Particle Model Motion

9 Key excerpts on "Particle Model Motion"

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  Engineering Mechanics

  • Ping YI, Jun LIU, Feng JIANG(Authors)
  • 2022(Publication Date)
  • EDP Sciences

    (Publisher)

  Chapter 6 Kinematics of Particles Objectives  Understand the kinematic concepts of position, displacement, velocity, and acceleration.  Investigate a particle’s curvilinear motion using different coordinate systems.  Study absolute dependent motion of two particles.  Analyze relative motion of two particles using a translating coordinate system. Statics deals with forces that lead to equilibrium of particles or bodies. However, kinematics cares about only the geometrical aspects of the motion, such as position, displacement, velocity and acceleration, without involving forces. The analysis object in kinematics can also be considered as a particle or a rigid body. If the motion analyzed is characterized by the motion of the object’s mass center and the dimensions (size and shape) have no or little influence on the motion, this object can be considered as a particle. For example, when we analyze a plane’s trajectory or its position as a function of time, figure 6.1, it can be considered as a particle although it is not small at all to our naked eyes. But if we analyze the fighter plane’s rotation and flipping, figure 6.2, it has to be considered as a body since the dimensions influence the motion greatly. So, we can see that whether an object can be considered as a particle or not is decided by its motion analysis, not by its physical dimensions. Following the cognitive process of learning from the simple to the complex, this chapter considers the motion of a particle, and the motion of a rigid body will be discussed in the next chapter. DOI: 10.1051/978-2-7598-2901-9.c006 © Science Press, EDP Sciences, 2022 6.1 General Curvilinear Motion When a particle moves along a straight line, it undergoes rectilinear motion; whereas a particle moving along a curved path undergoes curvilinear motion. Rectilinear motion has been considered extensively in physics and it will be treated as a special case of curvilinear motion.

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  Dynamics in Engineering Practice

  • Dara W. Childs, Andrew P. Conkey(Authors)
  • 2015(Publication Date)
  • CRC Press

    (Publisher)

  7 2.1 Introduction Dynamics generally involves motion of particles and rigid bodies and has been traditionally and usefully divided into “kinematics” and “kinetics.” Kinematics is geometric in nature, considering motion without regard to the forces that either cause or result from motion. Kinetics uses kinematics as a foundation for studying the motion of particles or rigid bodies due to prescribed forces or, conversely, examines reaction forces due to prescribed motion. By itself, kinematics is extremely useful in solving and analyzing engineering problems, and the mastery of kinematics is absolutely essential for the subsequent study of kinetics. The notion of a particle was covered in Chapter 1 and basically implies an idealized situation where the physical dimensions of a body (length, height, width, etc.) can be neglected, with all of the body’s mass con-centrated at a single point. In conventional terms, one can think of a physically small body such as the head of a pin or a “beebee.” However, the size of a body is, in most cases, a matter of perspective. For example, an aircraft at high altitude appears to be a dot in the sky. On a solar system scale, the Earth can be considered a particle. Kinematically, the nice thing about a particle is that we only need to keep track of its coordinates with respect to a reference system; for example, in a Cartesian coordinate system, the coordinates ( x , y , z ) would tell us the position of a particle. Cartesian coor-dinates are named after the French philosopher René Descartes (1596–1650) who invented analytical geom-etry. Descartes is known for the (philosophical) state-ment “I think; therefore, I am.” An appreciation of the relative ease of dealing with particles versus rigid bodies can be had by thinking of the more complicated problem of defining the position and orientation of a rigid body. For example, an aircraft can be changing position with respect to ground while pitching, rolling, and yawing.

