PHYS 419: Classical Mechanics Lecture Notes NEWTON's
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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 . -
Foundations of Newtonian Dynamics: an Axiomatic Approach For
Foundations of Newtonian Dynamics: 1 An Axiomatic Approach for the Thinking Student C. J. Papachristou 2 Department of Physical Sciences, Hellenic Naval Academy, Piraeus 18539, Greece Abstract. Despite its apparent simplicity, Newtonian mechanics contains conceptual subtleties that may cause some confusion to the deep-thinking student. These subtle- ties concern fundamental issues such as, e.g., the number of independent laws needed to formulate the theory, or, the distinction between genuine physical laws and deriva- tive theorems. This article attempts to clarify these issues for the benefit of the stu- dent by revisiting the foundations of Newtonian dynamics and by proposing a rigor- ous axiomatic approach to the subject. This theoretical scheme is built upon two fun- damental postulates, namely, conservation of momentum and superposition property for interactions. Newton’s laws, as well as all familiar theorems of mechanics, are shown to follow from these basic principles. 1. Introduction Teaching introductory mechanics can be a major challenge, especially in a class of students that are not willing to take anything for granted! The problem is that, even some of the most prestigious textbooks on the subject may leave the student with some degree of confusion, which manifests itself in questions like the following: • Is the law of inertia (Newton’s first law) a law of motion (of free bodies) or is it a statement of existence (of inertial reference frames)? • Are the first two of Newton’s laws independent of each other? It appears that -
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. -
Euler and the Dynamics of Rigid Bodies Sebastià Xambó Descamps
Euler and the dynamics of rigid bodies Sebastià Xambó Descamps Abstract. After presenting in contemporary terms an outline of the kinematics and dynamics of systems of particles, with emphasis in the kinematics and dynamics of rigid bodies, we will consider briefly the main points of the historical unfolding that produced this understanding and, in particular, the decisive role played by Euler. In our presentation the main role is not played by inertial (or Galilean) observers, but rather by observers that are allowed to move in an arbitrary (smooth, non-relativistic) way. 0. Notations and conventions The material in sections 0-6 of this paper is an adaptation of parts of the Mechanics chapter of XAMBÓ-2007. The mathematical language used is rather standard. The read- er will need a basic knowledge of linear algebra, of Euclidean geometry (see, for exam- ple, XAMBÓ-2001) and of basic calculus. 0.1. If we select an origin in Euclidean 3-space , each point can be specified by a vector , where is the Euclidean vector space associated with . This sets up a one-to-one correspondence between points and vectors . The inverse map is usually denoted . Usually we will speak of “the point ”, instead of “the point ”, implying that some point has been chosen as an origin. Only when conditions on this origin become re- levant will we be more specific. From now on, as it is fitting to a mechanics context, any origin considered will be called an observer. When points are assumed to be moving with time, their movement will be assumed to be smooth. -
THE EARTH's GRAVITY OUTLINE the Earth's Gravitational Field
GEOPHYSICS (08/430/0012) THE EARTH'S GRAVITY OUTLINE The Earth's gravitational field 2 Newton's law of gravitation: Fgrav = GMm=r ; Gravitational field = gravitational acceleration g; gravitational potential, equipotential surfaces. g for a non–rotating spherically symmetric Earth; Effects of rotation and ellipticity – variation with latitude, the reference ellipsoid and International Gravity Formula; Effects of elevation and topography, intervening rock, density inhomogeneities, tides. The geoid: equipotential mean–sea–level surface on which g = IGF value. Gravity surveys Measurement: gravity units, gravimeters, survey procedures; the geoid; satellite altimetry. Gravity corrections – latitude, elevation, Bouguer, terrain, drift; Interpretation of gravity anomalies: regional–residual separation; regional variations and deep (crust, mantle) structure; local variations and shallow density anomalies; Examples of Bouguer gravity anomalies. Isostasy Mechanism: level of compensation; Pratt and Airy models; mountain roots; Isostasy and free–air gravity, examples of isostatic balance and isostatic anomalies. Background reading: Fowler §5.1–5.6; Lowrie §2.2–2.6; Kearey & Vine §2.11. GEOPHYSICS (08/430/0012) THE EARTH'S GRAVITY FIELD Newton's law of gravitation is: ¯ GMm F = r2 11 2 2 1 3 2 where the Gravitational Constant G = 6:673 10− Nm kg− (kg− m s− ). ¢ The field strength of the Earth's gravitational field is defined as the gravitational force acting on unit mass. From Newton's third¯ law of mechanics, F = ma, it follows that gravitational force per unit mass = gravitational acceleration g. g is approximately 9:8m/s2 at the surface of the Earth. A related concept is gravitational potential: the gravitational potential V at a point P is the work done against gravity in ¯ P bringing unit mass from infinity to P. -
Newton.Indd | Sander Pinkse Boekproductie | 16-11-12 / 14:45 | Pag
