Statics Syllabus
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Key Concepts for Future QIS Learners Workshop Output Published Online May 13, 2020
Key Concepts for Future QIS Learners Workshop output published online May 13, 2020 Background and Overview On behalf of the Interagency Working Group on Workforce, Industry and Infrastructure, under the NSTC Subcommittee on Quantum Information Science (QIS), the National Science Foundation invited 25 researchers and educators to come together to deliberate on defining a core set of key concepts for future QIS learners that could provide a starting point for further curricular and educator development activities. The deliberative group included university and industry researchers, secondary school and college educators, and representatives from educational and professional organizations. The workshop participants focused on identifying concepts that could, with additional supporting resources, help prepare secondary school students to engage with QIS and provide possible pathways for broader public engagement. This workshop report identifies a set of nine Key Concepts. Each Concept is introduced with a concise overall statement, followed by a few important fundamentals. Connections to current and future technologies are included, providing relevance and context. The first Key Concept defines the field as a whole. Concepts 2-6 introduce ideas that are necessary for building an understanding of quantum information science and its applications. Concepts 7-9 provide short explanations of critical areas of study within QIS: quantum computing, quantum communication and quantum sensing. The Key Concepts are not intended to be an introductory guide to quantum information science, but rather provide a framework for future expansion and adaptation for students at different levels in computer science, mathematics, physics, and chemistry courses. As such, it is expected that educators and other community stakeholders may not yet have a working knowledge of content covered in the Key Concepts. -
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. -
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 .................................. -
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. -
Rotation: Moment of Inertia and Torque
Rotation: Moment of Inertia and Torque Every time we push a door open or tighten a bolt using a wrench, we apply a force that results in a rotational motion about a fixed axis. Through experience we learn that where the force is applied and how the force is applied is just as important as how much force is applied when we want to make something rotate. This tutorial discusses the dynamics of an object rotating about a fixed axis and introduces the concepts of torque and moment of inertia. These concepts allows us to get a better understanding of why pushing a door towards its hinges is not very a very effective way to make it open, why using a longer wrench makes it easier to loosen a tight bolt, etc. This module begins by looking at the kinetic energy of rotation and by defining a quantity known as the moment of inertia which is the rotational analog of mass. Then it proceeds to discuss the quantity called torque which is the rotational analog of force and is the physical quantity that is required to changed an object's state of rotational motion. Moment of Inertia Kinetic Energy of Rotation Consider a rigid object rotating about a fixed axis at a certain angular velocity. Since every particle in the object is moving, every particle has kinetic energy. To find the total kinetic energy related to the rotation of the body, the sum of the kinetic energy of every particle due to the rotational motion is taken. The total kinetic energy can be expressed as .. -
Quantum Information Science: Emerging No More
Quantum Information Science: Emerging No More Carlton M. Caves Center for Quantum Information and Control, University of New Mexico, Albuquerque, New Mexico 87131-0001, USA Final: 2013 January 30 Quantum information science (QIS) is a new field of enquiry, nascent in the 1980s, founded firmly in the 1990s, exploding in the 2010s, now established as a discipline for the 21st Century. Born in obscurity, then known as the foundations of quantum mechanics, the field began in the 60s and 70s with studies of Bell inequalities. These showed that the predictions of quantum mechanics cannot be squared with the belief, called local realism, that physical systems have realistic properties whose pre-existing values are revealed by measurements. The predictions of quantum mechanics for separate systems, correlated in the quantum way that we now call entanglement, are at odds with any version of local realism. Experiments in the early 80s demonstrated convincingly that the world comes down on the side of quantum mechanics. With local realism tossed out the window, it was natural to dream that quantum correlations could be used for faster-than-light communication, but this speculation was quickly shot down, and the shooting established the principle that quantum states cannot be copied. A group consisting of quantum opticians, electrical engineers, and mathematical physicists spent the 60s and 70s studying quantum measurements, getting serious about what can be measured and how well, going well beyond the description of observables that was (and often still is) taught in quantum-mechanics courses. This was not an empty exercise: communications engineers needed a more general description of quantum measurements to describe communications channels and to assess their performance. -
