History of Classical Mechanics
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Glossary Physics (I-Introduction)
1 Glossary Physics (I-introduction) - Efficiency: The percent of the work put into a machine that is converted into useful work output; = work done / energy used [-]. = eta In machines: The work output of any machine cannot exceed the work input (<=100%); in an ideal machine, where no energy is transformed into heat: work(input) = work(output), =100%. Energy: The property of a system that enables it to do work. Conservation o. E.: Energy cannot be created or destroyed; it may be transformed from one form into another, but the total amount of energy never changes. Equilibrium: The state of an object when not acted upon by a net force or net torque; an object in equilibrium may be at rest or moving at uniform velocity - not accelerating. Mechanical E.: The state of an object or system of objects for which any impressed forces cancels to zero and no acceleration occurs. Dynamic E.: Object is moving without experiencing acceleration. Static E.: Object is at rest.F Force: The influence that can cause an object to be accelerated or retarded; is always in the direction of the net force, hence a vector quantity; the four elementary forces are: Electromagnetic F.: Is an attraction or repulsion G, gravit. const.6.672E-11[Nm2/kg2] between electric charges: d, distance [m] 2 2 2 2 F = 1/(40) (q1q2/d ) [(CC/m )(Nm /C )] = [N] m,M, mass [kg] Gravitational F.: Is a mutual attraction between all masses: q, charge [As] [C] 2 2 2 2 F = GmM/d [Nm /kg kg 1/m ] = [N] 0, dielectric constant Strong F.: (nuclear force) Acts within the nuclei of atoms: 8.854E-12 [C2/Nm2] [F/m] 2 2 2 2 2 F = 1/(40) (e /d ) [(CC/m )(Nm /C )] = [N] , 3.14 [-] Weak F.: Manifests itself in special reactions among elementary e, 1.60210 E-19 [As] [C] particles, such as the reaction that occur in radioactive decay. -
A Brief History and Philosophy of Mechanics
A Brief History and Philosophy of Mechanics S. A. Gadsden McMaster University, [email protected] 1. Introduction During this period, several clever inventions were created. Some of these were Archimedes screw, Hero of Mechanics is a branch of physics that deals with the Alexandria’s aeolipile (steam engine), the catapult and interaction of forces on a physical body and its trebuchet, the crossbow, pulley systems, odometer, environment. Many scholars consider the field to be a clockwork, and a rolling-element bearing (used in precursor to modern physics. The laws of mechanics Roman ships). [5] Although mechanical inventions apply to many different microscopic and macroscopic were being made at an accelerated rate, it wasn’t until objects—ranging from the motion of electrons to the after the Middle Ages when classical mechanics orbital patterns of galaxies. [1] developed. Attempting to understand the underlying principles of 3. Classical Period (1500 – 1900 AD) the universe is certainly a daunting task. It is therefore no surprise that the development of ‘idea and thought’ Classical mechanics had many contributors, although took many centuries, and bordered many different the most notable ones were Galileo, Huygens, Kepler, fields—atomism, metaphysics, religion, mathematics, Descartes and Newton. “They showed that objects and mechanics. The history of mechanics can be move according to certain rules, and these rules were divided into three main periods: antiquity, classical, and stated in the forms of laws of motion.” [1] quantum. The most famous of these laws were Newton’s Laws of 2. Period of Antiquity (800 BC – 500 AD) Motion, which accurately describe the relationships between force, velocity, and acceleration on a body. -
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 . -
Wyoming an Interdisciplinary Forensic Science Unit Developed by the UW Science Posse
CSI: Wyoming An interdisciplinary forensic science unit developed by the UW Science Posse. Question: Who did it? From the Bullet to the Bear (and Tree!) Developed by: Sabrina L. Cales 4/6/08 1:58:04 PM Grade Level: 7-8th Estimated Time: ~1 hour (omit clue ii) Topics Covered: Physics, Ballistics Standards and Benchmarks: 1. CONCEPTS AND PROCESSES In the context of unifying concepts and processes, students develop an understanding of scientific content through inquiry. Science is a dynamic process; concepts and content are best learned through inquiry and investigation. EARTH, SPACE, AND PHYSICAL SYSTEMS 10. The Structure and Properties of Matter: Students identify characteristic properties of matter such as density, solubility, and boiling point and understand that elements are the basic components of matter. 11. Physical and Chemical Changes in Matter: Students evaluate chemical and physical changes, recognizing that chemical change forms compounds with different properties and that physical change alters the appearance but not the composition of a substance. 12. Forms and Uses of Energy: Students investigate energy as a property of substances in a variety of forms with a range of uses. 13. The Conservation of Matter and Energy: Students identify supporting evidence to explain conservation of matter and energy, indicating that matter or energy cannot be created or destroyed but is transferred from one object to another. 14. Effects of Motions and Forces: Students describe motion of an object by position, direction, and speed, and identify the effects of force and inertia on an object. 2. SCIENCE AS INQUIRY Students demonstrate knowledge, skills, and habits of mind necessary to safely perform scientific inquiry. -
