Kinematics of Particles

8 Key excerpts on "Kinematics of Particles"

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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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  Foundations and Applications of Engineering Mechanics

  • H. D. Ram, A. K. Chauhan(Authors)
  • 2015(Publication Date)
  • Cambridge University Press

    (Publisher)

  6 KINEMATICS OF PARTICLE AND RIGID BODY 6.1 Introduction Kinematics is the study of the motion of particles and rigid bodies disregarding the forces associated with these motions. 6.2 Kinematics of particle Kinematics of particle involves the study of position, velocity and acceleration of the particle without any consideration of the forces working on it. Particle can move on a straight line, on a plane or in space. We will restrict our study to plane motion only. The plane motion of the particle is classified as: (a) Straight line motion (b) Motion on curved path Following classification of motion on curved path is useful from application point of view (a) Position, velocity and acceleration in terms of Cartesian components (b) Position, velocity and acceleration in terms of path variables and (c) Position, velocity and acceleration in terms of polar coordinates 6.3 Straight line motion of particle The rectilinear motion of particle is the motion along straight line. Suppose a point is moving along x-axis, The position of the particle at time t is x The position of the particle at time ( t +  t ) is ( x +  x ). The displacement of the particle during the time interval  t is equal to {( x +  x ) – x or  x } The average velocity of the particle, Kinematics of Particle and Rigid Body | 341 v x t av = Δ Δ Instantaneous velocity of the particle is the limiting value of average velocity as time  t → 0. Therefore, v x t t = → lim Δ Δ Δ 0 or, v dx dt = (6.1) Suppose the velocity of the particle at time t is v and at time ( t +  t ) its velocity is ( v +  v ). Average acceleration is given by: a v t av = Δ Δ Instantaneous acceleration of the particle is the limiting value of average acceleration as time Δ t → 0 . Therefore, a v t t = → lim Δ Δ Δ 0 or, a dv dt = (6.2) Rectilinear motion of the particle with constant acceleration a : Suppose a particle is moving along x-axis. At t = 0, the position is x 0 and the velocity is v 0 .

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  Intermediate Dynamics for Engineers

  Newton-Euler and Lagrangian Mechanics

  • Oliver M. O'Reilly(Author)
  • 2020(Publication Date)
  • Cambridge University Press

    (Publisher)

  Part I A Single Particle 1 Kinematics of a Particle 1.1 Introduction One of the main goals of this book is to enable the reader to consider a mechanical system, model it as a system of particles and rigid bodies, and then interpret the results of the model. For this to happen, the reader needs to be equipped with an array of tools and techniques, the cornerstone of which is to be able to precisely formulate the kinematics of a particle. Without this foundation firmly in place, the conclusions from the model either do not hold up or lack conviction. Much of the material presented in this chapter will be used repeatedly throughout the book. We start the chapter with a discussion of coordinate systems for a particle moving in a three-dimensional space. This naturally leads us to a discussion of curvilinear coor- dinate systems. These systems encompass all of the familiar coordinate systems and the material presented is useful in many other contexts. At the conclusion of our discussion of coordinate systems and their application to particle mechanics, you should be able to establish expressions for gradient and acceleration vectors in any coordinate system. The other major topics of this chapter pertain to constraints on the motions of par- ticles. In earlier dynamics courses, these topics were intimately related to judicious choices of coordinate systems to solve particle problems. For such problems, a con- straint is usually imposed on the position vector of a particle. Here, we also discuss time-varying constraints on the velocity vector of the particle. Along with curvilinear coordinates, the topic of constraints is one most readers will not have seen before, and for many it will hopefully constitute an interesting thread that winds its way through this book. 1.2 Reference Frames To describe the Kinematics of Particles and rigid bodies, we presume the existence of a space with a set of three mutually perpendicular axes that meet at a common point P.

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  Engineering Mechanics

  Dynamics

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

    (Publisher)