Mechanics Lecture 3 Static Forces, Resultants, Equilibrium of a Particle Dr Philip Jackson
C.1 EE1.el3 (EEE1023): Electronics III Mechanics lecture 3 Static forces, resultants, equilibrium of a particle Dr Philip Jackson http://www.ee.surrey.ac.uk/Teaching/Courses/ee1.el3/ C.2 Mechanics • Mechanics is the study of the relationship between the motion of bodies and the forces applied to them It describes, measures and relates forces with motion • Statics: study of forces, resultant forces, bodies in equilibrium, no acceleration • Kinematics: study of motion, position and time • Dynamics: study of forces and motion C.3 Forces and equilibrium Forces magnitude and direction components of a force Equilibrium static or dynamic forces in balance Preparation What are the fundamental forces? List them What is a component of force? Give one example What is equilibrium? Draw a diagram with 3-4 forces C.4 Four fundamental forces All forces can be described by four fundamental forces: 1. Gravity: long range, v. weak 2. Electromagnetic: responsible for many electrical, magnetic and EM wave phenomena 3. Weak nuclear: short range, radioactive decay 4. Strong nuclear: short range C.5 Statics – what is a force? • A force is a directed action of one body on another • The action “tends to move” the body and depends on the magnitude, direction and point of application of the force • A force is represented by a vector: C.6 Properties of forces: opposition • Forces occur in equal and opposite pairs (Newton’s 3rd law), each having the same line of action and acting on different bodies • In mechanics, we consider two types of forces: – contact forces -
Geometry and Quantum Mechanics
Geometry and quantum mechanics Juan Maldacena IAS Strings 2014 Vision talk QFT, Lorentz symmetry and temperature • In relavisFc quantum field theory in the vacuum. • An accelerated observer sees a temperature. • Quantum effect • Due to the large degree of entanglement in the vacuum. • InteresFng uses of entanglement entropy in QFT: c, f theorems. Geometry & thermodynamics Geometry à thermodynamics • With gravity: • Hawking effect. Black holes à finite temperature. • We understand in great detail various aspects: • BPS counFng of states with exquisite detail. • 2nd Law • Exact descripFon from far away. (Matrix models, AdS/CFT) • We understand the black hole in a full quantum mechanical way from far away. • We understand the black hole in a full quantum mechanical way from far away. • Going inside: Quicksand Thermodynamics à geometry ? • How does the interior arise ? • Without smoothness at the horizon, the Hawking predicFon for the temperature is not obviously valid any more. (MP assumes ER à EPR ) • Is there an interior for a generic microstate ? • What is the interpretaon of the singularity ? Litmus test for your understanding of the interior Popular view • The interior is some kind of average. • Mathur, Marolf, Wall, AMPS(S): not as usual. The interior is not an average quanFty like the density of a liquid or the color of a material. o = M Oout M i | i | • Fixed operators • Observer is ``outside’’ the system. Predicon Predicon • We will understand it. • It will be simple. • Implicaons for cosmology. • A specially solvable string theory example will be key. • We can have it all: unitarity from the outside and a reasonably smooth horizon for the infalling observer. -
Quantum Mechanics and Motion: a Modern Perspective
Quantum Mechanics and Motion: A Modern Perspective Gerald E. Marsh Argonne National Laboratory (Ret) 5433 East View Park Chicago, IL 60615 E-mail: [email protected] Abstract. This essay is an attempted to address, from a modern perspective, the motion of a particle. Quantum mechanically, motion consists of a series of localizations due to repeated interactions that, taken close to the limit of the continuum, yields a world-line. If a force acts on the particle, its probability distribution is accordingly modified. This must also be true for macroscopic objects, although now the description is far more complicated by the structure of matter and associated surface physics. PACS: 03.65.Yz, 03.50.De, 03.65.Sq. Key Words: Quantum, motion. The elements that comprise this essay are based on well-founded and accepted physical principles—but the way they are put together, as well as the view of commonly accepted forces and the resulting motion of macroscopic objects that emerges, is unusual. What will be shown is that classical motion can be identified with collective quantum mechanical motion. Not very surprising, but the conception of motion that emerges is somewhat counterintuitive. After all, we all know that the term in the Schrödinger equation becomes ridiculously small for m corresponding to a macroscopic object. Space-time and Quantum Mechanics To deal with the concept of motion we must begin with the well-known problem of the inconsistency inherent in the melding of quantum mechanics and special relativity. One of the principal examples that can illustrate this incompatibility is the Minkowski diagram, where well-defined world-lines are used to represent the paths of elementary particles while quantum mechanics disallows the existence of any such well defined world-lines. -
