Lesson 7 - Basic of Angular Kinematics
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Rotational Motion (The Dynamics of a Rigid Body)
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Robert Katz Publications Research Papers in Physics and Astronomy 1-1958 Physics, Chapter 11: Rotational Motion (The Dynamics of a Rigid Body) Henry Semat City College of New York Robert Katz University of Nebraska-Lincoln, [email protected] Follow this and additional works at: https://digitalcommons.unl.edu/physicskatz Part of the Physics Commons Semat, Henry and Katz, Robert, "Physics, Chapter 11: Rotational Motion (The Dynamics of a Rigid Body)" (1958). Robert Katz Publications. 141. https://digitalcommons.unl.edu/physicskatz/141 This Article is brought to you for free and open access by the Research Papers in Physics and Astronomy at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Robert Katz Publications by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. 11 Rotational Motion (The Dynamics of a Rigid Body) 11-1 Motion about a Fixed Axis The motion of the flywheel of an engine and of a pulley on its axle are examples of an important type of motion of a rigid body, that of the motion of rotation about a fixed axis. Consider the motion of a uniform disk rotat ing about a fixed axis passing through its center of gravity C perpendicular to the face of the disk, as shown in Figure 11-1. The motion of this disk may be de scribed in terms of the motions of each of its individual particles, but a better way to describe the motion is in terms of the angle through which the disk rotates. -
The Rotation Group in Plate Tectonics and the Representation of Uncertainties of Plate Reconstructions
Geophys. J. In?. (1990) 101, 649-661 The rotation group in plate tectonics and the representation of uncertainties of plate reconstructions T. Chang’ J. Stock2 and P. Molnar3 ‘Department of Mathematics, University of Virginia, Charlottesville, Virginia 22903-3199, USA ‘Department of Earth and Planetary Sciences, Harvard University, Cambridge, MA 02138, USA ’Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Accepted 1989 December 27. Received 1989 December 27; in original form 1989 August 11 Downloaded from SUMMARY The calculation of the uncertainty in an estimated rotation requires a parametriza- tion of the rotation group; that is, a unique mapping of the rotation group to a point in 3-D Euclidean space, R3. Numerous parametrizations of a rotation exist, http://gji.oxfordjournals.org/ including: (1) the latitude and longitude of the axis of rotation and the angle of rotation; (2) a representation as a Cartesian vector with length equal to the rotation angle and direction parallel to the rotation axis; (3) Euler angles; or (4) unit length quaternions (or, equivalently, Cayley-Klein parameters). The uncertainty in a rotation is determined by the effect of nearby rotations on the rotated data. The uncertainty in a rotation is small, if rotations close to the best fitting rotation degrade the fit of the data by a large amount, and it is large, if only rotations differing by a large amount cause such a degradation. Ideally, we would at California Institute of Technology on August 29, 2014 like to parametrize the rotations in such a way so that their representation as points in R3 would have the property that the distance between two points in R3 reflects the effects of the corresponding rotations on the fit of the data. -
Modeling the Physics of RRAM Defects
Modeling the physics of RRAM defects A model simulating RRAM defects on a macroscopic physical level T. Hol Modeling the physics of RRAM defects A model simulating RRAM defects on a macroscopic physical level by T. Hol to obtain the degree of Master of Science at the Delft University of Technology, to be defended publicly on Friday August 28, 2020 at 13:00. Student number: 4295528 Project duration: September 1, 2019 – August 28, 2020 Thesis committee: Prof. dr. ir. S. Hamdioui TU Delft, supervisor Dr. Ir. R. Ishihara TU Delft Dr. Ir. S. Vollebregt TU Delft Dr. Ir. M. Taouil TU Delft Ir. M. Fieback TU Delft, supervisor An electronic version of this thesis is available at https://repository.tudelft.nl/. Abstract Resistive RAM, or RRAM, is one of the emerging non-volatile memory (NVM) technologies, which could be used in the near future to fill the gap in the memory hierarchy between dynamic RAM (DRAM) and Flash, or even completely replace Flash. RRAM operates faster than Flash, but is still non-volatile, which enables it to be used in a dense 3D NVM array. It is also a suitable candidate for computation-in-memory, neuromorphic computing and reconfigurable computing. However, the show stopping problem of RRAM is that it suffers from unique defects, which is the reason why RRAM is still not widely commercially adopted. These defects differ from those that appear in CMOS technology, due to the arbitrary nature of the forming process. They can not be detected by conventional tests and cause defective devices to go unnoticed. -