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  Classical Mechanics

  A Computational Approach with Examples Using Mathematica and Python

  • Christopher W. Kulp, Vasilis Pagonis(Authors)
  • 2020(Publication Date)
  • CRC Press

    (Publisher)

  CHAPTER 2

  Single-Particle Motion in One Dimension

  In this chapter, we will examine one-dimensional motion, i.e., motion along a line. It is sometimes the case that a particle’s motion need only to be described along one direction. Furthermore, a careful study of one-dimensional motion will be a useful foundation for understanding more general motion in higher dimensions. In this chapter, we will give several examples of solving Newton’s second law, F = ma in one dimension. We will consider several types of forces: both constant and those which depend on time F(t), velocity F(v), and position F(x). In addition, we will discuss and demonstrate two different uses of computers to solve physics problems: how to use computer algebra systems (CAS) to obtain the analytical solutions of Newton’s second law, and how to obtain numerical solutions of ordinary differential equations (ODE) using software packages and by using the Euler method.

  2.1Equations of motion

  To begin our study of one-dimesional motion, we first need to make some assumptions about the object whose motion we are examining. One fundamental assumption in this chapter is that the object being studied is a point particle. In order to mathematically describe the motion of a particle under the influence of a force, we need to find the particle’s equations of motion. The equations of motion of a particle are the equations which describe its position, velocity, and acceleration as a functions of time. Equations of motion can be in the form of algebraic equations, or in the form of differential equations.

  As we will see, the equations of motion of a particle can be found by solving Newton’s second law as a differential equation. In this chapter, we will focus on one-dimensional motion, where the force vector and the particle’s displacement are along the same line (but not necessarily in the same direction—the direction could be horizontal or vertical). Because all vectors in a given problem lay along the same line, we drop the vector notation in all the equations. A negative sign between two quantities will denote vectors that lay in opposite directions along the same line.

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  Meriam's Engineering Mechanics

  Dynamics

  • L. G. Kraige, J. N. Bolton(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  Kinematics is often described as the “geometry of motion.” Some en-gineering applications of kinematics include the design of cams, gears, linkages, and other machine elements to control or produce certain desired motions, and the calculation of flight trajectories for aircraft, rockets, and spacecraft. A thorough working knowledge of kinematics is a prerequisite to kinetics, which is the study of the relationships between motion and the corresponding forces which cause or accompany the motion. Particle Motion We begin our study of kinematics by first discussing in this chapter the motions of points or particles. A particle is a body whose physical dimensions are so small com-pared with the radius of curvature of its path that we may treat the motion of the particle as that of a point. For example, the wingspan of a jet transport flying be-tween Los Angeles and New York is of no consequence compared with the radius of curvature of its flight path, and thus the treatment of the airplane as a particle or point is an acceptable approximation. CHAPTER OUTLINE 2/1 Introduction 2/2 Rectilinear Motion 2/3 Plane Curvilinear Motion 2/4 Rectangular Coordinates ( x -y ) 2/5 Normal and Tangential Coordinates ( n -t ) 2/6 Polar Coordinates ( r -𝜽 ) 2/7 Space Curvilinear Motion 2/8 Relative Motion (Translating Axes) 2/9 Constrained Motion of Connected Particles 2/10 Chapter Review 16 Article 2/2 Rectilinear Motion 17 We can describe the motion of a particle in a number of ways, and the choice of the most convenient or appropriate way depends a great deal on experience and on how the data are given. Let us obtain an overview of the several methods developed in this chapter by referring to Fig. 2 ∕ 1 , which shows a particle P moving along some general path in space. If the particle is confined to a specified path, as with a bead sliding along a fixed wire, its motion is said to be constrained . If there are no physical guides, the motion is said to be unconstrained .

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  Dynamics of Mechanical Systems

  • Harold Josephs, Ronald Huston(Authors)
  • 2002(Publication Date)
  • CRC Press

    (Publisher)

