Motion in One Dimension

4 Key excerpts on "Motion in One Dimension"

• 

  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

  Doing Physics with Scientific Notebook

  A Problem Solving Approach

  • Joseph Gallant(Author)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  Chapter 2

  One-Dimensional Kinematics

  We start Doing Physics with SNB in the branch of physics known as Classical Mechanics. Our study of Classical Mechanics begins with one-dimensional kinematics, the description of motion in a straight line. This includes horizontal motion to the right or left and vertical motion straight up or down. This description will tell us where an object is, where it is going, and how much time it took to get there.

  Constant Acceleration

  Studying motion with a constant acceleration is a good place to start. We can describe this motion completely without using calculus. Let’s start with some definitions and important distinctions, and then we’ll solve some one-dimensional problems.

  Displacement and Position

  To describe an object’s motion, we need to establish a coordinate system so we can specify location. For 1-dimensional motion the coordinate system is just the x-axis with the origin at x = 0 and positive values to the right. The object’s position, specified by its x-coordinate, tells us how far from the origin it is and in which direction.

  The displacement is the object’s change in position.

  (2.1)

  This difference between the object’s final and initial position tells us how far from its original position it ends. A positive displacement means the object ends to the right of x0 and a negative displacement means the object ends to the left of x0 . For a round trip, the initial and final positions are equal and the displacement is zero. The SI unit for both position and displacement is the meter (m).

  Note The use of the upper case Greek letter delta (“Δ”) to mean “change in” is standard mathematical notation, but it has no special significance in SNB. In a calculation, SNB interprets the expression Δf as Δ x f, the product of two variables.

  Even though they have the same unit, displacement and distance are different. Suppose you start 2 meters from the origin, walk 9 meters to the right, turn around and walk 5 meters back toward the origin. Your initial position was x0 = 2 m and your final position is x = 6m, so your displacement is Δx

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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

  Understanding Physics

  • Michael M. Mansfield, Colm O'Sullivan(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  4 Motion in two and three dimensions

  AIMS

  • to show how, in two and three dimensions, physical quantities can be represented by mathematical entities called vectors
  • to rewrite the laws of dynamics in vector form
  • to study how the laws of dynamics may be applied to bodies which are constrained to move on specific paths in two and three dimensions
  • to describe how the effects of friction may be included in the analysis of dynamical problems
  • to study the motion of bodies which are moving on circular paths

  4.1 Vector physical quantities

  The material universe is a three‐dimensional world. In our investigation of the laws of motion in Chapter 3 , however, we considered only one‐dimensional motion, that is situations in which a body moves on a straight line and in which all forces applied to the body are directed along this line of motion. If a force is applied to a body in a direction other than the direction of motion the body will no longer continue to move along this line. In general, the body will travel on some path in three‐dimensional space, the detail of the trajectory depending on the magnitude and direction of the applied force at every instant. Equation (3.3) as it stands is not sufficiently general to deal with such situations, for example the motion of a pendulum bob (Figure 4.1 ) or the motion of a planet around the Sun (Figure 4.2 ). Newton's second law needs to be generalised from the simple one‐dimensional form discussed in Chapter 3 .

  Figure 4.1

  A pendulum comprising a mass attached to the end of a string; the mass can move on a path such that the distance from the fixed end of the string remains constant.

  A similar problem arises if two or more forces act on a body simultaneously, for example when a number of tugs are manoeuvring a large ship (Figure 4.3

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