Basic Mechanics
93 Chapter 6 Basic mechanics BASIC PRINCIPLES OF STATICS All objects on earth tend to accelerate toward the Statics is the branch of mechanics that deals with the centre of the earth due to gravitational attraction; hence equilibrium of stationary bodies under the action of the force of gravitation acting on a body with the mass forces. The other main branch – dynamics – deals with (m) is the product of the mass and the acceleration due moving bodies, such as parts of machines. to gravity (g), which has a magnitude of 9.81 m/s2: Static equilibrium F = mg = vρg A planar structural system is in a state of static equilibrium when the resultant of all forces and all where: moments is equal to zero, i.e. F = force (N) m = mass (kg) g = acceleration due to gravity (9.81m/s2) y ∑Fx = 0 ∑Fx = 0 ∑Fy = 0 ∑Ma = 0 v = volume (m³) ∑Fy = 0 or ∑Ma = 0 or ∑Ma = 0 or ∑Mb = 0 ρ = density (kg/m³) x ∑Ma = 0 ∑Mb = 0 ∑Mb = 0 ∑Mc = 0 Vector where F refers to forces and M refers to moments of Most forces have magnitude and direction and can be forces. shown as a vector. The point of application must also be specified. A vector is illustrated by a line, the length of Static determinacy which is proportional to the magnitude on a given scale, If a body is in equilibrium under the action of coplanar and an arrow that shows the direction of the force. forces, the statics equations above must apply. -
Advancing Quantum Information Science: National Challenges and Opportunities
ADVANCING QUANTUM INFORMATION SCIENCE: NATIONAL CHALLENGES AND OPPORTUNITIES A JOINT REPORT OF THE Committee on Science and Committee on Homeland and National Security OF THE NATIONAL SCIENCE AND TECHNOLOGY COUNCIL Produced by the Interagency Working Group on Quantum Information Science of the Subcommittee on Physical Sciences July 2016 Advancing Quantum Information Science: National Challenges and Opportunities EXECUTIVE OFFICE OF THE PRESIDENT NATIONAL SCIENCE AND TECHNOLOGY COUNCIL WASHINGTON, D.C. 20502 July 22, 2016 Dear Colleagues: I am pleased to provide to you this report on Advancing Quantum Information Science: National Challenges and Opportunities. Recent advances in quantum information science (QIS) have increased interest and motivated new activity both domestically and internationally, making this report particularly timely. The document has been produced by an Interagency Working Group (IWG) on Quantum Information Science under the Subcommittee on Physical Sciences (PSSC) of the National Science and Technology Council. The report represents a coordinated effort across the numerous agencies and Executive Office of the President components participating in the IWG on QIS, and was informed by interactions with industry, academia, and other stakeholders. It is presented as a joint report of the PSSC’s parent group, the Committee on Science, and of the parallel Committee on Homeland and National Security. Quantum information science offers tremendous promise for both qualitative and quantitative leaps forward in information acquisition, processing, and transmission by exploiting quantum- mechanical phenomena. The field has seen rapid progress and growing attention in recent years, with ramifications across a broad range of fundamental sciences and for both near-term and long- term applications. -
Basic Mechanics: Kinematic Variables
CHAPTER 1 Basic Mechanics: © GlobalStock/iStockphoto.com© Kinematic Variables Chapter Objectives At the end of this chapter, you will be able to: • Describe the three-dimensional reference frame used to analyze human movements • Explain the importance of the center of mass of a body • Describe diff erent types of motion • Explain the relationships between the kinematic variables of position, velocity, and acceleration • Use the equations of motion to determine the outcome of movement tasks • Describe the diff erent technologies that can be used to record linear and angular kinematic variables and the relative merits of each 9781284022124_CH01_001_028.indd 1 11/02/15 9:59 PM 2 Chapter 1 Basic Mechanics: Kinematic Variables Key Terms Acceleration First central difference Linear displacement Second central difference method method Angular displacement Position General motion Translation Central difference formulae Radial acceleration Kinematic variables Velocity Equations of motion Rotation 9781284022124_CH01_001_028.indd 2 11/02/15 9:59 PM The Mechanical Reference Frame 3 Chapter Overview Mechanics is the study of the forces that act on a body and the changes in motion aris- ing from these forces. This chapter will focus on the kinematic variables associated with a biomechanical analysis of human movements. Kinematic variables allow the description of the motion of a body without reference to variables that act to change the motion of the body. (The variables that act to change the motion of a body, kinetic variables, will be covered elsewhere.) An appreciation of the kinematic variables associated with an athlete’s movements will allow the strength and conditioning prac- titioner to use specific technologies to assess an athlete’s performance in a variety of movement tasks.