  57 3 Kinematics of a Particle 3.1 Introduction Kinematics is a study of motion without regard to the cause of the motion. Often this motion occurs in three dimensions. For such cases, and even when the motion is restricted to two dimensions, it is convenient to describe the motion using vector quantities. Indeed, the principal kinematic quantities of position, velocity, and acceleration are vector quantities. In this chapter, we will study the kinematics of particles. We will think of a “particle” as an object that is sufficiently small that it can be identified by and represented by a point. Hence, we can study the kinematics of particles by studying the movement of points. In the next chapter, we will extend our study to rigid bodies and will think of a rigid body as being simply a collection of particles. We begin our study with a discussion of vector differentiation. 3.2 Vector Differentiation Consider a vector V whose characteristics (magnitude and direction) are dependent upon a parameter t (time). Let the functional dependence of V on t be expressed as: (3.2.1) Then, as with scalar functions, the derivative of V with respect to t is defined as: (3.2.2) The manner in which V depends upon t depends in turn upon the reference frame in which V is observed. For example, if V is fixed in a reference frame ˆ R, then in ˆ R, V is independent of t . If, however, ˆ R moves relative to a second reference frame, R, then in R V depends upon t (time). Hence, even though the rate of change of V relative to an observer in ˆ R is zero, the rate of change of V relative to an observer in R is not necessarily zero. Therefore, in general, the derivative of a vector function will depend upon the reference frame in which that derivative is calculated. Hence, to avoid ambiguity, a super-script is usually added to the derivative symbol to designate the reference frame in which V V = ( ) t d dt t t t t t V V V = + ( ) − ( ) → D Lim ∆ ∆ ∆ 0

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  Engineering Mechanics

  Dynamics

  • L. G. Kraige, J. N. Bolton(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  The only way to reduce both the acceleration of gravity and the corresponding weight of a body to zero is to take the body to an infinite distance from the earth. SAMPLE PROBLEM 1/1 (CONTINUED) Even if this car maintains a constant speed along the winding road, it accelerates laterally, and this acceleration must be considered in the design of the car, its tires, and the roadway itself. Sean Cayton∕The Image Works CHAPTER 2 Kinematics of Particles 2/1 Introduction Kinematics is the branch of dynamics which describes the motion of bodies without reference to the forces which either cause the motion or are generated as a result of the motion. Kinematics is often described as the “geometry of motion.” Some en- gineering applications of kinematics include the design of cams, gears, linkages, and other machine elements to control or produce certain desired motions, and the calculation of flight trajectories for aircraft, rockets, and spacecraft. A thorough working knowledge of kinematics is a prerequisite to kinetics, which is the study of the relationships between motion and the corresponding forces which cause or accompany the motion. Particle Motion We begin our study of kinematics by first discussing in this chapter the motions of points or particles. A particle is a body whose physical dimensions are so small com- pared with the radius of curvature of its path that we may treat the motion of the particle as that of a point. For example, the wingspan of a jet transport flying be- tween Los Angeles and New York is of no consequence compared with the radius of curvature of its flight path, and thus the treatment of the airplane as a particle or point is an acceptable approximation.

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  Arthur Haas: Introduction to Theoretical Physics. Volume 1

  • Arthur Haas, T. Verschoyle(Authors)
  • 2020(Publication Date)
  • De Gruyter

    (Publisher)

  To simplify the following investigations we shall assume that the body moves as if its whole mass were concentrated at a single point. The following considerations are therefore based on an imaginary so-called particle. 2 1 Newton's statement of the Second Law of Motion is : Mutationem motus proportionalem esse vi motrici impressse et fieri secundum lineam rectam qua vis ilia imprimitur ( Change of motion is proportional to the impressed force and takes place in the direction in which that force is impressed ). Newton adds to this, as a second definition, the statement : Quantitas motus est mensura eius orta ex velocitate et quantitate materiae coniunctim (The quantity of motion is measured by the product of the velocity and the mass ). Theoretical physics takes as unit of mass the gram, i.e., the thousandth part of a standard mass which is preserved at Sèvres (near Paris), and which, with very fair accuracy, is equal to the mass of a cubic decimeter of water at 4° C. The unit of length is the centimetre, i.e., the hundredth part of a standard measure of length preserved at Sèvres. As unit of time we have the second, i .e., the 24 x 60 x 60th part of the mean solar day. From these three units we can readily derive the units of velocity and acceleration, as well as that of force, which is termed a dyne. 2 The imaginary concentration of the mass in a single point greatly simplifies the problem, especially when the acting force is a space function, and so changes from place to place, in which case consideration of the distribution of the body MOTION OF A F R E E MATERIAL PARTICLE 17 The position of the moving particle at any moment can be determined by a radius-vector r which is drawn from any given origin to the particle, and whose components are the coordinates x, y, z of the latter. Let the values of the radius-vector at times t and (t + dt) be r x and r 2 ; then (2) r 2 = r x + ^ c i < .

