Motion of a Particle

4 Key excerpts on "Motion of a Particle"

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  eBook - ePub

  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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  eBook - ePub

  A Modern Approach to Classical Mechanics

  • Harald Iro(Author)
  • 2015(Publication Date)
  • WSPC

    (Publisher)

  3

  One-dimensional Motion of a Particle

  The simplest system consists of a single particle whose motion only has one degree of freedom. In the following, we investigate the constants of the motion and the conserved quantities for Newton’s equation in one dimension. Since the representation in phase space here is two-dimensional, the particle’s motion in phase space can be visualized graphically.

  3.1  Examples of one-dimensional motion

  Figure 3.1: The inclined track.

  i) The inclined track

  : A particle of mass m slides without friction on a track inclined at angle α to the direction of the gravitational force F = mg. Since forces obey vector addition (see Page 17), the force of gravity can be split into components parallel and perpendicular to the track1 :

  where (see Fig. 3.1 )

  Motion along the track is influenced only by F|| ; the perpendicular component F is exactly counterbalanced by the track2 (see Fig. 3.1 ). Let s be the distance along the track (with respect to some initial point s0 = 0). The momentum of the particle is p = m . Hence, the equation of motion is

  Figure 3.2: The plane pendulum.

  ii) The plane mathematical pendulum

  : A particle of mass m is attached to the end of a massless bar of length l that can swing freely about a fixed point. The particle is pushed such that its motion always remains in the plane (i.e. the initial velocity vector lies in the plane containing the bar and gravitational force vector). Since lϕ is the arclength, where ϕ is the angle between the bar and the vertical, the equation of motion reads (see also Section 3.3.2 below):

  Figure 3.3: An oscillating mass.

  iii) The harmonic oscillator

  : A particle of mass m is confined to move along the x-axis. It is attached to a spring, with equilibrium position x

  equ

  . Assume that the spring obeys Hooke’s Law, |F| ∝ |x − x

  equ

  |, meaning that the force is harmonic, i.e. it is given by F = −k(x − x

  equ

  ). If x

  equ

  is chosen as the origin, i.e. x

  equ

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  eBook - ePub

  Reasoning About Luck

  • Vinay Ambegaokar(Author)
  • 2017(Publication Date)
  • Dover Publications

    (Publisher)

  6

  Forces, motion, and energy

  ... the whole burden of philosophy seems to consist in this – from the phenomena of motions to investigate the forces of nature, and then from these forces to demonstrate the other phenomena ... Isaac Newton

  Probability enters theoretical physics in two important ways: in the theory of heat, which is a manifestation of the irregular motions of the microscopic constituents of matter; and, in quantum mechanics, where it plays the bizarre but, as far as we know, fundamental role already briefly mentioned in the discussion of radioactive decay.

  Before we can understand heat, we have to understand motion. What makes objects move, and how do they move? Isaac Newton, in the course of explaining the motion of planets and of things around us that we can see and feel with our unaided senses, answered these questions for such motions three centuries ago. The science he founded has come to be called classical or Newtonian mechanics, to distinguish it from quantum mechanics, the theory of motion in the atomic and sub-atomic world.

  Classical mechanics is summarized in Newton’s laws’ of motion. These will here be illustrated by an example involving the gravitational attraction, described by Newton’s ‘law’ of gravitation. In the brief description that follows, I shall attempt to show that these laws of nature: (i) unify apparently unrelated phenomena – like the motion of the moon and the fall of an apple; (ii) make simple but astonishing quantitative connections – between, for example, the rate of the moon’s revolution around the earth and the numerical value of the acceleration in an apple’s fall; and (iii), suggest new and useful concepts – here kinetic energy, potential energy, and, more generally, energy – which provide access to unexplored territory.

  To say that an object moves is to say that its position in space changes with time. This can only be given meaning by referring it to some fixed point. Imagine a straight line directed from such a fixed point to the place where our object is. This directed line is called the position ‘vector’ of the object. The word vector is used to indicate something that has a magnitude and a direction. We shall encounter other vector quantities soon. Two vectors are said to be equal when they have the same magnitude and

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  eBook - ePub

  An Elementary Treatise on Theoretical Mechanics

  • Sir James H. Jeans(Author)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  CHAPTER IX

  MOTION OF SYSTEMS OF PARTICLES

  EQUATIONS OF MOTION

  176. The present chapter will deal with the motion of systems of particles, taking account of the actions and reactions which may be set up between the different pairs of particles. As a preliminary to this, it will be convenient to recapitulate the results which have been obtained for a single particle, stating these results in a more analytical form than before.

  The whole system of forces which act on a particle must, since they act at a point, have a single force as resultant. Let us call this resultant P, and denote its components along three rectangular axes by X, Y, Z.

  Also the particle, being regarded as a point, must have a definite acceleration f, and, since f is a vector, this acceleration may be supposed to be compounded of three components ƒ x , ƒ y , ƒ z along the three coordinate axes.

  The second law of motion supplies the relation

  (65)

  We are, however, told more than this by the second law of motion: we are told that the directions of P and of ƒ are the same. Let λ, μ, v be the direction cosines of this single direction, then we have

  X = λP , Y = μP , Z = vP,

  and also

  ƒ x = λƒ, ƒ y = μƒ ,

  ƒz

  = vƒ.

  From these relations, combined with relation (65), we clearly have

  (66)

  These are the equations of Motion of a Particle in analytical form. They simply express the second law of motion in mathematical language.

  177. Let x, y, z be the coordinates of the particle at any instant, and let u , v, w be the three components of its velocity. The component u is the velocity, along the axis of x, of the projection of the moving point on the axis of x, and the distance of this point from the origin at any instant is simply x. Thus, by the definition of velocity, we have

  (67)

  and similarly, of course,

  The rate at which the x -component of velocity increases is , but it has also been supposed to be ƒ x , for this is the x

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