DOCSLIB.ORG

Quantum Physics Explains Newton's Laws of Motion

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Quantum Physics Explains Newton's Laws of Motion FEATURES www.iop.org/journals/physed Quantum physics explains Newton’s laws of motion Jon Ogborn1 and Edwin F Taylor2 1 Institute of Education, University of London, UK 2 Massachusetts Institute of Technology, Cambridge, MA 02139, USA Abstract Newton was obliged to give his laws of motion as fundamental axioms. But today we know that the quantum world is fundamental, and Newton’s laws can be seen as consequences of fundamental quantum laws. This article traces this transition from fundamental quantum mechanics to derived classical mechanics. Explaining Fermat’s principle: source of the key It is common to present quantum physics and the quantum idea behaviour of quantum objects such as electrons or Geometrical optics predicts the formation of photons as mysterious and peculiar. And indeed, images by light rays. Examples are images due in Richard Feynman’s words, electrons do ‘behave to reflection and images formed by eyeglasses and in their own inimitable way . in a way that is like camera lenses. All of geometrical optics—every nothing that you have ever seen before.’ (Feynman path of every light ray—can be predicted from a 1965, p 128). single principle: Between source and reception But it was also Richard Feynman who point light travels along a path that takes the devised a way to describe quantum behaviour with shortest possible time. This is called Fermat’s astounding simplicity and clarity (Feynman 1985). principle after the Frenchman Pierre de Fermat There is no longer any need for the mystery that (1601–1665). comes from trying to describe quantum behaviour A simple example of Fermat’s principle is the as some strange approximation to the classical law of reflection: angle of incidence equals angle behaviour of waves and particles. Instead we of reflection. Fermat’s principle also accounts turn the job of explaining around. We start from for the action of a lens: a lens places different quantum behaviour and show how this explains classical behaviour. thicknesses of glass along different paths so that This may make you uncomfortable at first. every ray takes the same time to travel from a Why explain familiar things in terms of something point on the source to the corresponding point on unfamiliar? But this is the way explanations have the image. Fermat’s principle accounts for how to work. Explanations that don’t start somewhere a curved mirror in a telescope works: the mirror else than what they explain don’t explain! is bent so that each path takes the same time to In this article we show that quantum reach the focus. These and other examples are mechanics actually explains why Newton’s laws discussed in the Advancing Physics AS Student’s of motion are good enough to predict how footballs Book (Ogborn and Whitehouse 2000). and satellites move. For Newton, fundamental Fermat’s contemporaries had a fundamental laws had to be axioms—starting points. For us, objection to his principle, asking him a profound Newton’s laws are seen to be consequences of the question, ‘How could the light possibly know in fundamental way the quantum world works. advance which path is the quickest?’ The answer 26 P HYSICS E DUCATION 40 (1) 0031-9120/05/010026+09$30.00 © 2005 IOP Publishing Ltd Quantum physics explains Newton’s laws of motion goes very deep and was delivered fully only in the emission of a photon at one place and time and its twentieth century. Here is the key idea: The light detection at a different place and time. (There are explores all possible paths between emission and also rules for how the length of the arrow changes reception. Later we will find a similar rule for with distance, which yield an inverse square law motion of atomic particles such as the electron: a of intensity with distance. For simplicity we particle explores all possible paths between source consider only cases where the distances vary little and detector. This is the basic idea behind the and changes in arrow length can be ignored.) formulation of quantum mechanics developed by The resultant arrow determines the probabil- Richard Feynman (1985). ity of the event. The probability is equal to the The idea of exploring all possible paths raises (suitably normalized) square of the length of the two deep questions: (1) What does it mean to arrow. In this way the classical result that the in- explore all paths? (2) How can ‘explore all tensity is proportional to the square of the wave paths’ be reconciled with the fact that everyday amplitude is recovered. objects (such as footballs) and light rays follow We have outlined Feynman’s simple and vivid unique single paths? To answer these questions description of ‘quantum behaviour’ for a photon. is to understand the bridge that connects quantum In effect he steals the mathematics of Huygens’ mechanics to Newton’s mechanics. wavelets without assuming that there are waves. For Huygens, wavelets go everywhere because What does Explore all paths! mean? that is what waves do. For Feynman, photons ‘go everywhere’ because that is what photons do. The idea of exploring all paths descends from Christiaan Huygens’ idea of wavelets (1690). Huygens explained the propagation of a wavefront The bridge from quantum to classical by imagining that each point on the wavefront physics sends out a spherical wavelet. He could then The next question is this: How can Nature’s show that the wavelets reconstitute the wavefront command Explore all paths! be made to fit