Physics
Motion of a Particle
The motion of a particle refers to its change in position over time. It can be described in terms of displacement, velocity, and acceleration. Understanding the motion of a particle is fundamental in physics and is often analyzed using mathematical equations and graphical representations to study its behavior and predict future motion.
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11 Key excerpts on "Motion of a Particle"
No longer available |Learn more- 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.
eBook - PDF- 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
eBook - PDFMeriam'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 .
eBook - PDF- James Shipman, Jerry Wilson, Charles Higgins, Bo Lou, James Shipman(Authors)
- 2020(Publication Date)
- Cengage Learning EMEA(Publisher)
This chapter focuses on describing motion and on defining and discussing terms such as speed, velocity, and acceleration. These concepts will be considered without the forces that produce motion. The discussion of forces is reserved for Chapter 3. [Note: If you are not familiar with powers-of-10 notation (scientific notation) or if you need a math review, then see the appropriate sections in the Appendices.] 2.1 Defining Motion Key Questions ● ● What is needed to designate a position? ● ● What is motion? The term position refers to the location of an object. To designate the position of an object, a reference point and a unit measurement scale are needed. For example, the entrance to campus is 1.6 km (1 mi) from the intersection with a particular traffic light. The book is 15 cm from the corner of the table. The Cartesian coordinates of the point on a graph are (x, y) = (2.0 cm, 3.0 cm). Here the reference point is the origin of the coordinate system (Chapter 1.4). If an object changes position, we say that motion has occurred. That is, an object is in motion when it is undergoing a continuous change in position. Consider an automobile traveling on a straight highway. The motion of the auto- mobile may or may not be occurring at a constant rate. In either case, the motion is described by using the fundamental units of length and time. The description of motion by length and time is evident in running. For example, as shown in ●●Fig. 2.1, the cheetah runs a certain distance at full speed in the shortest possible time. Combining length and time to give the time rate of change of position is the basis of describing motion in terms of speed and velocity, as discussed in the following section. PHYSICS FAC TS ● ● It takes 8.33 minutes for light to travel from the Sun to the Earth. (See Example 2.2.) ● ● Electrical signals between your brain and muscles travel at about 435 km/h (270 mi/h).
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 - PDFMechanics
Lectures on Theoretical Physics, Vol. 1
- Arnold Sommerfeld(Author)
- 2016(Publication Date)
- Academic Press(Publisher)
CHAPTER I MECHANICS OF A PARTICLE § 1. Newton's Axioms The laws of motion will be introduced in axiomatic form; they summarize in precise form the whole body of experience. First law: Every material body remains in its state of rest or of uniform rectilinear motion unless compelled by forces acting on it to change its state. 1 We shall at first withhold explanation of the concept of force introduced in this law. We notice that the states of rest and of uniform {rectilinear) motion are treated on equal footing and are regarded as natural states of the body. The law postulates a tendency of the body to remain in such a natural state; this tendency is called the inertia of the body. One often speaks of Galileo's law of inertia instead of Newton's first law in referring to the above axiom. We must say in this connection that while it is perfectly true that Galileo arrived at this law long before Newton (as a limiting result of his experiments with sliding bodies on planes of vanishing inclination), we find it characteristic of Newton that the law holds top position in his system. Newton's word body will, for the time being, be replaced by the words particle or mass point. To formulate the first law mathematically we shall make use of definitions 1 and 2 preceding it in the Principia. Definition 2 : The quantity of motion is the measure of the same, arising from the velocity and the quantity of matter conjunctly. 2 The quantity of motion is hence the product of two factors, the velocity, whose meaning is geometrically evident, 3 and the quantity of 1 We mention here, and in connection with what is to follow, the book Die Mechanik in ihrer Entwickelung (8th ed., F. A. Brockhaus, Leipzig, 1923; translated into English under the title The Science of Mechanics, Open Court Publishing Co., LaSalle, 111., 1942) by Ernst Mach.
eBook - PDF- John Matolyak, Ajawad Haija(Authors)
- 2013(Publication Date)
- CRC Press(Publisher)
25 © 2010 Taylor & Francis Group, LLC Motion in One Dimension This chapter addresses deriving and using the equations of motion that describe the time depen-dence of an object’s displacement, velocity, and acceleration. Relationships between displacement, velocity, and acceleration are also of importance and will be derived. This chapter starts with the basic definitions of displacement and average velocity of an object moving in one dimension. Such a simplified start will help to lead a complete set of equations of motion for an object moving along one dimension, east–west, north–south, or up and down. In the context of coordinate systems that were treated in the previous chapter, the one-dimensional motion will reduce the time and effort needed on the study of motion in two dimensions, which is the subject of the next chapter. 2.1 DISPLACEMENT Motion can be defined as a continuous change in position and that change could occur in one, two, or three dimensions. As the treatment here will be limited to motions only in one dimension, one axis of the coordinate system described in Chapter 1 will suffice. Choosing this axis as the x-axis, we depict on it a point, O, which will be considered as an origin of zero coordinate. Any point on the right of the origin O will have a positive coordinate and any point on the left of O will have a negative coordinate (Figure 2.1). The displacement, denoted by Δ x, of an object as it moves from an initial position x i to a final position x f along the x-axis can be defined as the change in the object’s position along the x-axis. That is Δ x = x f – x i . (2.1) 2.1.1 S PECIAL R EMARKS The quantities in Equation 2.1 are treated as vectors. x i is the initial position vector, x f the final posi-tion vector, and Δ x the displacement vector. Accordingly, all these quantities have a direction and magnitude.