Kinematics, Kinetics, Dynamics, Inertia Kinematics Is a Branch of Classical
Course: PGPathshala-Biophysics Paper 1:Foundations of Bio-Physics Module 22: Kinematics, kinetics, dynamics, inertia Kinematics is a branch of classical mechanics in physics in which one learns about the properties of motion such as position, velocity, acceleration, etc. of bodies or particles without considering the causes or the driving forces behindthem. Since in kinematics, we are interested in study of pure motion only, we generally ignore the dimensions, shapes or sizes of objects under study and hence treat them like point objects. On the other hand, kinetics deals with physics of the states of the system, causes of motion or the driving forces and reactions of the system. In a particular state, namely, the equilibrium state of the system under study, the systems can either be at rest or moving with a time-dependent velocity. The kinetics of the prior, i.e., of a system at rest is studied in statics. Similarly, the kinetics of the later, i.e., of a body moving with a velocity is called dynamics. Introduction Classical mechanics describes the area of physics most familiar to us that of the motion of macroscopic objects, from football to planets and car race to falling from space. NASCAR engineers are great fanatics of this branch of physics because they have to determine performance of cars before races. Geologists use kinematics, a subarea of classical mechanics, to study ‘tectonic-plate’ motion. Even medical researchers require this physics to map the blood flow through a patient when diagnosing partially closed artery. And so on. These are just a few of the examples which are more than enough to tell us about the importance of the science of motion. -
Linear and Angular Velocity Examples
Linear and Angular Velocity Examples Example 1 Determine the angular displacement in radians of 6.5 revolutions. Round to the nearest tenth. Each revolution equals 2 radians. For 6.5 revolutions, the number of radians is 6.5 2 or 13 . 13 radians equals about 40.8 radians. Example 2 Determine the angular velocity if 4.8 revolutions are completed in 4 seconds. Round to the nearest tenth. The angular displacement is 4.8 2 or 9.6 radians. = t 9.6 = = 9.6 , t = 4 4 7.539822369 Use a calculator. The angular velocity is about 7.5 radians per second. Example 3 AMUSEMENT PARK Jack climbs on a horse that is 12 feet from the center of a merry-go-round. 1 The merry-go-round makes 3 rotations per minute. Determine Jack’s angular velocity in radians 4 per second. Round to the nearest hundredth. 1 The merry-go-round makes 3 or 3.25 revolutions per minute. Convert revolutions per minute to radians 4 per second. 3.25 revolutions 1 minute 2 radians 0.3403392041 radian per second 1 minute 60 seconds 1 revolution Jack has an angular velocity of about 0.34 radian per second. Example 4 Determine the linear velocity of a point rotating at an angular velocity of 12 radians per second at a distance of 8 centimeters from the center of the rotating object. Round to the nearest tenth. v = r v = (8)(12 ) r = 8, = 12 v 301.5928947 Use a calculator. The linear velocity is about 301.6 centimeters per second. -
Hamilton Description of Plasmas and Other Models of Matter: Structure and Applications I
Hamilton description of plasmas and other models of matter: structure and applications I P. J. Morrison Department of Physics and Institute for Fusion Studies The University of Texas at Austin [email protected] http://www.ph.utexas.edu/ morrison/ ∼ MSRI August 20, 2018 Survey Hamiltonian systems that describe matter: particles, fluids, plasmas, e.g., magnetofluids, kinetic theories, . Hamilton description of plasmas and other models of matter: structure and applications I P. J. Morrison Department of Physics and Institute for Fusion Studies The University of Texas at Austin [email protected] http://www.ph.utexas.edu/ morrison/ ∼ MSRI August 20, 2018 Survey Hamiltonian systems that describe matter: particles, fluids, plasmas, e.g., magnetofluids, kinetic theories, . \Hamiltonian systems .... are the basis of physics." M. Gutzwiller Coarse Outline William Rowan Hamilton (August 4, 1805 - September 2, 1865) I. Today: Finite-dimensional systems. Particles etc. ODEs II. Tomorrow: Infinite-dimensional systems. Hamiltonian field theories. PDEs Why Hamiltonian? Beauty, Teleology, . : Still a good reason! • 20th Century framework for physics: Fluids, Plasmas, etc. too. • Symmetries and Conservation Laws: energy-momentum . • Generality: do one problem do all. • ) Approximation: perturbation theory, averaging, . 