Physics
Dynamics
Dynamics in physics refers to the study of forces and motion, particularly how objects move and interact with each other. It involves analyzing the causes of motion and the effects of forces on objects. Dynamics encompasses concepts such as acceleration, velocity, and the laws of motion formulated by Isaac Newton.
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8 Key excerpts on "Dynamics"
eBook - PDF- Pierluigi Zotto, Sergio Lo Russo, Paolo Sartori(Authors)
- 2022(Publication Date)
- Società Editrice Esculapio(Publisher)
Dynamics of Point-like Particles 4.1 Introduction Kinematics addresses the study of the motion of bodies using a merely descriptive ap- proach. Dynamics introduces the cause and effect relationship, i.e. it studies the causes of motion, called forces † , and the methods that allow to forecast the motion they generate starting from their knowledge. The assumption of Dynamics is that the motion of a body is determined by an interaction with other bodies, which is mediated by a force. The formulae relating the forces acting on a body to its motion, or better to the variation of its motion, are called equations of motion. In other term, the problem of Dynamics is the solution of the equations of motion of a body by considering its interaction with the rest of the universe. 4.2 Newton’s Laws The classical model of Mechanics, discipline including kinematics, Dynamics and statics of bodies, is described by Newton’s theory (1642–1727). The model presumes that the mechanical processes that happen in nature are the conse- quence of three fundamental laws (or principles), obtained as a synthesis of centuries of experimental measurements of uncountable observers ‡ . • Newton’s First Law or Law of Inertia Any body keeps its state of motion unless external causes intervene to modify it. The index of a body state of motion is its velocity in a given reference system. If we define as free particle a body which is not subject to any interaction, i.e. to any force, the law establishes that a free particle can move only with constant vector velocity, i.e. with constant magnitude and direction of velocity. The modification of the state of motion, i.e. a change in velocity, is possible only by an external intervention, i.e. only if a force is applied to the particle. The law is obvious for rest, a state of null velocity, which changes only if an action is performed on the body.
eBook - PDF- Joaquim A. Batlle, Ana Barjau Condomines(Authors)
- 2022(Publication Date)
- Cambridge University Press(Publisher)
This correlation is sometimes described in a simplistic way as a causality relationship: “Dynamics is the study of the motion of bodies caused by the action of forces.” The equations of Newtonian Dynamics are differential equations relating the second order derivative of the objects generalized coordinates to the coordinates and their first derivatives, all of them at a same time instant. This simultaneity of motion variables makes it difficult (and formally impossible) to distinguish “causes” and “effects,” as the former should actually precede the latter. That distinction becomes a convention. Principle of Absolute Simultaneity The principle of absolute simultaneity 1 states, in short, that a sequence (order of succession) of events is the same for all observers (or independent from the reference frame). In Newton’s conception of the universe, absolute time goes hand in hand with absolute space: they constitute an immutable stage where physical events occur, they are independent external realities. Other Assumptions Last but not least: as in any scientific theory, Newtonian Dynamics have a limited field of application (directly related to the experimental limitations of the seventeenth century!). Outside that field, it yields inaccurate results. The main limitations are: Low-speed Dynamics: objects moving with speeds comparable to the speed of light cannot be treated successfully with this theory. Medium length scale Dynamics: molecular Dynamics and long-reach astronomy are also out of scope. Electromagnetic phenomena are excluded. 1 Batlle, J.A. and Barjau Condomines, A. (2019) Rigid Body Kinematics, Cambridge University Press, chapter 1. 2 Particle Dynamics 1.2 Galilean and Non-Galilean Reference Frames When we enter the field of Newtonian Dynamics, we are confronted with a surprising fact: all reference frames are not equivalent for the formulation of the dynamical equations (or of the fundamental laws of Newtonian Dynamics).
