Christiaan Huygens and the Problem of the Hanging Chain John Bukowski
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Vector Mechanics: Statics
PDHOnline Course G492 (4 PDH) Vector Mechanics: Statics Mark A. Strain, P.E. 2014 PDH Online | PDH Center 5272 Meadow Estates Drive Fairfax, VA 22030-6658 Phone & Fax: 703-988-0088 www.PDHonline.org www.PDHcenter.com An Approved Continuing Education Provider www.PDHcenter.com PDHonline Course G492 www.PDHonline.org Table of Contents Introduction ..................................................................................................................................... 1 Vectors ............................................................................................................................................ 1 Vector Decomposition ................................................................................................................ 2 Components of a Vector ............................................................................................................. 2 Force ............................................................................................................................................... 4 Equilibrium ..................................................................................................................................... 5 Equilibrium of a Particle ............................................................................................................. 6 Rigid Bodies.............................................................................................................................. 10 Pulleys ...................................................................................................................................... -
Knowledge Assessment in Statics: Concepts Versus Skills
Session 1168 Knowledge Assessment in Statics: Concepts versus skills Scott Danielson Arizona State University Abstract Following the lead of the physics community, engineering faculty have recognized the value of good assessment instruments for evaluating the learning of their students. These assessment instruments can be used to both measure student learning and to evaluate changes in teaching, i.e., did student-learning increase due different ways of teaching. As a result, there are significant efforts underway to develop engineering subject assessment tools. For instance, the Foundation Coalition is supporting assessment tool development efforts in a number of engineering subjects. These efforts have focused on developing “concept” inventories. These concept inventories focus on determining student understanding of the subject’s fundamental concepts. Separately, a NSF-supported effort to develop an assessment tool for statics was begun in the last year by the authors. As a first step, the project team analyzed prior work aimed at delineating important knowledge areas in statics. They quickly recognized that these important knowledge areas contained both conceptual and “skill” components. Both knowledge areas are described and examples of each are provided. Also, a cognitive psychology-based taxonomy of declarative and procedural knowledge is discussed in relation to determining the difference between a concept and a skill. Subsequently, the team decided to focus on development of a concept-based statics assessment tool. The ongoing Delphi process to refine the inventory of important statics concepts and validate the concepts with a broader group of subject matter experts is described. However, the value and need for a skills-based assessment tool is also recognized. -
Classical Mechanics