SIMON STEVIN (1548 – 1620) by HEINZ KLAUS STRICK, Germany
SIMON STEVIN (1548 – 1620) by HEINZ KLAUS STRICK, Germany The Flemish mathematician, physicist and engineer SIMON STEVIN is one of the lesser-known personalities in the history of science. However, his work has left many traces. We do not even know his exact date of birth and death; his birthplace was Bruges; where he died is uncertain: either Leiden or the Hague. Raised in the Calvinist tradition, he grew up in Flanders, became an accountant and cashier for a trading firm in Antwerp, travelled for several years through Poland, Prussia and Norway until he took a job at the tax office in Bruges in 1577. Around this time, the 17 provinces of the Netherlands, which also include the area of present-day Belgium, Luxembourg and parts of northern France, belong to the Spanish dominion. Large parts of the population, especially in the northern provinces, converted to the Calvinist faith. In 1567 King PHILIP II OF SPAIN appointed the DUKE OF ALBA as governor. When ALBA carried out a punitive expedition against the Protestants, a war began which only ended in 1648 with the Peace Treaty of Münster (partial treaty of the Peace of Westphalia). In 1579 the Protestant provinces in the north of the Netherlands united to form the Union of Utrecht and declared their independence as the Republic of the United Netherlands; they elected WILLIAM THE SILENT or WILLIAM OF ORANGE as regent. As the political situation came to a head, SIMON STEVIN's life situation also changed: although he was already 33 years old, he still attended a Latin school and then took up studies at the newly founded University of Leiden. -
Engineering Physics I Syllabus COURSE IDENTIFICATION Course
Engineering Physics I Syllabus COURSE IDENTIFICATION Course Prefix/Number PHYS 104 Course Title Engineering Physics I Division Applied Science Division Program Physics Credit Hours 4 credit hours Revision Date Fall 2010 Assessment Goal per Outcome(s) 70% INSTRUCTION CLASSIFICATION Academic COURSE DESCRIPTION This course is the first semester of a calculus-based physics course primarily intended for engineering and science majors. Course work includes studying forces and motion, and the properties of matter and heat. Topics will include motion in one, two, and three dimensions, mechanical equilibrium, momentum, energy, rotational motion and dynamics, periodic motion, and conservation laws. The laboratory (taken concurrently) presents exercises that are designed to reinforce the concepts presented and discussed during the lectures. PREREQUISITES AND/OR CO-RECQUISITES MATH 150 Analytic Geometry and Calculus I The engineering student should also be proficient in algebra and trigonometry. Concurrent with Phys. 140 Engineering Physics I Laboratory COURSE TEXT *The official list of textbooks and materials for this course are found on Inside NC. • COLLEGE PHYSICS, 2nd Ed. By Giambattista, Richardson, and Richardson, McGraw-Hill, 2007. • Additionally, the student must have a scientific calculator with trigonometric functions. COURSE OUTCOMES • To understand and be able to apply the principles of classical Newtonian mechanics. • To effectively communicate classical mechanics concepts and solutions to problems, both in written English and through mathematics. • To be able to apply critical thinking and problem solving skills in the application of classical mechanics. To demonstrate successfully accomplishing the course outcomes, the student should be able to: 1) Demonstrate knowledge of physical concepts by their application in problem solving. -
Exercises in Classical Mechanics 1 Moments of Inertia 2 Half-Cylinder