- T. Crouch(Author)
- 2016(Publication Date)
- Pergamon(Publisher)
Chapter 2 Dynamics 2.1.Newton's Laws of Motion Dynamics is concerned with the relationships between force, mass, energy and motion. For Engineering applications, except those dealing with nuclear and fast moving electron phenomena, the Newtonian model of mass, space, time and force is adequate. Newton (1642 - 1727) in his Philosophiae Naturalis Prinoipia Mathem-atical of 1687 enunciated three laws or axioms relating force and motion which can be stated as follows: 1 A particle will continue in a state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by forces impressed upon it. 2 A change of motion with respect to time is proportional to the motive force impressed. 3 For every force acting on a particle, there is a corresponding force exerted by the particle. These forces are equal in magnitude, but opposite in direction. The first law implies the existence of an inertial frame of reference. Consider the following hypothetical experiment. Erect a set of co-ordinate axes in deep space remote from any other matter and project a particle successively along each axis. If the axes are not accel-erating and not rotating, then the force free motion will persist along the axis which it was projected. Such a set of axes is said to be inertial. No set of axes is truly inertial, but a set of axis fixed in the f fixed' stars are very nearly inertial and must be used, for example,in space ballistics. For most Engineering applications forces can be predicted assuming that a reference frame fixed in the earth is inertial. In this text the inertial reference is always designated 1. The motion of the second law is measured by the momentum of the particle, which is the product of its mass and inertial velocity.Thus, by Newton's second law 38
eBook - PDF- Bogdan Skalmierski(Author)
- 2013(Publication Date)
- Elsevier(Publisher)
CHAPTER 2 The Dynamics of a Particle 2.1 Fundamental definitions and theorems In this chapter we shall discuss the dynamic aspects of a particle in motion. We begin with the basic laws of Dynamics. Axioi 1 (Newton's second law). If a force P acts on a particle, the acceleration thus produced is proportional to that force, which can be written as follows: P= m a, (1) where m is the mass of the particle. We shall treat mass as a primary concept. The force P should be regarded as the resultant of the forces acting on the particle, that is, n P = R j . (la) f= 1 The cited law brings into association three basic concepts (force, mass and motion) of mechanics. It is valid in inertial systems (see Chapter 5). Under the SI system, the unit of force is the newton: 1 kgms -2 = 11. For a unit of force we can also take the force with which the earth attracts 1 kg of mass: 1 kgf = 1 kg • g, where g = 9.80665 m s — 2 and is the normal value of acceleration of gravity. The equality sign is valid between inert and heavy mass. Axioi 2 (Newton's third law). If a particle A acts on another particle B with force P AD , then simultaneously B acts on A with force PBA of equal absolute value but with an opposite sense, i.e. R A B + PBa = = 0 . (2) (4) a V P i = 70 THE Dynamics OF A PARTICLE Ch. 2 This is known as the law of action and reaction. We shall now introduce the definition of work. By work one should understand a process in which resistance is being overcome along a certain route. This definition is, however, imprecise, and for that reason it is better to define work in concise mathematical notation: Work analytically formulated is a curvilinear integral df B W = P • ds . (3) A If under the integral (3) a total differential occurs, then the forces doing the work are said to have potential V. A decrease in potential is tantamount to an increase in work: — ~ V = d W. Therefore Potential forces, as it will easily be seen, act in the direction of the maximum drop of the potential.
eBook - PDF- Paul Peter Urone, Roger Hinrichs(Authors)
- 2012(Publication Date)
- Openstax(Publisher)
Motion itself can be beautiful, causing us to marvel at the forces needed to achieve spectacular motion, such as that of a dolphin jumping out of the water, or a pole vaulter, or the flight of a bird, or the orbit of a satellite. The study of motion is kinematics, but kinematics only describes the way objects move—their velocity and their acceleration. Dynamics considers the forces that affect the motion of moving objects and systems. Newton’s laws of motion are the foundation of Dynamics. These laws provide an example of the breadth and simplicity of principles under which nature functions. They are also universal laws in that they apply to similar situations on Earth as well as in space. Isaac Newton’s (1642–1727) laws of motion were just one part of the monumental work that has made him legendary. The development of Newton’s laws marks the transition from the Renaissance into the modern era. This transition was characterized by a revolutionary change in the way people thought about the physical universe. For many centuries natural philosophers had debated the nature of the universe based largely on certain rules of logic with great weight given to the thoughts of earlier classical philosophers such as Aristotle (384–322 BC). Among the many great thinkers who contributed to this change were Newton and Galileo. Figure 4.2 Isaac Newton’s monumental work, Philosophiae Naturalis Principia Mathematica, was published in 1687. It proposed scientific laws that are still used today to describe the motion of objects. (credit: Service commun de la documentation de l'Université de Strasbourg) Galileo was instrumental in establishing observation as the absolute determinant of truth, rather than “logical” argument. Galileo’s use of the telescope was his most notable achievement in demonstrating the importance of observation. He discovered moons orbiting Jupiter and made other observations that were inconsistent with certain ancient ideas and religious dogma.