with at a later time; the parts of the wavelets going our everyday observation that an object such as a everywhere else just cancel each other out. In 1819 football or a light ray follows a single path? the French road and bridge engineer Augustin The short answer is It does not follow a Fresnel put the idea on a sound mathematical single path! There is no clean limit between basis and used it to explain optical diffraction and particles that can be shown to explore many paths interference effects in precise detail. and everyday objects. What do you mean by In the 1940s Richard Feynman (following a everyday objects anyway? Things with a wide hint from Dirac) adapted Huygens’ idea to give range of masses and structural complexity are quantum physics a new foundation, starting with ‘everyday’ objects of study by many scientists. the quantum of light: the photon. Nature’s simple A recent example is interference observed for three-word command to the photon is Explore all the large molecules of the fullerene carbon-70, paths!; try every possible route from source to which has the approximate mass of 840 protons detector. Each possible path is associated with a (www.quantum.at). change of phase. One can imagine a photon having In fact quantum behaviour tapers off gradually a ‘stopclock’ whose hand rotates at the classical into classical behaviour. This and the following frequency of the light. The rotation starts when sections show you how to predict the range of the photon is emitted; the rotation stops when the this taper for various kinds of observations. In the photon arrives at the detector. The final position meantime, as you look around you, think about the of the hand gives an ‘arrow’ for that path. deep sense in which the football goes from foot to The photon explores all paths between goal by way of Japan. So do the photons by which emission event and a possible detection event. The you see your nearby friend! arrows for all paths are to be added head-to-tail The key idea is illustrated in figure 1 for the (that is, taking account of their phases, just as case of photons reflected from a mirror. The mirror wavelets are to be superposed) to find the total is conceptually divided into little segments, sub- resultant quantum amplitude (resultant arrow) for mirrors labelled A to M. The little arrows shown an event. This resultant arrow describes the under each section in the middle panel correspond January 2005 P HYSICS E DUCATION 27 J Ogborn andEFTaylor in many directions, so total contributions from SPthese end-segments never amount to much. Even if we extend the mirror AM on each side to make it longer, the contributions to the resulting arrow made by reflections from the sections of these ABCDEFGHI J KLM right-and-left extensions add almost nothing to the total resulting arrow. Why? Because they will all curl even more tightly than arrows ABC and KLM at the two ends of the resulting arrow shown in the bottom panel of the figure. Most of the time resulting arrow comes from the small proportion of arrows that ‘line up’, and almost nothing comes from those that ‘curl up’. ABCDEFGHI J KLM Now you see how Explore all paths! leads to a narrow spread of paths that contribute significantly to the resulting arrow at P. And that narrow spread I J must lie near the path of minimum time of travel, H M B because that is where the time, and so the phase, C K L varies only slightly from path to nearby path. A G Here then is the answer to Fermat’s critics. D F E They asked ‘How could the light possibly know in Figure 1. Many paths account of reflection at a mirrror advance which path is the quickest?’ Answer: the (adapted from Feynman 1985, p 43). Each path the light photon does not know in advance: it explores all could go (in this simplified situation) is shown at the top, with a point on the graph below showing the time paths.
Load more

Recommended publications
  • Path Integrals in Quantum Mechanics
    Path Integrals in Quantum Mechanics Dennis V. Perepelitsa MIT Department of Physics 70 Amherst Ave. Cambridge, MA 02142 Abstract We present the path integral formulation of quantum mechanics and demon- strate its equivalence to the Schr¨odinger picture. We apply the method to the free particle and quantum harmonic oscillator, investigate the Euclidean path integral, and discuss other applications. 1 Introduction A fundamental question in quantum mechanics is how does the state of a particle evolve with time? That is, the determination the time-evolution ψ(t) of some initial | i state ψ(t ) . Quantum mechanics is fully predictive [3] in the sense that initial | 0 i conditions and knowledge of the potential occupied by the particle is enough to fully specify the state of the particle for all future times.1 In the early twentieth century, Erwin Schr¨odinger derived an equation specifies how the instantaneous change in the wavefunction d ψ(t) depends on the system dt | i inhabited by the state in the form of the Hamiltonian. In this formulation, the eigenstates of the Hamiltonian play an important role, since their time-evolution is easy to calculate (i.e. they are stationary). A well-established method of solution, after the entire eigenspectrum of Hˆ is known, is to decompose the initial state into this eigenbasis, apply time evolution to each and then reassemble the eigenstates. That is, 1In the analysis below, we consider only the position of a particle, and not any other quantum property such as spin. 2 D.V. Perepelitsa n=∞ ψ(t) = exp [ iE t/~] n ψ(t ) n (1) | i − n h | 0 i| i n=0 X This (Hamiltonian) formulation works in many cases.