Classical Mechanics Hyoungsoon Choi Spring, 2014 Contents 1 Introduction4 1.1 Kinematics and Kinetics . .5 1.2 Kinematics: Watching Wallace and Gromit ............6 1.3 Inertia and Inertial Frame . .8 2 Newton's Laws of Motion 10 2.1 The First Law: The Law of Inertia . 10 2.2 The Second Law: The Equation of Motion . 11 2.3 The Third Law: The Law of Action and Reaction . 12 3 Laws of Conservation 14 3.1 Conservation of Momentum . 14 3.2 Conservation of Angular Momentum . 15 3.3 Conservation of Energy . 17 3.3.1 Kinetic energy . 17 3.3.2 Potential energy . 18 3.3.3 Mechanical energy conservation . 19 4 Solving Equation of Motions 20 4.1 Force-Free Motion . 21 4.2 Constant Force Motion . 22 4.2.1 Constant force motion in one dimension . 22 4.2.2 Constant force motion in two dimensions . 23 4.3 Varying Force Motion . 25 4.3.1 Drag force . 25 4.3.2 Harmonic oscillator . 29 5 Lagrangian Mechanics 30 5.1 Configuration Space . 30 5.2 Lagrangian Equations of Motion . 32 5.3 Generalized Coordinates . 34 5.4 Lagrangian Mechanics . 36 5.5 D'Alembert's Principle . 37 5.6 Conjugate Variables . 39 1 CONTENTS 2 6 Hamiltonian Mechanics 40 6.1 Legendre Transformation: From Lagrangian to Hamiltonian . 40 6.2 Hamilton's Equations . 41 6.3 Configuration Space and Phase Space . 43 6.4 Hamiltonian and Energy . 45 7 Central Force Motion 47 7.1 Conservation Laws in Central Force Field . 47 7.2 The Path Equation . -
Plotting Graphs of Parametric Equations
Parametric Curve Plotter This Java Applet plots the graph of various primary curves described by parametric equations, and the graphs of some of the auxiliary curves associ- ated with the primary curves, such as the evolute, involute, parallel, pedal, and reciprocal curves. It has been tested on Netscape 3.0, 4.04, and 4.05, Internet Explorer 4.04, and on the Hot Java browser. It has not yet been run through the official Java compilers from Sun, but this is imminent. It seems to work best on Netscape 4.04. There are problems with Netscape 4.05 and it should be avoided until fixed. The applet was developed on a 200 MHz Pentium MMX system at 1024×768 resolution. (Cost: $4000 in March, 1997. Replacement cost: $695 in June, 1998, if you can find one this slow on sale!) For various reasons, it runs extremely slowly on Macintoshes, and slowly on Sun Sparc stations. The curves should move as quickly as Roadrunner cartoons: for a benchmark, check out the circular trigonometric diagrams on my Java page. This applet was mainly done as an exercise in Java programming leading up to more serious interactive animations in three-dimensional graphics, so no attempt was made to be comprehensive. Those who wish to see much more comprehensive sets of curves should look at: http : ==www − groups:dcs:st − and:ac:uk= ∼ history=Java= http : ==www:best:com= ∼ xah=SpecialP laneCurve dir=specialP laneCurves:html http : ==www:astro:virginia:edu= ∼ eww6n=math=math0:html Disclaimer: Like all computer graphics systems used to illustrate mathematical con- cepts, it cannot be error-free. -
Differential Geometry
Differential Geometry J.B. Cooper 1995 Inhaltsverzeichnis 1 CURVES AND SURFACES—INFORMAL DISCUSSION 2 1.1 Surfaces ................................ 13 2 CURVES IN THE PLANE 16 3 CURVES IN SPACE 29 4 CONSTRUCTION OF CURVES 35 5 SURFACES IN SPACE 41 6 DIFFERENTIABLEMANIFOLDS 59 6.1 Riemannmanifolds .......................... 69 1 1 CURVES AND SURFACES—INFORMAL DISCUSSION We begin with an informal discussion of curves and surfaces, concentrating on methods of describing them. We shall illustrate these with examples of classical curves and surfaces which, we hope, will give more content to the material of the following chapters. In these, we will bring a more rigorous approach. Curves in R2 are usually specified in one of two ways, the direct or parametric representation and the implicit representation. For example, straight lines have a direct representation as tx + (1 t)y : t R { − ∈ } i.e. as the range of the function φ : t tx + (1 t)y → − (here x and y are distinct points on the line) and an implicit representation: (ξ ,ξ ): aξ + bξ + c =0 { 1 2 1 2 } (where a2 + b2 = 0) as the zero set of the function f(ξ ,ξ )= aξ + bξ c. 1 2 1 2 − Similarly, the unit circle has a direct representation (cos t, sin t): t [0, 2π[ { ∈ } as the range of the function t (cos t, sin t) and an implicit representation x : 2 2 → 2 2 { ξ1 + ξ2 =1 as the set of zeros of the function f(x)= ξ1 + ξ2 1. We see from} these examples that the direct representation− displays the curve as the image of a suitable function from R (or a subset thereof, usually an in- terval) into two dimensional space, R2. -