eBook - PDFDynamics of Particles and the Electromagnetic Field
(With CD-ROM)
- Slobodan Danko Bosanac(Author)
- 2005(Publication Date)
- WSPC(Publisher)
Newtonian Dynamics 3 shift of emphases in the traditional Dynamics of particles is made in this chapter, without going further than the first three axioms allow. This step is essential when at a later stage additional axioms are introduced, in particular Axiom 5. Axiom 2 is in the form of equation, which for a long time epitomized physics. In a somewhat expanded form this equation (Newton equation) is and in essence it relates the rate at which a particle changes its position to the two quantities that are assumed known: the force F' on and the mass m of the particle. This relationship is somewhat obscured by introducing an intermediate variable @, called momentum. It is a vector quantity that is proportional to the velocity, and the factor of proportionality is mass m. The significance of momentum is in its very convenient properties, as it will become evident in further developments, but apart from that there is no deeper meaning to it. The space that is spanned by the coordinate vector r' and the momentum vector 9 is called phase space The equations (1.1) define initial value problem, meaning that given initial position and velocity of the particle its past and future movement is entirely determined, of course if the force is known and the mass. Based on this character of equations classical mechanics is associated with the deterministic view of Nature, regardless of the fact that initial conditions are always determined from experiment, and therefore they are never given precisely. This is overlooked when applying classical mechanics in the im- plicit belief that these uncertainties do not affect very much the prediction for the time movement of the particle. In other words, it is believed that if from the experiment the mean value for the position and velocity is chosen for the solution of the equations then their uncertainty stays the same in the course of time.
eBook - PDF- Richard C. Hill, Kirstie Plantenberg(Authors)
- 2013(Publication Date)
- SDC Publications(Publisher)
Kinetic analysis can be performed in three distinctly different ways: through Newtonian mechanics, energy methods or using impulse and momentum principles. The approach covered in this chapter is Newtonian mechanics as applied to a particle. This method uses Newton’s laws to analyze the translational motion of an object that results from the applied forces. Conceptual Dynamics Kinetics: Chapter 5 – Particle Newtonian Mechanics 5 - 3 5.1) NEWTONIAN MECHANICS 5.1.1) KINETICS Kinematics is an analysis technique that relates the position, velocity and acceleration of a body through the use of defining relations and calculus. A common question posed in kinematic analysis is, “Given the position of a particle as a function of time, find the acceleration.” Kinetics, on the other hand, is the study of how forces generate motion. A common question posed in kinetic analysis is, “A force F is applied to a particle. What is the resulting acceleration of the particle?” In this chapter, we will answer this type of question using Newtonian mechanics. In subsequent chapters we will learn other approaches for answering the same question. 5.1.2) NEWTONIAN MECHANICS If you look around, you will see things in motion. In 1687, Newton published three laws of motion that are so powerful that they are still taught in every college engineering program nearly 300 years after they were conceived. One of Newton's assumptions was that space and time are absolute. If you are a student of Einstein, you know that this is not true. Newton's laws are very good at predicting how an object will move in most practical instances. However, they do break down at the atomic level or at velocities approaching the speed of light. It should be further noted that Newton's laws only hold true when the particle’s acceleration is described with respect to an inertial reference frame, that is, a reference frame that is not accelerating or rotating.
eBook - PDF- Ronald Huston, C Q Liu(Authors)
- 2001(Publication Date)
- CRC Press(Publisher)
Chapter 5 PARTICLE Dynamics 5.1 Introduction In this chapter we list commonly used formulas for dynamic analyses of particles. These formulas arise from the basic principles of Dynamics and the corresponding laws of motion. In later chapters we review additional, more advanced procedures for application with rigid bodies and mechanical systems. Initially, we will review the Dynamics principles themselves. We then consider a few elementary applications, followed by a listing of formulas for impact/collision analysis. We conclude with a comparison of the relative advantages and disadvantages of the various methods. 5.2 Principles of Dynamics/Laws of Motion Most principles of Dynamics and the resulting laws of motion have their roots in Newtoris laws of motion. In this sense they are all equivalent and can be developed one from another. Table 5.2.1 provides a listing of these principles and laws commonly used in particle Dynamics analyses. Table 5.2.1 Dynamics Principles/Laws of Motion for Particle Dynamics Name Principle Formulas/Equations 1. Newton's Laws First Law: A particle at rest or in uniform motion remains at rest or in uniform motion unless acted upon by a force. If F = 0 v = c (a constant) (5.2.1) 148 Particle Dynamics 149 Name Principle Formulas/Equations 1. Newton's Laws Second Law: The accelera- tion of a particle is propor tional to the force applied to the particle and inversely proportional to the mass of the particle. F = ma (5.2.2) Third Law: If a particle P exerts a force on a particle Q, then Q exerts an equal and opposite force on P (Action-Reaction). f p / q = -F q / p (5.2.3) 2. d'Alembert's Principle A particle with mass m and acceleration a in an inertial frame R experiences an inertial force F* propor tional to m and a and directed opposite to a. The sum of the applied and inertia forces is zero. F* = -m a (5.2.4) F + F* = 0 (5.2.5) 3.
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