    [Show full text]
  • Motion and Time Study the Goals of Motion Study
    Motion and Time Study The Goals of Motion Study • Improvement • Planning / Scheduling (Cost) •Safety Know How Long to Complete Task for • Scheduling (Sequencing) • Efficiency (Best Way) • Safety (Easiest Way) How Does a Job Incumbent Spend a Day • Value Added vs. Non-Value Added The General Strategy of IE to Reduce and Control Cost • Are people productive ALL of the time ? • Which parts of job are really necessary ? • Can the job be done EASIER, SAFER and FASTER ? • Is there a sense of employee involvement? Some Techniques of Industrial Engineering •Measure – Time and Motion Study – Work Sampling • Control – Work Standards (Best Practices) – Accounting – Labor Reporting • Improve – Small group activities Time Study • Observation –Stop Watch – Computer / Interactive • Engineering Labor Standards (Bad Idea) • Job Order / Labor reporting data History • Frederick Taylor (1900’s) Studied motions of iron workers – attempted to “mechanize” motions to maximize efficiency – including proper rest, ergonomics, etc. • Frank and Lillian Gilbreth used motion picture to study worker motions – developed 17 motions called “therbligs” that describe all possible work. •GET G •PUT P • GET WEIGHT GW • PUT WEIGHT PW •REGRASP R • APPLY PRESSURE A • EYE ACTION E • FOOT ACTION F • STEP S • BEND & ARISE B • CRANK C Time Study (Stopwatch Measurement) 1. List work elements 2. Discuss with worker 3. Measure with stopwatch (running VS reset) 4. Repeat for n Observations 5. Compute mean and std dev of work station time 6. Be aware of allowances/foreign element, etc Work Sampling • Determined what is done over typical day • Random Reporting • Periodic Reporting Learning Curve • For repetitive work, worker gains skill, knowledge of product/process, etc over time • Thus we expect output to increase over time as more units are produced over time to complete task decreases as more units are produced Traditional Learning Curve Actual Curve Change, Design, Process, etc Learning Curve • Usually define learning as a percentage reduction in the time it takes to make a unit.
    [Show full text]
  • 1 the Principle of Wave–Particle Duality: an Overview
    3 1 The Principle of Wave–Particle Duality: An Overview 1.1 Introduction In the year 1900, physics entered a period of deep crisis as a number of peculiar phenomena, for which no classical explanation was possible, began to appear one after the other, starting with the famous problem of blackbody radiation. By 1923, when the “dust had settled,” it became apparent that these peculiarities had a common explanation. They revealed a novel fundamental principle of nature that wascompletelyatoddswiththeframeworkofclassicalphysics:thecelebrated principle of wave–particle duality, which can be phrased as follows. The principle of wave–particle duality: All physical entities have a dual character; they are waves and particles at the same time. Everything we used to regard as being exclusively a wave has, at the same time, a corpuscular character, while everything we thought of as strictly a particle behaves also as a wave. The relations between these two classically irreconcilable points of view—particle versus wave—are , h, E = hf p = (1.1) or, equivalently, E h f = ,= . (1.2) h p In expressions (1.1) we start off with what we traditionally considered to be solely a wave—an electromagnetic (EM) wave, for example—and we associate its wave characteristics f and (frequency and wavelength) with the corpuscular charac- teristics E and p (energy and momentum) of the corresponding particle. Conversely, in expressions (1.2), we begin with what we once regarded as purely a particle—say, an electron—and we associate its corpuscular characteristics E and p with the wave characteristics f and of the corresponding wave.