eBook - PDFEngineering 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.
eBook - PDFApplied Mechanics
Made Simple
- George E. Drabble(Author)
- 2013(Publication Date)
- Made Simple(Publisher)
CHAPTER TWO THE NATURE OF MOTION Without concerning ourselves about the reasons why motion takes place (these we shall deal with in Chapter Four), let us consider a familiar case of a moving object: an aircraft crossing the Atlantic. The first consideration is where it is going to and where it has come from. In terms of applied mechanics, we call this the displacement of the aircraft. The second consideration (a very important one) is how long it takes to perform the journey. This is deter-mined by the average rate of travel along the route chosen, and we call this the average velocity. Thirdly, and finally, we know that, although it is con-venient for our schedules to speak of an average speed of, say 550 miles an hour, under practical conditions the actual speed is bound to vary from this. To take the two most obvious divergencies, the plane cannot start at 550 miles an hour, nor can it finish at this speed. The velocity, then, has to change from time to time, and the rate at which it does this is termed the acceleration. Let us look at each of these aspects of motion in turn. (1) Displacement Displacement seems to be a very simple concept. But if we are going to deal mathematically with it, which is exactly what the study of applied mechanics purports to do with physical situations, we have to be very careful to make sure that the rules of mathematics work when applied to each and every situation. Let us try to apply a simple mathematical rule of addition to displacement. We know that, mathematically, 2 added to 3 makes 5. This is easy; but implicit in this simple statement are quite a few important conditions. First, the two things added must be of the same kind. Two years added to three months does not add up to five of anything. Secondly, applying the rule to displacement, it seems obvious at first that two feet added to three feet makes five feet; and indeed this is true if, for instance, we are measuring out material or merely measuring distance travelled.
- Arthur Haas, T. Verschoyle(Authors)
- 2020(Publication Date)
- De Gruyter(Publisher)
INTRODUCTION TO THEORETICAL PHYSICS PART I MECHANICS TYogether with the General Theory of Vector Fields, ot Vibrations, and of Potential CHAPTER I THE MOTION OF A FREE MATERIAL PARTICLE § 1. The Principle of Inertia, and the Conception of Force. THE simplest form of motion is rectilinear motion in which equal distances are traversed in equal times ; such motion is termed uniform. The velocity of motion is defined as the ratio of any given length of the path to the time required in traversing it, and, in the case of uniform motion, its value remains constant and independent of the length of path considered. If, now, a body describes an entirely arbitrary path, which will in general be curved, and if its motion is likewise arbitrary, then for a short length of path, the motion can also be regarded as being approximately uniform. The shorter the length we consider, the closer the approximation of the imaginary uniform and rectilinear motion to the actual non-uniform and curved motion between two neigh-bouring points along the path. Hence if ds is the element of length along the path which is described in the element of time dt, we may define the differential coefficient dsjdt as the instantaneous value of the velocity. We may also ascribe an instantaneous direction to the velocity, namely, the direction of the element of length, or in other words, that of the tangent to the path. The first fundamental principle of mechanics is the principle of inertia, originally propounded by Descartes (1644) and later formulated by Newton (1687) as the First Law of Motion -1 According to this law, every body maintains its 1 The principle of inertia is really due to Galileo. He originated the concep-tion of ideal motion free from all obstacles, and created for this ideal motion a supreme axiom in the principle of the complete reversibility of the ideal mechanical process.
eBook - PDFMechanics
Lectures on Theoretical Physics
- Arnold Sommerfeld(Author)
- 2013(Publication Date)
- Academic Press(Publisher)
CHAPTER I MECHANICS OF A PARTICLE § 1. Newton's Axioms The laws of motion will be introduced in axiomatic form; they summarize in precise form the whole body of experience. First law : Every material body remains in its state of rest or of uniform rectilinear motion unless compelled by forces acting on it to change its state. 1 We shall at first withhold explanation of the concept of force introduced in this law. We notice that the states of rest and of uniform (rectilinear) motion are treated on equal footing and are regarded as natural states of the body. The law postulates a tendency of the body to remain in such a natural state; this tendency is called the inertia of the body. One often speaks of Galileo's law of inertia instead of Newton's first law in referring to the above axiom. We must say in this connection that while it is perfectly true that Galileo arrived at this law long before Newton (as a limiting result of his experiments with sliding bodies on planes of vanishing inclination), we find it characteristic of Newton that the law holds top position in his system. Newton's word body will, for the time being, be replaced by the words particle or mass point. To formulate the first law mathematically we shall make use of definitions 1 and 2 preceding it in the Principia. Definition 2 : The quantity of motion is the measure of the same, arising from the velocity and the quantity of matter conjunctly. 2 The quantity of motion is hence the product of two factors, the velocity, whose meaning is geometrically evident, 8 and the quantity of 1 We mention here, and in connection with what is to follow, the book Die Mechanik in ihrer Entwickelung (8th ed., F. A. Brockhaus, Leipzig, 1923; translated into English under the title The Science of Mechanics, Open Court Publishing Co., LaSalle, 111., 1942) by Ernst Mach.
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