MATH 4030 Problem Set 11 Due Date: Sep 19, 2019 Reading
MATH 4030 Problem Set 11 Due date: Sep 19, 2019 Reading assignment: Preliminary materials, and do Carmo's Section 1.2, 1.3, 1.5 Problems: (Those marked with y are optional.) 1. Compute the Frenet frame fT; N; Bg, curvature κ, and torsion τ of the space curves below (Note that some of the curves may not be parametrized by arc length!): (a) α(s) = 1 (1 + s)3=2; 1 (1 − s)3=2; p1 s , s 2 (−1; 1) 3 3 2 p p (b) α(t) = 1 + t2; t; log(t + 1 + t2) , t 2 R 2. Let α(t) = (t; f(t)), t 2 [a; b], be a parametrization of the graph of a smooth function f :[a; b] ! R. (a) Prove that α is a regular plane curve. R b p 0 2 (b) Show that the arc length of α from t = a to t = b is given by a 1 + (f (x)) dx. (c) Prove that the signed curvature k of the plane curve α is given by f 00(t) k(t) = : [1 + (f 0(t))2]3=2 3. Consider the catenary parametrized by α(t) = (t; cosh t), t 2 [0; b] for some b > 0, (a) Show that the arc length of α(t), t 2 [0; b], is given by sinh b. (b) Reparametrize α(t) by arc length β(s), where s 2 [0; s0]. Find s0. (c) Find the signed curvature kβ of the catenary using the arc length parametrization β(s). 4. Consider the tractrix parametrized by α(t) = (cos t + log tan(t=2); sin t), t 2 [π=2; π], (a) Write down the expression of the arc length of α(t), t 2 [π=2; π=2 + s] for any given s 2 (0; π=2). -
Evolute-Involute Partner Curves According to Darboux Frame in the Euclidean 3-Space E3
Fundamentals of Contemporary Mathematical Sciences (2020) 1(2) 63 { 70 Evolute-Involute Partner Curves According to Darboux Frame in the Euclidean 3-space E3 Abdullah Yıldırım 1,∗ Feryat Kaya 2 1 Harran University, Faculty of Arts and Sciences, Department of Mathematics S¸anlıurfa, T¨urkiye 2 S¸ehit Abdulkadir O˘guzAnatolian Imam Hatip High School S¸anlıurfa, T¨urkiye, [email protected] Received: 29 February 2020 Accepted: 29 June 2020 Abstract: In this study, evolute-involute curves are researched. Characterization of evolute-involute curves lying on the surface are examined according to Darboux frame and some curves are obtained. Keywords: Curve, surface, geodesic, curvature, frame. 1. Introduction The interest of special curves has increased recently. Some of these are associated curves. They are curves where one of the Frenet vectors at opposite points is linearly dependent to the other curve. One of the best examples of these curves is the evolute-involute partner curves. An involute thought known to have been used in his optical work came up in 1658 by C. Huygens. C. Huygens discovered involute curves while trying to make more accurate measurement studies [5]. Many researches have been conducted about evolute-involute partner curves. Some of them conducted recently are Bilici and C¸alı¸skan [4], Ozyılmaz¨ and Yılmaz [9], As and Sarıo˘glugil[2]. Bekta¸sand Y¨uceconsider the notion of the involute-evolute curves lying on the surfaces for a special situation. They determine the special involute-evolute partner D−curves in E3: By using the Darboux frame of the curves they obtain the necessary and sufficient conditions between κg , ∗ − ∗ ∗ τg; κn and κn for a curve to be the special involute partner D curve. -
Computer-Aided Design and Kinematic Simulation of Huygens's