Exercises in Classical Mechanics CUNY GC, Prof. D. Garanin No.1 Solution ||||||||||||||||||||||||||||||||||||||||| 1 Moments of inertia (5 points) Calculate tensors of inertia with respect to the principal axes of the following bodies: a) Hollow sphere of mass M and radius R: b) Cone of the height h and radius of the base R; both with respect to the apex and to the center of mass. c) Body of a box shape with sides a; b; and c: Solution: a) By symmetry for the sphere I®¯ = I±®¯: (1) One can ¯nd I easily noticing that for the hollow sphere is 1 1 X ³ ´ I = (I + I + I ) = m y2 + z2 + z2 + x2 + x2 + y2 3 xx yy zz 3 i i i i i i i i 2 X 2 X 2 = m r2 = R2 m = MR2: (2) 3 i i 3 i 3 i i b) and c): Standard solutions 2 Half-cylinder ϕ (10 points) Consider a half-cylinder of mass M and radius R on a horizontal plane. a) Find the position of its center of mass (CM) and the moment of inertia with respect to CM. b) Write down the Lagrange function in terms of the angle ' (see Fig.) c) Find the frequency of cylinder's oscillations in the linear regime, ' ¿ 1. Solution: (a) First we ¯nd the distance a between the CM and the geometrical center of the cylinder. With σ being the density for the cross-sectional surface, so that ¼R2 M = σ ; (3) 2 1 one obtains Z Z 2 2σ R p σ R q a = dx x R2 ¡ x2 = dy R2 ¡ y M 0 M 0 ¯ 2 ³ ´3=2¯R σ 2 2 ¯ σ 2 3 4 = ¡ R ¡ y ¯ = R = R: (4) M 3 0 M 3 3¼ The moment of inertia of the half-cylinder with respect to the geometrical center is the same as that of the cylinder, 1 I0 = MR2: (5) 2 The moment of inertia with respect to the CM I can be found from the relation I0 = I + Ma2 (6) that yields " µ ¶ # " # 1 4 2 1 32 I = I0 ¡ Ma2 = MR2 ¡ = MR2 1 ¡ ' 0:3199MR2: (7) 2 3¼ 2 (3¼)2 (b) The Lagrange function L is given by L('; '_ ) = T ('; '_ ) ¡ U('): (8) The potential energy U of the half-cylinder is due to the elevation of its CM resulting from the deviation of ' from zero. -
Ballistics: Concepts and Connections with Applied Physics
Orissa Journal of Physics © Orissa Physical Society Vol. 22, No.1 ISSN 0974-8202 February 2015 pp. 27-38 Ballistics: Concepts and Connections with Applied Physics K K CHAND1 and M C ADHIKARY2 1Scientist, Proof & Experimental Es tablishment (PXE), DRDO, Chandipur, Balasore 1 emails: [email protected] 2Reader in Applied Physics and Ballistics, FM University, Vyasavihar, Balasore, 2 [email protected] Received : 1.11.2014 ; Accepted : 10.1.2015 Abstract : Ballistics, a generic term, is intended for various physical applications, which deals with the properties and interactions of matter and energy, space and time. Discoveries theories and experiments provide an essential link between applied physics and ballistics problems. "Applied" is distinguished from "pure" by a subtle combination of factors such as the motivation and attitude of researchers and the nature of the relationship to the ballistics. It usually differs from ballistics in that an applied physicist may not be designing something in particular, but rather research on physical concepts and connected laws/theories with the aim of understanding or solving ballistics related problems. This approach is similar to that of ballistics, which is the name of the applied scientific field. Competence in Applied Physics and Ballistics (APAB) is important multidisciplinary research areas in armaments science. Because of their multidisciplinary nature, the APAB is inseparable from physical, mathematical, experimental and computational aspects. In this context, this paper discusses a brief review into the APAB, their important roles in the armament research. And also summarize some of the current APAB activities in academia, armament research institute and industry. 1. Introduction: Objectives and Scopes Applied physics is a branch of physics that concerns itself with the applications of physical laws and theories with experimental its knowledge to other domains. -
The Celestial Mechanics of Newton
GENERAL I ARTICLE The Celestial Mechanics of Newton Dipankar Bhattacharya Newton's law of universal gravitation laid the physical foundation of celestial mechanics. This article reviews the steps towards the law of gravi tation, and highlights some applications to celes tial mechanics found in Newton's Principia. 1. Introduction Newton's Principia consists of three books; the third Dipankar Bhattacharya is at the Astrophysics Group dealing with the The System of the World puts forth of the Raman Research Newton's views on celestial mechanics. This third book Institute. His research is indeed the heart of Newton's "natural philosophy" interests cover all types of which draws heavily on the mathematical results derived cosmic explosions and in the first two books. Here he systematises his math their remnants. ematical findings and confronts them against a variety of observed phenomena culminating in a powerful and compelling development of the universal law of gravita tion. Newton lived in an era of exciting developments in Nat ural Philosophy. Some three decades before his birth J 0- hannes Kepler had announced his first two laws of plan etary motion (AD 1609), to be followed by the third law after a decade (AD 1619). These were empirical laws derived from accurate astronomical observations, and stirred the imagination of philosophers regarding their underlying cause. Mechanics of terrestrial bodies was also being developed around this time. Galileo's experiments were conducted in the early 17th century leading to the discovery of the Keywords laws of free fall and projectile motion. Galileo's Dialogue Celestial mechanics, astronomy, about the system of the world was published in 1632. -