1 function. • Stability: built-in principle, Lagrange-Dirichlet, δW ,.... • Beacon: -dim KAM theorem? Krein with Cont. Spec.? • 9 1 Numerical Methods: structure preserving algorithms: • symplectic, conservative, Poisson integrators, -
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. -
4-2 Degrees and Radians
4-2 Degrees and Radians Write each degree measure in radians as a multiple of π and each radian measure in degrees. 10. 30° SOLUTION: To convert a degree measure to radians, multiply by ANSWER: 14. SOLUTION: To convert a radian measure to degrees, multiply by ANSWER: 120° Identify all angles that are coterminal with the given angle. Then find and draw one positive and one negative angle coterminal with the given angle. 20. 225° SOLUTION: All angles measuring are coterminal with a angle. Sample answer: Let n = 1 and −1. eSolutions Manual - Powered by Cognero Page 1 ANSWER: 225° + 360n°; Sample answer: 585°, −135° 24. SOLUTION: All angles measuring are coterminal with a angle. Sample answer: Let n = 1 and −1. ANSWER: Find the length of the intercepted arc with the given central angle measure in a circle with the given radius. Round to the nearest tenth. 29. , r = 4 yd SOLUTION: ANSWER: 5.2 yd 30. 105°, r = 18.2 cm SOLUTION: Method 1 Convert 105° to radian measure, and then use s = rθ to find the arc length. Substitute r = 18.2 and θ = . Method 2 Use s = to find the arc length. ANSWER: 33.4 cm Find the rotation in revolutions per minute given the angular speed and the radius given the linear speed and the rate of rotation. 36. = 104π rad/min SOLUTION: The angular speed is 104π radians per minute. Each revolution measures 2π radians 104π ÷ 2π = 52 The angle of rotation is 52 revolutions per minute. ANSWER: 52 rev/min 37. v = 82.3 m/s, 131 rev/min SOLUTION: 131 × 2π = 262π The linear speed is 82.3 meters per second with an angle of rotation of 262π radians per minute. -
Model CIB-L Inertia Base Frame
KINETICS® Inertia Base Frame Model CIB-L Description Application Model CIB-L inertia base frames, when filled with KINETICSTM CIB-L inertia base frames are specifically concrete, meet all specifications for inertia bases, designed and engineered to receive poured concrete, and when supported by proper KINETICSTM vibration for use in supporting mechanical equipment requiring isolators, provide the ultimate in equipment isolation, a reinforced concrete inertia base. support, anchorage, and vibration amplitude control. Inertia bases are used to support mechanical KINETICSTM CIB-L inertia base frames incorporate a equipment, reduce equipment vibration, provide for unique structural design which integrates perimeter attachment of vibration isolators, prevent differential channels, isolator support brackets, reinforcing rods, movement between driving and driven members, anchor bolts and concrete fill into a controlled load reduce rocking by lowering equipment center of transfer system, utilizing steel in tension and concrete gravity, reduce motion of equipment during start-up in compression, resulting in high strength and stiff- and shut-down, act to reduce reaction movement due ness with minimum steel frame weight. Completed to operating loads on equipment, and act as a noise inertia bases using model CIB-L frames are stronger barrier. and stiffer than standard inertia base frames using heavier steel members. Typical uses for KINETICSTM inertia base frames, with poured concrete and supported by KINETICSTM Standard CIB-L inertia base frames -
KINETICS Tool of ANSA for Multi Body Dynamics