    [Show full text]
  • Quantum Mechanics
    Quantum Mechanics Richard Fitzpatrick Professor of Physics The University of Texas at Austin Contents 1 Introduction 5 1.1 Intendedaudience................................ 5 1.2 MajorSources .................................. 5 1.3 AimofCourse .................................. 6 1.4 OutlineofCourse ................................ 6 2 Probability Theory 7 2.1 Introduction ................................... 7 2.2 WhatisProbability?.............................. 7 2.3 CombiningProbabilities. ... 7 2.4 Mean,Variance,andStandardDeviation . ..... 9 2.5 ContinuousProbabilityDistributions. ........ 11 3 Wave-Particle Duality 13 3.1 Introduction ................................... 13 3.2 Wavefunctions.................................. 13 3.3 PlaneWaves ................................... 14 3.4 RepresentationofWavesviaComplexFunctions . ....... 15 3.5 ClassicalLightWaves ............................. 18 3.6 PhotoelectricEffect ............................. 19 3.7 QuantumTheoryofLight. .. .. .. .. .. .. .. .. .. .. .. .. .. 21 3.8 ClassicalInterferenceofLightWaves . ...... 21 3.9 QuantumInterferenceofLight . 22 3.10 ClassicalParticles . .. .. .. .. .. .. .. .. .. .. .. .. .. .. 25 3.11 QuantumParticles............................... 25 3.12 WavePackets .................................. 26 2 QUANTUM MECHANICS 3.13 EvolutionofWavePackets . 29 3.14 Heisenberg’sUncertaintyPrinciple . ........ 32 3.15 Schr¨odinger’sEquation . 35 3.16 CollapseoftheWaveFunction . 36 4 Fundamentals of Quantum Mechanics 39 4.1 Introduction ..................................
    [Show full text]
  • Zero-Point Energy and Interstellar Travel by Josh Williams
    ;;;;;;;;;;;;;;;;;;;;;; itself comes from the conversion of electromagnetic Zero-Point Energy and radiation energy into electrical energy, or more speciÞcally, the conversion of an extremely high Interstellar Travel frequency bandwidth of the electromagnetic spectrum (beyond Gamma rays) now known as the zero-point by Josh Williams spectrum. ÒEre many generations pass, our machinery will be driven by power obtainable at any point in the universeÉ it is a mere question of time when men will succeed in attaching their machinery to the very wheel work of nature.Ó ÐNikola Tesla, 1892 Some call it the ultimate free lunch. Others call it absolutely useless. But as our world civilization is quickly coming upon a terrifying energy crisis with As you can see, the wavelengths from this part of little hope at the moment, radical new ideas of usable the spectrum are incredibly small, literally atomic and energy will be coming to the forefront and zero-point sub-atomic in size. And since the wavelength is so energy is one that is currently on its way. small, it follows that the frequency is quite high. The importance of this is that the intensity of the energy So, what the hell is it? derived for zero-point energy has been reported to be Zero-point energy is a type of energy that equal to the cube (the third power) of the frequency. exists in molecules and atoms even at near absolute So obviously, since weÕre already talking about zero temperatures (-273¡C or 0 K) in a vacuum. some pretty high frequencies with this portion of At even fractions of a Kelvin above absolute zero, the electromagnetic spectrum, weÕre talking about helium still remains a liquid, and this is said to be some really high energy intensities.
    [Show full text]
  • Relativistic Quantum Mechanics 1
    Relativistic Quantum Mechanics 1 The aim of this chapter is to introduce a relativistic formalism which can be used to describe particles and their interactions. The emphasis 1.1 SpecialRelativity 1 is given to those elements of the formalism which can be carried on 1.2 One-particle states 7 to Relativistic Quantum Fields (RQF), which underpins the theoretical 1.3 The Klein–Gordon equation 9 framework of high energy particle physics. We begin with a brief summary of special relativity, concentrating on 1.4 The Diracequation 14 4-vectors and spinors. One-particle states and their Lorentz transforma- 1.5 Gaugesymmetry 30 tions follow, leading to the Klein–Gordon and the Dirac equations for Chaptersummary 36 probability amplitudes; i.e. Relativistic Quantum Mechanics (RQM). Readers who want to get to RQM quickly, without studying its foun- dation in special relativity can skip the first sections and start reading from the section 1.3. Intrinsic problems of RQM are discussed and a region of applicability of RQM is defined. Free particle wave functions are constructed and particle interactions are described using their probability currents. A gauge symmetry is introduced to derive a particle interaction with a classical gauge field. 1.1 Special Relativity Einstein’s special relativity is a necessary and fundamental part of any Albert Einstein 1879 - 1955 formalism of particle physics. We begin with its brief summary. For a full account, refer to specialized books, for example (1) or (2). The- ory oriented students with good mathematical background might want to consult books on groups and their representations, for example (3), followed by introductory books on RQM/RQF, for example (4).