  Jupiterimages/Getty Images, Inc. CHAPTER 3 Kinetics of Particles 3/1 Introduction According to Newton’s second law, a particle will accelerate when it is subjected to unbalanced forces. Kinetics is the study of the relations between unbalanced forces and the resulting changes in motion. In Chapter 3 we will study the kinetics of particles. This topic requires that we combine our knowledge of the properties of CHAPTER OUTLINE 3/1 Introduction SECTION A Force, Mass, and Acceleration 3/2 Newton’s Second Law 3/3 Equation of Motion and Solution of Problems 3/4 Rectilinear Motion 3/5 Curvilinear Motion SECTION B Work and Energy 3/6 Work and Kinetic Energy 3/7 Potential Energy SECTION C Impulse and Momentum 3/8 Introduction 3/9 Linear Impulse and Linear Momentum 3/10 Angular Impulse and Angular Momentum SECTION D Special Applications 3/11 Introduction 3/12 Impact 3/13 Central-Force Motion 3/14 Relative Motion 3/15 Chapter Review 56 Article 3/2 Newton’s Second Law 57 forces, which we developed in statics, and the kinematics of particle motion just covered in Chapter 2. With the aid of Newton’s second law, we can combine these two topics and solve engineering problems involving force, mass, and motion. The three general approaches to the solution of kinetics problems are: (A) direct application of Newton’s second law (called the force-mass-acceleration method), (B) use of work and energy principles, and (C) solution by impulse and momentum methods. Each approach has its special characteristics and advan- tages, and Chapter 3 is subdivided into Sections A, B, and C, according to these three methods of solution. In addition, a fourth section, Section D, treats special applications and combinations of the three basic approaches. Before proceeding, you should review carefully the definitions and concepts of Chapter 1, because they are fundamental to the developments which follow.

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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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  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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  Conceptual Dynamics

  • Richard C. Hill, Kirstie Plantenberg(Authors)
  • 2013(Publication Date)
  • SDC Publications

    (Publisher)

  A particle may be treated as a point; it has mass but no size. Therefore, we will only need to consider translational motion and not worry about rotation. Conceptual Dynamics Kinematics: Chapter 2 – Kinematics of Particles - Rectilinear Motion 2 - 3 2.1) RECTILINEAR MOTION 2.1.1) RECTILINEAR MOTION The motion of a real object with size and mass is very complex. Take, for example, the stunt airplane shown in Figure 2.1-1. A plane can move in the forward direction, turn around and move in the opposite direction. It can gain altitude, lose altitude, and also turn left and right. It becomes even more complex when we consider that the plane can yaw, pitch and roll. These are the various rotations the plane can undergo. In total, there are six variables, or degrees of freedom, that are needed to describe the position and orientation of this plane. If we include velocities and accelerations, the number of variables needed increases to eighteen. As students who are just learning dynamics, you don’t want to jump in and learn how to analyze the most complex body and system first. Therefore, we are going to start with a simple body and constrain this body to move in a particular way. First, we will start by analyzing a particle. A particle is a body that has mass but no size. This means that it can translate but not rotate. Using a particle to describe the body of interest, we simplify our calculations by removing all of the rotational degrees of freedom. Second, we will constrain our body to move along a straight line. This means that the body can only move left and right, up and down or forwards and backwards, depending on the situation. This simplifies the calculations even further because now we only need three variables or degrees of freedom to describe how the body is moving. One variable to describe where the body is on the line, one variable to describe how fast it is moving along the line and one variable to describe how its speed is changing.

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  Intermediate Dynamics

  • Patrick Hamill(Author)
  • 2022(Publication Date)
  • Cambridge University Press

    (Publisher)

  The Lagrangian (which you will study in Chapter 4) is the basis of Elementary Particle Physics and the Hamiltonian (also in Chapter 4) is fundamental in Quantum Mechanics. As you may recall, the mechanics section of your introductory physics book covered the following topics: • Kinematics • Newton’s second law • Work and energy • Momentum • Rotational motion. We now very briefly review some concepts from each of these items. 1.1 Kinematics Kinematics is the study of motion. Essentially, kinematics involves determining the relationships between position, velocity, acceleration, and time. 1 If you are a particularly well-prepared student and feel that you know the material in this chapter, I suggest that you go to the end of each section where you will find a few exercises. If you can solve them, then skip to the next section, but if you feel uncertain or even somewhat confused, read the section. You might also try solving a few of the problems at the end of the chapter. 4 1 A BRIEF REVIEW OF INTRODUCTORY CONCEPTS Position is denoted by the vector 2 r and the change in position (or displacement) can be written as r. Velocity is defined as the displacement with respect to time, so the average velocity is given by v = r t , where t is the time during which the object had a displacement r. As the time interval becomes very small, we replace the difference (represented by ) with the derivative and write v = d r dt . (1.1) The change in velocity with respect to time is called the acceleration and is given by a = d v dt . (1.2) We can use the definitions of acceleration and velocity to write the inverse relation:  d v =  adt . Integrating we get v =  adt + constant. Integrating again, r =  vdt + constant. These are vector relationships and are valid in any coordinate system. In Chapter 2 you will find the relations between acceleration, velocity, and position in various coordinate systems.

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