physics on screen KINETICS tool of ANSA for Multi Body Dynamics Introduction During product manufacturing processes it is important for engineers to have an overview of their design prototypes’ motion behavior to understand how moving bodies interact with respect to each other. Addressing this, the KINETICS tool has been incorporated in ANSA as a Multi-Body Dynamics (MBD) solution. Through its use, engineers can perform motion analysis to study and analyze the dynamics of mechanical systems that change their response with respect to time. Through the KINETICS tool ANSA offers multiple capabilities that are accomplished by the functionality presented below. Model definition Multi-body models can be defined on any CAD data or on existing FE models. Users can either define their Multi-body models manually from scratch or choose to automatically convert an existing FE model setup (e.g. ABAQUS, LSDYNA) to a Multi- body model. Contact Modeling The accuracy and robustness of contact modeling are very important in MBD simulations. The implementation of algorithms based on the non-smooth dynamics theory of unilateral contacts allows users to study contact incidents providing accurate solutions that the ordinary regularized methodologies cannot offer. Furthermore, the selection between different friction types maximizes the realism in the model behavior. Image 1: A Seat-dummy FE model converted to a multibody model Image 2: A Bouncing of a piston inside a cylinder BETA CAE Systems International AG D4 Business Village Luzern, Platz 4 T +41 415453650 [email protected] CH-6039 Root D4, Switzerland F +41 415453651 www.beta-cae.com Configurator In many cases, engineers need to manipulate mechanisms under the influence of joints, forces and contacts and save them in different positions. -
Rotation Matrix - Wikipedia, the Free Encyclopedia Page 1 of 22
Rotation matrix - Wikipedia, the free encyclopedia Page 1 of 22 Rotation matrix From Wikipedia, the free encyclopedia In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. For example the matrix rotates points in the xy -Cartesian plane counterclockwise through an angle θ about the origin of the Cartesian coordinate system. To perform the rotation, the position of each point must be represented by a column vector v, containing the coordinates of the point. A rotated vector is obtained by using the matrix multiplication Rv (see below for details). In two and three dimensions, rotation matrices are among the simplest algebraic descriptions of rotations, and are used extensively for computations in geometry, physics, and computer graphics. Though most applications involve rotations in two or three dimensions, rotation matrices can be defined for n-dimensional space. Rotation matrices are always square, with real entries. Algebraically, a rotation matrix in n-dimensions is a n × n special orthogonal matrix, i.e. an orthogonal matrix whose determinant is 1: . The set of all rotation matrices forms a group, known as the rotation group or the special orthogonal group. It is a subset of the orthogonal group, which includes reflections and consists of all orthogonal matrices with determinant 1 or -1, and of the special linear group, which includes all volume-preserving transformations and consists of matrices with determinant 1. Contents 1 Rotations in two dimensions 1.1 Non-standard orientation -
Rotational Motion CONTENTS CHAPTER-OPENING QUESTION—Guess Now! 8–1 Angular Quantities a Solid Ball and a Solid Cylinder Roll Down a Ramp
You too can experience rapid rotation—if your stomach can take the high angular velocity and centripetal acceleration of some of the faster amusement park rides. If not, try the slower merry-go-round or Ferris wheel. Rotating carnival rides have rotational kinetic energy as well as angular momentum. Angular acceleration is produced by a net torque, and rotating objects have rotational kinetic energy. P T A E H R C 8 Rotational Motion CONTENTS CHAPTER-OPENING QUESTION—Guess now! 8–1 Angular Quantities A solid ball and a solid cylinder roll down a ramp. They both start from rest at the 8–2 Constant Angular Acceleration same time and place. Which gets to the bottom first? 8–3 Rolling Motion (a) They get there at the same time. (Without Slipping) (b) They get there at almost exactly the same time except for frictional differences. 8–4 Torque (c) The ball gets there first. 8–5 Rotational Dynamics; (d) The cylinder gets there first. Torque and Rotational Inertia (e) Can’t tell without knowing the mass and radius of each. 8–6 Solving Problems in Rotational Dynamics ntil now, we have been concerned mainly with translational motion. We 8–7 Rotational Kinetic Energy discussed the kinematics and dynamics of translational motion (the role 8–8 Angular Momentum and of force). We also discussed the energy and momentum for translational Its Conservation U *8–9 Vector Nature of motion. In this Chapter we will deal with rotational motion. We will discuss the Angular Quantities kinematics of rotational motion and then its dynamics (involving torque), as well as rotational kinetic energy and angular momentum (the rotational analog of linear momentum).