Physics
Translation Vector
In physics, a translation vector represents the displacement of an object from one position to another in space. It is a mathematical quantity that describes the magnitude and direction of the movement. Translation vectors are used to analyze the motion of particles and objects, and they play a crucial role in understanding the dynamics of physical systems.
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9 Key excerpts on "Translation Vector"
eBook - PDF- Richard L. Myers(Author)
- 2005(Publication Date)
- Greenwood(Publisher)
D Translational Motion Introduction Chapter 2 presented the historical development of ideas about motion. These ideas were syn- thesized by Newton at the end of the sev- enteenth century in his Principia and have shaped our thinking of motion ever since. Concepts, once questioned and debated, are now commonly accepted, and each of us has a common understanding of the basic prin- ciples of motion. Terms describing motion, such as velocity and acceleration, are used colloquially to describe motion, often with little thought of their precise physical mean- ing. In this chapter the principles of mechan- ics introduced in the previous chapter will be examined in greater detail. The focus in this chapter will be translational, or linear, motion. Translational motion is characterized by movement along a linear path between two points. Scalars and Vectors Before translational motion can be exam- ined, it is important to distinguish between scalar and vector quantities. A scalar quan- tity, or just scalar, is defined by a magnitude and appropriate units. Ten dollars, 3 meters, and 32 °F are examples of scalar quantities. A quantity that is defined by a magnitude and direction with appropriate units is a vector quantity, or vector. Wind velocity is a good example of a vector. When wind velocity is reported, both the magnitude and direction are given, for example, 10 miles per hour out of the north. Vectors require a specified or assumed direction. A vector quantity is incomplete without a direction. The importance of includ- ing the direction for a vector can be illustrated by considering a person standing at the end of a narrow dock. If the person were told to move three steps forward versus three steps directly backward, it would probably make the differ- ence between falling in the water and staying dry. Three steps is a scalar, whereas three steps backward is a vector. Throughout this book, physical quantities will be defined as scalars or vectors.
eBook - PDF- Daniel A. Fleisch(Author)
- 2011(Publication Date)
- Cambridge University Press(Publisher)
1 Vectors 1.1 Definitions (basic) There are many ways to define a vector. For starters, here’s the most basic: A vector is the mathematical representation of a physical entity that may be characterized by size (or “magnitude”) and direction. In keeping with this definition, speed (how fast an object is going) is not rep-resented by a vector, but velocity (how fast and in which direction an object is going) does qualify as a vector quantity. Another example of a vector quantity is force, which describes how strongly and in what direction something is being pushed or pulled. But temperature, which has magnitude but no direction, is not a vector quantity. The word “vector” comes from the Latin vehere meaning “to carry;” it was first used by eighteenth-century astronomers investigating the mechanism by which a planet is “carried” around the Sun. 1 In text, the vector nature of an object is often indicated by placing a small arrow over the variable representing the object (such as F ), or by using a bold font (such as F ), or by underlining (such as F or F ∼ ). When you begin hand-writing equations involving vectors, it’s very important that you get into the habit of denoting vectors using one of these techniques (or another one of your choosing). The important thing is not how you denote vectors, it’s that you don’t simply write them the same way you write non-vector quantities. A vector is most commonly depicted graphically as a directed line seg-ment or an arrow, as shown in Figure 1.1 (a). And as you’ll see later in this section, a vector may also be represented by an ordered set of N numbers, 1 The Oxford English Dictionary . 2nd ed. 1989. 1 2 Vectors (b) (a) Figure 1.1 Graphical depiction of a vector (a) and a vector field (b). where N is the number of dimensions in the space in which the vector resides. Of course, the true value of a vector comes from knowing what it represents.