omslag Newton.indd | Sander Pinkse Boekproductie | 16-11-12 / 14:45 | Pag. 1 e Dutch Republic proved ‘A new light on several to be extremely receptive to major gures involved in the groundbreaking ideas of Newton Isaac Newton (–). the reception of Newton’s Dutch scholars such as Willem work.’ and the Netherlands Jacob ’s Gravesande and Petrus Prof. Bert Theunissen, Newton the Netherlands and van Musschenbroek played a Utrecht University crucial role in the adaption and How Isaac Newton was Fashioned dissemination of Newton’s work, ‘is book provides an in the Dutch Republic not only in the Netherlands important contribution to but also in the rest of Europe. EDITED BY ERIC JORINK In the course of the eighteenth the study of the European AND AD MAAS century, Newton’s ideas (in Enlightenment with new dierent guises and interpre- insights in the circulation tations) became a veritable hype in Dutch society. In Newton of knowledge.’ and the Netherlands Newton’s Prof. Frans van Lunteren, sudden success is analyzed in Leiden University great depth and put into a new perspective. Ad Maas is curator at the Museum Boerhaave, Leiden, the Netherlands. Eric Jorink is researcher at the Huygens Institute for Netherlands History (Royal Dutch Academy of Arts and Sciences). / www.lup.nl LUP Newton and the Netherlands.indd | Sander Pinkse Boekproductie | 16-11-12 / 16:47 | Pag. 1 Newton and the Netherlands Newton and the Netherlands.indd | Sander Pinkse Boekproductie | 16-11-12 / 16:47 | Pag. 2 Newton and the Netherlands.indd | Sander Pinkse Boekproductie | 16-11-12 / 16:47 | Pag. -
The Newton-Leibniz Controversy Over the Invention of the Calculus
The Newton-Leibniz controversy over the invention of the calculus S.Subramanya Sastry 1 Introduction Perhaps one the most infamous controversies in the history of science is the one between Newton and Leibniz over the invention of the infinitesimal calculus. During the 17th century, debates between philosophers over priority issues were dime-a-dozen. Inspite of the fact that priority disputes between scientists were ¡ common, many contemporaries of Newton and Leibniz found the quarrel between these two shocking. Probably, what set this particular case apart from the rest was the stature of the men involved, the significance of the work that was in contention, the length of time through which the controversy extended, and the sheer intensity of the dispute. Newton and Leibniz were at war in the later parts of their lives over a number of issues. Though the dispute was sparked off by the issue of priority over the invention of the calculus, the matter was made worse by the fact that they did not see eye to eye on the matter of the natural philosophy of the world. Newton’s action-at-a-distance theory of gravitation was viewed as a reversion to the times of occultism by Leibniz and many other mechanical philosophers of this era. This intermingling of philosophical issues with the priority issues over the invention of the calculus worsened the nature of the dispute. One of the reasons why the dispute assumed such alarming proportions and why both Newton and Leibniz were anxious to be considered the inventors of the calculus was because of the prevailing 17th century conventions about priority and attitude towards plagiarism. -
Lecture Notes for PHY 405 Classical Mechanics
Lecture Notes for PHY 405 Classical Mechanics From Thorton & Marion's Classical Mechanics Prepared by Dr. Joseph M. Hahn Saint Mary's University Department of Astronomy & Physics November 25, 2003 revised November 29 Chapter 11: Dynamics of Rigid Bodies rigid body = a collection of particles whose relative positions are fixed. We will ignore the microscopic thermal vibrations occurring among a `rigid' body's atoms. The inertia tensor Consider a rigid body that is composed of N particles. Give each particle an index α = 1 : : : N. Particle α has a velocity vf,α measured in the fixed reference frame and velocity vr,α = drα=dt in the reference frame that rotates about axis !~ . Then according to Eq. 10.17: vf,α = V + vr,α + !~ × rα where V = velocity of the moving origin relative to the fixed origin. Note that the particles in a rigid body have zero relative velocity, ie, vr,α = 0. 1 Thus vα = V + !~ × rα after dropping the f subscript 1 1 and T = m v2 = m (V + !~ × r )2 is the particle's KE α 2 α α 2 α α N so T = Tα the system's total KE is α=1 X1 1 = MV2 + m V · (!~ × r ) + m (!~ × r )2 2 α α 2 α α α α X X 1 1 = MV2 + V · !~ × m r + m (!~ × r )2 2 α α 2 α α α ! α X X where M = α mα = the system's total mass. P Now choose the system's center of mass R as the origin of the moving frame. -
Guide for the Use of the International System of Units (SI)
Guide for the Use of the International System of Units (SI) m kg s cd SI mol K A NIST Special Publication 811 2008 Edition Ambler Thompson and Barry N. Taylor NIST Special Publication 811 2008 Edition Guide for the Use of the International System of Units (SI) Ambler Thompson Technology Services and Barry N. Taylor Physics Laboratory National Institute of Standards and Technology Gaithersburg, MD 20899 (Supersedes NIST Special Publication 811, 1995 Edition, April 1995) March 2008 U.S. Department of Commerce Carlos M. Gutierrez, Secretary National Institute of Standards and Technology James M. Turner, Acting Director National Institute of Standards and Technology Special Publication 811, 2008 Edition (Supersedes NIST Special Publication 811, April 1995 Edition) Natl. Inst. Stand. Technol. Spec. Publ. 811, 2008 Ed., 85 pages (March 2008; 2nd printing November 2008) CODEN: NSPUE3 Note on 2nd printing: This 2nd printing dated November 2008 of NIST SP811 corrects a number of minor typographical errors present in the 1st printing dated March 2008. Guide for the Use of the International System of Units (SI) Preface The International System of Units, universally abbreviated SI (from the French Le Système International d’Unités), is the modern metric system of measurement. Long the dominant measurement system used in science, the SI is becoming the dominant measurement system used in international commerce. The Omnibus Trade and Competitiveness Act of August 1988 [Public Law (PL) 100-418] changed the name of the National Bureau of Standards (NBS) to the National Institute of Standards and Technology (NIST) and gave to NIST the added task of helping U.S.