applied sciences Article Computer-Aided Design and Kinematic Simulation of Huygens’s Pendulum Clock Gloria Del Río-Cidoncha 1, José Ignacio Rojas-Sola 2,* and Francisco Javier González-Cabanes 3 1 Department of Engineering Graphics, University of Seville, 41092 Seville, Spain; [email protected] 2 Department of Engineering Graphics, Design, and Projects, University of Jaen, 23071 Jaen, Spain 3 University of Seville, 41092 Seville, Spain; [email protected] * Correspondence: [email protected]; Tel.: +34-953-212452 Received: 25 November 2019; Accepted: 9 January 2020; Published: 10 January 2020 Abstract: This article presents both the three-dimensional modelling of the isochronous pendulum clock and the simulation of its movement, as designed by the Dutch physicist, mathematician, and astronomer Christiaan Huygens, and published in 1673. This invention was chosen for this research not only due to the major technological advance that it represented as the first reliable meter of time, but also for its historical interest, since this timepiece embodied the theory of pendular movement enunciated by Huygens, which remains in force today. This 3D modelling is based on the information provided in the only plan of assembly found as an illustration in the book Horologium Oscillatorium, whereby each of its pieces has been sized and modelled, its final assembly has been carried out, and its operation has been correctly verified by means of CATIA V5 software. Likewise, the kinematic simulation of the pendulum has been carried out, following the approximation of the string by a simple chain of seven links as a composite pendulum. The results have demonstrated the exactitude of the clock. -
Tractor Trailer Cornering
TRACTOR TRAILER CORNERING by RAMGOPAL BATTU B. S. , Osmania University, 1963 A MASTER'S REPORT submitted in partial fulfillment of the requirements for the degree MASTER OF SCIENCE Department of Mechanical Engineering KANSAS STATE UNIVERSITY Manhattan, Kansas 1965 Approved by: Major Professor %6 ii TABLE OF CONTENTS NOMENCLATURE lii INTRODUCTION 1 DESCRIPTION OF CORNERING PROBLEM 2 DERIVATION OF EQUATIONS FOR FIRST PORTION OF TRACTRIX ... 4 EQUATIONS FOR SECOND PORTION OF TRACTRIX 11 PRESENTATION OF RESULTS 15 NUMERICAL RESULTS l8 DISCUSSION OF RESULTS 30 CONCLUSIONS 31 ACKNOWLEDGMENT 32 REFERENCES & APPENDIX 34 A. Listing of Fortran Program to solve for the distance between the tractrix curve and the leading curve. Ill NOMENCLATURE R Radius Of path of fifth wheel Ft a x-Co-ordinate of center of curvature Ft in x-y plane b y-Co-ordinate of center of curvature Ft in x-y plane x x-Co-ordinate of path of Trailer in Ft x-y plane Ft y y-Co-ordinate of path of Trailer in x-y plane K x-Co-ordinate of path of Fifth wheel Ft in x-y plane Ft f) y-Co-ordinate of path of Fifth wheel in x-y plane Y Angle indicated in Figure 2 Degrees 9 Angle indicated in Figure 2 Degrees L Length of Trailer Ft V Distance between "the tractrix and the leading curve Ft iv LIST OF FIGURES PAGE FIGURE 2 1 SCHEMATIC OF GENERAL TRACT RIX 2 SCHEMATIC OF A 90° CIRCULAR TURN 2 11 3 TRACTRIX OF A STRAIGHT LINE . 4 TOTAL TRACTRIX INCLUDING THE STRAIGHT LINE MOTION OF THE HITCH POINT 12 5 DIAGRAM SHOWING PARAMETERS USED IN PRESENTING RESULTS 14 6 DISTANCE BETWEEN THE TRACTRIX AND THE LEADING CURVE 16 7 MAXIMUM DIMENSIONLESS DISTANCE V/L VERSUS R/L . -
Variation Analysis of Involute Spline Tooth Contact
Brigham Young University BYU ScholarsArchive Theses and Dissertations 2006-02-22 Variation Analysis of Involute Spline Tooth Contact Brian J. De Caires Brigham Young University - Provo Follow this and additional works at: https://scholarsarchive.byu.edu/etd Part of the Mechanical Engineering Commons BYU ScholarsArchive Citation De Caires, Brian J., "Variation Analysis of Involute Spline Tooth Contact" (2006). Theses and Dissertations. 375. https://scholarsarchive.byu.edu/etd/375 This Thesis is brought to you for free and open access by BYU ScholarsArchive. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of BYU ScholarsArchive. For more information, please contact [email protected], [email protected]. VARIATION ANALYSIS OF INVOLUTE SPLINE TOOTH CONTACT By Brian J. K. DeCaires A thesis submitted to the faculty of Brigham Young University in partial fulfillment of the requirements for the degree of Master of Science Department of Mechanical Engineering Brigham Young University April 2006 Copyright © 2006 Brian J. K. DeCaires All Rights Reserved BRIGHAM YOUNG UNIVERSITY GRADUATE COMMITTEE APPROVAL of a thesis submitted by Brian J. K. DeCaires This thesis has been read by each member of the following graduate committee and by majority vote has been found to be satisfactory. ___________________________ ______________________________________ Date Kenneth W. Chase, Chair ___________________________ ______________________________________ Date Carl D. Sorensen ___________________________ -
