistic mechanics shows how to use covariant Hamiltonian theory to write the Klein-Gordon and Dirac equations. The chapter on Hamilton-Jacobi theory includes a discussion of the closely related Bohm hidden variable model of quantum mechanics. Classical mechanics itself is presented with an emphasis on methods, such as linear vector operators and dyadics, that will familiarise the student with similar techniques in quantum theory. Several of the current fundamental problems in theoretical physics, such as the development of quantum information technology and the problem of quantising the gravitational field, require a rethinking of the quantum-classical connection.
The Calculus of Variations
Authors: Johns Oliver 
Published: January 2010

Abstract
This chapter introduces the calculus of variations in the context of the finite-dimensional configuration space discussed previously. The calculus of variations is concerned with the comparison of line integrals along different paths. The difference between the integral along some chosen path and the integral of the same quantity along other paths is called the variation of that integral. This chapter discusses paths in an N-dimensional space, variations of coordinates, variations of functions, variation of a line integral, finding extremum paths, example of an extremum path calculation, invariance and homogeneity, the Brachistochrone problem, calculus of variations with constraints, example with constraints, reduction of degrees of freedom, example of a reduction, example of a better reduction, and the coordinate parametric method.
10 Small Vibrations About Equilibrium
Authors: Johns Oliver 
Published: January 2010

Abstract
A number of interesting mechanical systems have one or more essentially stable equilibrium configurations. When disturbed slightly, they vibrate about equilibrium in characteristic patterns called normal modes. This chapter presents the Lagrangian theory of these small vibrations for the simple case of systems with a finite number of degrees of freedom. The theory has wide application. For example, the normal mode oscillations of crystalline solids underlie both the overtone structure of a church bell and the definition of phonons in solid state physics. A similar formalism leads to photons as the quanta of modes of the electromagnetic field. This chapter defines equilibrium points in the configuration space of a mechanical system and discusses how to find them, along with small coordinates, normal modes, generalised eigenvalue problem, stability, initial conditions, energy of small vibrations, single mode excitations, and zero-frequency modes.
Relativity and Spacetime
Authors: Johns Oliver 
Published: January 2010

Abstract
When it was proposed at the start of the 20th century, special relativity was revolutionary and controversial. Now, some hundred years later, the consensus in physics is that, at least in local spaces where the effects of curvature are negligible, standard special relativity will reliably predict the outcome of any experiment to which it is applied. The Lorentz transformation is considered a symmetry of nature, and new theories are crafted with Lorentz invariance as a necessary feature. This chapter introduces the ideas of special relativity with the assumption that it is an established theory whose experimental efficacy is unquestioned. The goal is to help the reader understand the theory more deeply, to see what the relativistic effects may be telling us about the world and the nature of space-time. Galileo’s principle of relativity is discussed, along with the conflict between the Galilean principle of relativity and the aether, Albert Einstein’s relativity, coordinate systems, Lorentz transformation, Lorentzian relativity, and link between mechanism and relativity.
Fourvectors and Operators
Authors: Johns Oliver 
Published: January 2010

Abstract
This chapter develops techniques that allow relativistically covariant calculations to be done in an elegant manner and introduces what are known as fourvectors. Fourvectors are analogous to the familiar vectors in three-dimensional Cartesian space (termed threevectors), except that, in additional to the three spatial components, fourvectors will have an additional zeroth component associated with time. This additional component allows us to deal with the fact that the Lorentz transformation of special relativity transforms time as well as spatial coordinates. The theory of fourvectors and operators is presented using an invariant notation. The concept of fourvectors and tensors is discussed in the simple context of special relativity, as well as the choice of metric, relativistic interval, space-time diagram, general fourvectors and construction of new fourvectors, covariant and contravariant components, general Lorentz transformations, transformation of components, examples of Lorentz transformations, gradient fourvector, manifest covariance, formal covariance, fourvector operators, fourvector dyadics, wedge products, and manifestly covariant form of Maxwell’s equations.
Hamilton–Jacobi Theory
Authors: Johns Oliver 
Published: January 2010

Abstract
The Hamilton-Jacobi theory is the apotheosis of Lagrangian and Hamiltonian mechanics: action functions encode all of the possible trajectories of a mechanical system satisfying certain criteria. These action functions are the solutions of a nonlinear, first-order partial differential equation, called the Hamilton-Jacobi equation. The characteristic equations of this differential equation are the extended Hamilton equations. Solution of a class of mechanics problems is thus reduced to the solution of a single partial differential equation. Aside from its use as a problem-solving tool, the Hamilton-Jacobi theory has particular importance because of its close relation to the Schroedinger formulation of quantum mechanics. This chapter discusses the connection between the Hamilton-Jacobi theory and the Schroedinger formulation, the Bohm hidden variable model and Feynman path integral method that are derived from it, Hamilton’s characteristic equations, complete integrals, separation of variables, canonical transformations, general integrals, mono-energetic integrals, relativistic Hamilton-Jacobi equation, quantum Cauchy problem, quantum mechanics, and classical mechanics.
Quantum chemistry
Authors: GAVEZZOTTI ANGELO 
Published: January 2010

Abstract
In the early decades of the 20th century, a large and convincing body of experimental evidence pointed to the fact that atomic particles share some properties of massive bodies and some properties of waves. The founders of the new mechanics of the microscopic world, in search of the appropriate equations for the description of what seemed then a very weird universe, turned to the equations of vibrating strings — material waves whose state and energy change in leaps with the number of nodes. There is a certain resemblance between the quantum mechanical Schrodinger equation and the classical dynamic equation of vibrating strings. This chapter discusses the fundamentals of quantum mechanics, dynamic variables, wavefunctions, operators, the Schrödinger equation and stationary states, hydrogen atom and atomic orbitals, approximate quantum chemical methods, evolution of quantum chemical calculations, dimerisation energies and basis set superposition error, and early experiences in quantum chemistry.
The physical nature and the computer simulation of the intermolecular potential
Authors: GAVEZZOTTI ANGELO 
Published: January 2010

Abstract
Everyday objects display an almost incredible variety of properties that may be felt with animal senses and that human intellect wishes to apprehend, categorise, understand, and — possibly — predict. We have gaseous bodies such as the Earth's atmosphere, liquid bodies such as a drop of water, solid bodies such as glasses or crystals, semi-solid bodies such as gels, or even such exotic states of condensation as liquid crystals. The distribution of electric charges in a molecule plays a central role in all discussions of intermolecular interactions. This chapter discusses the physical nature and computer simulations of the intermolecular potential, the representation of the molecular charge distribution and of the electric potential, full electron density, central multipoles and distributed multipoles, point charges, Coulombic potential energy, polarisation (electrostatic induction) energy, dispersion energy, Pauli (exchange) repulsion energy, total energies versus partitioned energies, intermolecular hydrogen bonding, simulation methods, and force field fitting from ab initio calculations.