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  Game Physics Engine Development

  How to Build a Robust Commercial-Grade Physics Engine for your Game

  • Ian Millington(Author)
  • 2010(Publication Date)
  • CRC Press

    (Publisher)

  Copyright © 2010, Elsevier Inc. All rights reserved. DOI: 10.1016/B978-0-12-381976-5.00003-6 47 48 Chapter 3 The Laws of Motion 3.1 The Particle A particle has a position, but no orientation. In other words, we can’t tell what direc-tion a particle is pointing: it either doesn’t matter or it doesn’t make sense. In the former category are bullets: in a game we don’t really care which direction a bul-let is pointing in, we just care what direction it is traveling and whether it hits the target. In the second cateogry are sparks of light, from an explosion for example— because the spark is a dot of light, it doesn’t make sense to ask which direction it is pointing. For each particle we’ll need to keep track of various properties: current position, velocity, and acceleration. We will add properties to the particle as we go. Position, velocity, and acceleration are all vectors. The particle can be implemented with the following structure: Excerpt from file include/cyclone/particle.h /** * A particle is the simplest object that can be simulated in the * physics system. */ class Particle { protected: /** * Holds the linear position of the particle in * world space. */ Vector3 position; /** * Holds the linear velocity of the particle in * world space. */ Vector3 velocity; /** * Holds the acceleration of the particle. This value * can be used to set acceleration due to gravity (its primary * use), or any other constant acceleration. */ Vector3 acceleration; }; Using this structure, we can apply some basic physics to create our first physics engine. 3.2 The First Two Laws 49 3.2 The First Two Laws There are three laws of motion put forward by Newton; for now we will need only the first two. They deal with the way an object behaves in the presence and absence of forces. The first two laws of motion follow: 1. An object continues with a constant velocity unless a force acts upon it. 2. A force acting on an object produces acceleration that is proportional to the object’s mass.

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  Mechanics and Strength of Materials

  • Bogdan Skalmierski(Author)
  • 2013(Publication Date)
  • Elsevier

    (Publisher)

  CHAPTER 1 Kinematics The primary concepts in mechanics are space and time. Regarding space, we shall assume that it is Euclidean (plane), therefore an orthogonal frame of reference can be applied to it. Furthermore, it will be assumed that space is uniform and isotropic, i.e., there are no distinct loci nor preferred directions in it. With regard to time, it will be recognized as uniform, i.e., preferred instants are non-existent in it. Some parts of this book deal with problems within the purview of classical mechanics, therefore the following two hypotheses have been accepted: 1. Time is absolute, i.e. it runs identically in all frames of reference (moving and fixed). 2. The distance between two arbitrary points in space, irrespective of the frame of reference in which it is measured, is identical (it is an invariant). Kinematics is a science of motion but it is not concerned with the causes liable to induce or disturb motion. By motion we shall understand changes in time in the position of a body referred to a system treated as stationary. The position of a body or configuration is an area of Euclidean space, in which the particles of that body have been mapped one-to-one and continuously. This mapping we shall call homeomorphism. Accordingly, motion is the change of mapping in time. Note that the concept of motion is relative, since it depends on the adopted frame of reference. 1.1 Motion of a single particle We assume a right-handed frame of reference Oxyz in a space (Fig. 1.1), embracing a triplet of unit vectors, i, j, and k, which correspond to x, y, Fig. 1.1 12 KINEMATICS Ch. 1 and z. In a space endowed with such orientation we shall consider the motion of a single particle determined by a position vector r(t). This vector, being the function of time t, can be written as follows: r(t) = x(t)i+y(t)j+z(t)k (1) or more concisely, using the summation convention, r(t) = x`(t)e i .

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