    [Show full text]
  • Newtonian Mechanics Is Most Straightforward in Its Formulation and Is Based on Newton’S Second Law
    CLASSICAL MECHANICS D. A. Garanin September 30, 2015 1 Introduction Mechanics is part of physics studying motion of material bodies or conditions of their equilibrium. The latter is the subject of statics that is important in engineering. General properties of motion of bodies regardless of the source of motion (in particular, the role of constraints) belong to kinematics. Finally, motion caused by forces or interactions is the subject of dynamics, the biggest and most important part of mechanics. Concerning systems studied, mechanics can be divided into mechanics of material points, mechanics of rigid bodies, mechanics of elastic bodies, and mechanics of fluids: hydro- and aerodynamics. At the core of each of these areas of mechanics is the equation of motion, Newton's second law. Mechanics of material points is described by ordinary differential equations (ODE). One can distinguish between mechanics of one or few bodies and mechanics of many-body systems. Mechanics of rigid bodies is also described by ordinary differential equations, including positions and velocities of their centers and the angles defining their orientation. Mechanics of elastic bodies and fluids (that is, mechanics of continuum) is more compli- cated and described by partial differential equation. In many cases mechanics of continuum is coupled to thermodynamics, especially in aerodynamics. The subject of this course are systems described by ODE, including particles and rigid bodies. There are two limitations on classical mechanics. First, speeds of the objects should be much smaller than the speed of light, v c, otherwise it becomes relativistic mechanics. Second, the bodies should have a sufficiently large mass and/or kinetic energy.
    [Show full text]
  • Motion and Forces Notes Key Speed • to Describe How Fast Something Is Traveling, You Have to Know Two Things About Its Motion
    Motion and Forces Notes Key Speed • To describe how fast something is traveling, you have to know two things about its motion. • One is the distance it has traveled, or how far it has gone. • The other is how much time it took to travel that distance. Average Speed • To calculate average speed, divide the distance traveled by the time it takes to travel that distance. • Speed Equation Average speed (in m/s) = distance traveled (in m) divided by time (in s) it took to travel the distance s = d t • Because average speed is calculated by dividing distance by time, its units will always be a distance unit divided by a time unit. • For example, the average speed of a bicycle is usually given in meters per second. • Average speed is useful if you don't care about the details of the motion. • When your motion is speeding up and slowing down, it might be useful to know how fast you are going at a certain time. Instantaneous Speed • The instantaneous speed is the speed of an object at any instant of time. Constant Speed • Sometimes an object is moving such that its instantaneous speed doesn’t change. • When the instantaneous speed doesn't change, an object is moving with constant speed. • The average speed and the instantaneous speed are the same. Calculating Distance • If an object is moving with constant speed, then the distance it travels over any period of time can be calculated using the equation for average speed. • When both sides of this equation are multiplied by the time, you have the equation for distance.
    [Show full text]
  • Gravity, Orbital Motion, and Relativity
    Gravity, Orbital Motion,& Relativity Early Astronomy Early Times • As far as we know, humans have always been interested in the motions of objects in the sky. • Not only did early humans navigate by means of the sky, but the motions of objects in the sky predicted the changing of the seasons, etc. • There were many early attempts both to describe and explain the motions of stars and planets in the sky. • All were unsatisfactory, for one reason or another. The Earth-Centered Universe • A geocentric (Earth-centered) solar system is often credited to Ptolemy, an Alexandrian Greek, although the idea is very old. • Ptolemy’s solar system could be made to fit the observational data pretty well, but only by becoming very complicated. Copernicus’ Solar System • The Polish cleric Copernicus proposed a heliocentric (Sun centered) solar system in the 1500’s. Objections to Copernicus How could Earth be moving at enormous speeds when we don’t feel it? . (Copernicus didn’t know about inertia.) Why can’t we detect Earth’s motion against the background stars (stellar parallax)? Copernicus’ model did not fit the observational data very well. Galileo • Galileo Galilei - February15,1564 – January 8, 1642 • Galileo became convinced that Copernicus was correct by observations of the Sun, Venus, and the moons of Jupiter using the newly-invented telescope. • Perhaps Galileo was motivated to understand inertia by his desire to understand and defend Copernicus’ ideas. Orbital Motion Tycho and Kepler • In the late 1500’s, a Danish nobleman named Tycho Brahe set out to make the most accurate measurements of planetary motions to date, in order to validate his own ideas of planetary motion.