eBook - PDFMathematical Methods for Physicists
A Concise Introduction
- Tai L. Chow(Author)
- 2000(Publication Date)
- Cambridge University Press(Publisher)
1 Vector and tensor analysis Vectors and scalars Vector methods have become standard tools for the physicists. In this chapter we discuss the properties of the vectors and vector fields that occur in classical physics. We will do so in a way, and in a notation, that leads to the formation of abstract linear vector spaces in Chapter 5. A physical quantity that is completely specified, in appropriate units, by a single number (called its magnitude) such as volume, mass, and temperature is called a scalar. Scalar quantities are treated as ordinary real numbers. They obey all the regular rules of algebraic addition, subtraction, multiplication, division, and so on. There are also physical quantities which require a magnitude and a direction for their complete specification. These are called vectors if their combination with each other is commutative (that is the order of addition may be changed without aecting the result). Thus not all quantities possessing magnitude and direction are vectors. Angular displacement, for example, may be characterised by magni-tude and direction but is not a vector, for the addition of two or more angular displacements is not, in general, commutative (Fig. 1.1). In print, we shall denote vectors by boldface letters (such as A ) and use ordin-ary italic letters (such as A ) for their magnitudes; in writing, vectors are usually represented by a letter with an arrow above it such as ~ A . A given vector A (or ~ A ) can be written as A A ^ A ; 1 : 1 where A is the magnitude of vector A and so it has unit and dimension, and ^ A is a dimensionless unit vector with a unity magnitude having the direction of A . Thus ^ A A = A . 1 A vector quantity may be represented graphically by an arrow-tipped line seg-ment. The length of the arrow represents the magnitude of the vector, and the direction of the arrow is that of the vector, as shown in Fig.
eBook - PDF- Norman Gray(Author)
- 2022(Publication Date)
- Cambridge University Press(Publisher)
6 Vectors and Kinematics In Chapter 5, we used the axioms of Chapter 2 to obtain the Lorentz trans- formation. That allowed us to describe events in two different frames in relative motion. That part was rather mathematical in style. Now we are going to return to the physics, and describe motion: velocity, acceleration, momentum, energy and mass. Aims: you should: 6.1. understand the concept of a 4-vector as a geometrical object, and the distinction between a vector and its components. Chapter 5 was concerned with static events as observed from moving frames. In this part, we are concerned with particle motion. Before we can explain motion, we must first be able to describe it. This is the subject of kinematics. We will first have to define the vectors of four- dimensional Minkowski space, and specifically the velocity and acceleration vectors. 6.1 Three-Vectors You are familiar with 3-vectors – the vectors of ordinary three-dimensional euclidean space. To an extent, 3-vectors are merely an ordered triple of numbers, but they are interesting to us as physicists because they represent a more fundamental geometrical object: the three numbers are not just picked at random, but are the vector’s components – the projections of the vector onto three orthogonal axes (that the axes are orthogonal is not essential to the definition of a vector, but it is almost always simpler than the alternative). That is, the components of a vector are functions of both the vector and our 103 104 6 Vectors and Kinematics A θ A x A y A x A y x x y y Figure 6.1 A displacement vector in 3-d euclidean space. choice of axes, and if we change the axes, then the components will change in a systematic way. For example, consider a prototype displacement vector (Δ, Δ, Δ). These are the components of a vector with respect to the usual axes , and .
eBook - PDF- Bernard Schutz(Author)
- 2022(Publication Date)
- Cambridge University Press(Publisher)
2 Vector analysis in special relativity 2.1 Definition of a vector For the moment we will use the notion of a vector that carries over from Euclidean geometry: that a vector is something whose components transform like coordinates do under a coordinate transformation. Later on we shall define vectors in a more satisfactory manner. A typical vector is the displacement vector, which points from one event to another and has components equal to the coordinate differences: Δ x → O (Δt , Δx , Δy, Δz ) . (2.1) Here we have introduced some new notation: an arrow over a symbol denotes a vector (so that x is a vector having nothing in particular to do with the coordinate x ); the arrow after Δ x means ‘has components’ and the O underneath it means ‘in the frame O’; the components will always be in the order t , x , y, z (equivalently, indices in the order 0, 1, 2, 3). The notation → O is used in order to emphasize the distinction between the vector and its components. The vector Δ x can be regarded as an arrow between two events, while the collection of components is a set of four coordinate-dependent numbers. We shall always emphasize the notion of a vector (and, later, any tensor) as a geometrical object: something which can be defined and (sometimes) visualized without referring to a specific coordinate system. Another important notation is Δ x → O {Δx α }, (2.2) where by {Δx α } we mean all of Δx 0 , Δx 1 , Δx 2 , Δx 3 . If we ask for this vector’s components in another coordinate system, say the frame ¯ O, we write Δ x → ¯ O {Δx ¯ α }. That is, we put a bar over the index to denote the new coordinates. The vector Δ x is the same, and no new notation is needed for it when the frame is changed. Only its 33 34 2 Vector analysis in special relativity components change. 1 What are the new components Δx ¯ α ? We get them from the Lorentz transformation: Δx ¯ 0 = Δx 0 √ (1 − v 2 ) − vΔx 1 √ (1 − v 2 ) , etc.