Luckenbach Ballistics (Back to the Basics)
Luckenbach Ballistics (Back to the Basics) Most of you have probably heard the old country‐ such as the Kestrel that puts both a weather western song, Luckenbach, Texas (Back to the station and a powerful ballistics computer in my Basics of Love). There is really a place named palm. We give credit to Sir Isaac Newton for Luckenbach near our family home in the Texas formulating the basics of ballistics and calculus Hill Country. There’s the old dance hall, there’s we still use. Newton’s greatest contribution is the general store, and there are a couple of old that he thought of ballistics in terms of time, not homes. The few buildings are near a usually‐ distance. One of Newton’s best known equations flowing creek with grassy banks and nice shade tells us the distance an object falls depending on trees. The gravel parking lot is often filled with gravity and fall time. With distance in meters and Harleys while the riders sit in the shade and enjoy time in seconds, Newton’s equation is a beer from the store. Locals drive by and shake their heads as they remember the town before it Distance = 4.9 x Time x Time. was made famous by the song and Willie Nelson’s Fourth‐of‐July Picnics. At a time of 2 seconds the distance is 19.6 meters. A rock dropped from 19.6 meters Don’t think of Luckenbach as a place. The song takes 2 seconds to reach the ground. A rock really describes an attitude or way of life. -
TOPICS in CELESTIAL MECHANICS 1. the Newtonian N-Body Problem
TOPICS IN CELESTIAL MECHANICS RICHARD MOECKEL 1. The Newtonian n-body Problem Celestial mechanics can be defined as the study of the solution of Newton's differ- ential equations formulated by Isaac Newton in 1686 in his Philosophiae Naturalis Principia Mathematica. The setting for celestial mechanics is three-dimensional space: 3 R = fq = (x; y; z): x; y; z 2 Rg with the Euclidean norm: p jqj = x2 + y2 + z2: A point particle is characterized by a position q 2 R3 and a mass m 2 R+. A motion of such a particle is described by a curve q(t) where t runs over some interval in R; the mass is assumed to be constant. Some remarks will be made below about why it is reasonable to model a celestial body by a point particle. For every motion of a point particle one can define: velocity: v(t) =q _(t) momentum: p(t) = mv(t): Newton formulated the following laws of motion: Lex.I. Corpus omne perservare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus a viribus impressis cogitur statum illum mutare 1 Lex.II. Mutationem motus proportionem esse vi motrici impressae et fieri secundem lineam qua vis illa imprimitur. 2 Lex.III Actioni contrarium semper et aequalem esse reactionem: sive corporum duorum actiones in se mutuo semper esse aequales et in partes contrarias dirigi. 3 The first law is statement of the principle of inertia. The second law asserts the existence of a force function F : R4 ! R3 such that: p_ = F (q; t) or mq¨ = F (q; t): In celestial mechanics, the dependence of F (q; t) on t is usually indirect; the force on one body depends on the positions of the other massive bodies which in turn depend on t. -
Simon Stevin
II THE PRINCIPAL WORKS OF SIMON STEVIN E D IT E D BY ERNST CRONE, E. J. DIJKSTERHUIS, R. J. FORBES M. G. J. MINNAERT, A. PANNEKOEK A M ST E R D A M C. V. SW ETS & Z E IT L IN G E R J m THE PRINCIPAL WORKS OF SIMON STEVIN VOLUME II MATHEMATICS E D IT E D BY D. J. STRUIK PROFESSOR AT THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY, CAMBRIDGE (MASS.) A M S T E R D A M C. V. SW ETS & Z E IT L IN G E R 1958 The edition of this volume II of the principal works of SIMON STEVIN devoted to his mathematical publications, has been rendered possible through the financial aid of the Koninklijke. Nederlandse Akademie van Wetenschappen (Royal Netherlands Academy of Science) Printed by Jan de Lange, Deventer, Holland The following edition of the Principal Works of SIMON STEVIN has been brought about at the initiative of the Physics Section of the Koninklijke Nederlandse Akademie van Weten schappen (Royal Netherlands Academy of Sciences) by a committee consisting of the following members: ERNST CRONE, Chairman of the Netherlands Maritime Museum, Amsterdam E. J. DIJKSTERHUIS, Professor of the History of Science at the Universities of Leiden and Utrecht R. J. FORBES, Professor of the History of Science at the Municipal University of Amsterdam M. G. J. M INNAERT, Professor of Astronomy at the University of Utrecht A. PANNEKOEK, Former Professor of Astronomy at the Municipal University of Amsterdam The Dutch texts of STEVIN as well as the introductions and notes have been translated into English or revised by Miss C.