Raphael, the Catenary-Tractrix Principle of the Transfiguration
1 of 13 From the desk of Pierre Beaudry RAPHAEL SANZIO: THE CATENARY/TRACTRIX PRINCIPLE OF THE TRANSFIGURATION by Pierre Beaudry 7/21/2009 Raphael’s Transfiguration contains a very disturbing discontinuity representing two different and opposite processes that have baffled critics and admirers alike for centuries. The perplexity of the spectator before this painting lies in the fact that the moment depicted in the upper part of the painting, the actual transfiguration of Christ, does not occur at the same time as the tragic event of the curing of the epileptic boy, in the lower part of the fresco, so that the two scenes appear to be completely foreign, separate, and even opposite subjects. This first impression is neurotic, incomplete, and misleading merely because a true cognitive connection of unity has not been discovered between the two events. On the other hand, if one looks at the whole scene as a metaphor of the creative process, the perplexity is dissipated, and the unity of effect is reestablished. In other words, if the Transfiguration is considered as the reflection of a catenary/tractrix envelopment by inversion of two different times, mortality and immortality, and two opposite movements, local and infinite, woven into an extraordinary singular unity of effect, the enigma of Raphael is resolved. Here, Leonardo da Vinci’s conception of the catenary/tractrix function shows how the human mind works at the same time that one discovers the process of its discovery in a Cusa contracted infinite. That is the central irony of Raphael’s Transfiguration, and the object of this report: to show how Raphael treated the external appearance of the subject in such a manner that it corresponded to the internal nature of the event. -
Recognition and Wonder – Huygens, Tractional Motion and Some
RECOGNITION AND WONDER Huygens, Tractional Motion and Some Thoughts on the History of Mathematics H.J.M. Bos Note: This is the translation of my inaugural lecture, held .March 20, 1987, as exlraordinar)' professor in the history' of mathematics at the University of Utrecht. The present version differs from the original Dutch text in that the salutation at the beginning and the personal words at the end have been left out and that some errors have been corrected. The original version was separately published (II.J.M. Bos, Vamtit Hcrkt'nning en Verbazing (Utrecht: OMI Grafisch Bcdrijf, 1987); the main text has also been published in Euclides 63, 1987. pp. 65-76. Recognition and wonder. - These two experiences are essential motives behind interest in the past. For that reason I have chosen them as guidelines in presenting my specialty, the history of mathematics, today, upon my accession to office as extraordinary professor. Recognition makes it possible to distinguish historical events and thus initiates the link of past to present. If recognition or affinity is absent, earlier events can hardly, if at all, be historically described. Wonder, on the other hand, is indispensable too. The unexpected, the essentially different nature of occurrences in the past excites the interest and raises the expectation that something can be discovered and learned. History studied without wonder reduces to a mere listing of recognizable past events, which differ from what is familiar only by having another date. Let me illustrate this by two examples which for me strongly evoke the two experien ces. In the first example recognition is foremost.