    [Show full text]
  • The Von Neumann/Stapp Approach
    4. The von Neumann/Stapp Approach Von Neumann quantum theory is a formulation in which the entire physical universe, including the bodies and brains of the conscious human participant/observers, is represented in the basic quantum state, which is called the state of the universe. The state of a subsystem, such as a brain, is formed by averaging (tracing) this basic state over all variables other than those that describe the state of that subsystem. The dynamics involves three processes. Process 1 is the choice on the part of the experimenter about how he will act. This choice is sometimes called “The Heisenberg Choice,” because Heisenberg emphasized strongly its crucial role in quantum dynamics. At the pragmatic level it is a “free choice,” because it is controlled, at least in practice, by the conscious intentions of the experimenter/participant, and neither the Copenhagen nor von Neumann formulations provide any description of the causal origins of this choice, apart from the mental intentions of the human agent. Each intentional action involves an effort that is intended to result in a conceived experiential feedback, which can be an immediate confirmation of the success of the action, or a delayed monitoring the experiential consequences of the action. Process 2 is the quantum analog of the equations of motion of classical physics. As in classical physics, these equations of motion are local (i.e., all interactions are between immediate neighbors) and deterministic. They are obtained from the classical equations by a certain quantization procedure, and are reduced to the classical equations by taking the classical approximation of setting to zero the value of Planck’s constant everywhere it appears.
    [Show full text]
  • On the Logic Resolution of the Wave Particle Duality Paradox in Quantum Mechanics (Extended Abstract) M.A
    On the logic resolution of the wave particle duality paradox in quantum mechanics (Extended abstract) M.A. Nait Abdallah To cite this version: M.A. Nait Abdallah. On the logic resolution of the wave particle duality paradox in quantum me- chanics (Extended abstract). 2016. hal-01303223 HAL Id: hal-01303223 https://hal.archives-ouvertes.fr/hal-01303223 Preprint submitted on 18 Apr 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. On the logic resolution of the wave particle duality paradox in quantum mechanics (Extended abstract) M.A. Nait Abdallah UWO, London, Canada and INRIA, Paris, France Abstract. In this paper, we consider three landmark experiments of quantum physics and discuss the related wave particle duality paradox. We present a formalization, in terms of formal logic, of single photon self- interference. We show that the wave particle duality paradox appears, from the logic point of view, in the form of two distinct fallacies: the hard information fallacy and the exhaustive disjunction fallacy. We show that each fallacy points out some fundamental aspect of quantum physical systems and we present a logic solution to the paradox.
    [Show full text]
  • Kinematics Definition: (A) the Branch of Mechanics Concerned with The
    Chapter 3 – Kinematics Definition: (a) The branch of mechanics concerned with the motion of objects without reference to the forces that cause the motion. (b) The features or properties of motion in an object, regarded in such a way. “Kinematics can be used to find the possible range of motion for a given mechanism, or, working in reverse, can be used to design a mechanism that has a desired range of motion. The movement of a crane and the oscillations of a piston in an engine are both simple kinematic systems. The crane is a type of open kinematic chain, while the piston is part of a closed four‐bar linkage.” Distinguished from dynamics, e.g. F = ma Several considerations: 3.2 – 3.2.2 How do wheels motions affect robot motion (HW 2) 3.2.3 Wheel constraints (rolling and sliding) for Fixed standard wheel Steered standard wheel Castor wheel Swedish wheel Spherical wheel 3.2.4 Combine effects of all the wheels to determine the constraints on the robot. “Given a mobile robot with M wheels…” 3.2.5 Two examples drawn from 3.2.4: Differential drive. Yields same result as in 3.2 (HW 2) Omnidrive (3 Swedish wheels) Conclusion (pp. 76‐77): “We can see from the preceding examples that robot motion can be predicted by combining the rolling constraints of individual wheels.” “The sliding constraints can be used to …evaluate the maneuverability and workspace of the robot rather than just its predicted motion.” 3.3 Mobility and (vs.) Maneuverability Mobility: ability to directly move in the environment.
    [Show full text]