eBook - PDF- Michael Tammaro(Author)
- 2019(Publication Date)
- Wiley(Publisher)
Examples of vectors: position (e.g., 1.5 km north or 3.5 m in the positive x direction), velocity (2.0 m/s in the negative y direction), and force (5.5 N in the negative z direction). Going Deeper Scalar Quantities and Vector Quantities I N T E R A C T I V E F E A T U R E Position and Displacement: Vectors in One Dimension | 45 Displacement is the difference between two vectors, so it is also a vector. In mathematics, the symbol ∆ usually means change in or difference in and is equal to the final value minus the initial value of the changing quantity. Figure 2.1.2 When the initial position is +9.00 m and the final position is +2.00 m, the displace- ment is −7.00 m. +x 0 x = +2.00 m x 0 = +9.00 m Definition of Displacement When an object moves from place to place, the magnitude of its displacement is equal to the straight-line distance between the initial and final positions. The direction of the displacement is measured from the initital position toward the final position. The displacement depends only on the initial and final positions, and not on the details of the path taken. In Figure 2.1.2, the bear is initially facing right, suggesting that the bear first rode some unspecified distance in the positive direction before turning around and riding back to the final position. We distinguish between the distance (the total length of travel) and the magnitude of the displacement. Unless the path from the initial to the final position was a straight line with no backtracking, the distance traveled will be greater than the magnitude of the displacement. Adding and Subtracting Vectors in One Dimension Sometimes it is necessary or useful to depict vectors graphically, in which case they are represented as arrows. The direction of the arrow should be consistent with the coordinate system in the diagram, and the length of the arrow should be proportional to the vector’s magnitude.
eBook - PDF- Daniel Kleppner, Robert Kolenkow(Authors)
- 2013(Publication Date)
- Cambridge University Press(Publisher)
Rather than interrupt the flow of discussion later, we are taking time now to ensure they are on hand when required. 1.2 Vectors The topic of vectors provides a natural introduction to the role of math-ematics in physics. By using vector notation, physical laws can often be written in compact and simple form. Modern vector notation was invented by a physicist, Willard Gibbs of Yale University, primarily to simplify the appearance of equations. For example, here is how New-ton’s second law appears in nineteenth century notation: F x = ma x F y = ma y F z = ma z . In vector notation, one simply writes F = m a , where the bold face symbols F and a stand for vectors. Our principal motivation for introducing vectors is to simplify the form of equations. However, as we shall see in Chapter 14 , vectors have a much deeper significance. Vectors are closely related to the fundamen-tal ideas of symmetry and their use can lead to valuable insights into the possible forms of unknown laws. 1.2.1 Definition of a Vector Mathematicians think of a vector as a set of numbers accompanied by rules for how they change when the coordinate system is changed. For our purposes, a down to earth geometric definition will do: we can think of a vector as a directed line segment . We can represent a vector graphi-cally by an arrow, showing both its scale length and its direction. Vectors are sometimes labeled by letters capped by an arrow, for instance A , but we shall use the convention that a bold face letter, such as A , stands for a vector. To describe a vector we must specify both its length and its direction. Unless indicated otherwise, we shall assume that parallel translation does not change a vector. Thus the arrows in the sketch all represent the same vector. 1.3 THE ALGEBRA OF VECTORS 3 If two vectors have the same length and the same direction they are equal.
eBook - PDF- William Moebs, Samuel J. Ling, Jeff Sanny(Authors)
- 2016(Publication Date)
- Openstax(Publisher)
We can apply the same basic equations for displacement, velocity, and acceleration we derived in Motion Along a Straight Line to describe the motion of the jets in two and three dimensions, but with some modifications—in particular, the inclusion of vectors. In this chapter we also explore two special types of motion in two dimensions: projectile motion and circular motion. Last, we conclude with a discussion of relative motion. In the chapter-opening picture, each jet has a relative motion with respect Chapter 4 | Motion in Two and Three Dimensions 157 to any other jet in the group or to the people observing the air show on the ground. 4.1 | Displacement and Velocity Vectors Learning Objectives By the end of this section, you will be able to: • Calculate position vectors in a multidimensional displacement problem. • Solve for the displacement in two or three dimensions. • Calculate the velocity vector given the position vector as a function of time. • Calculate the average velocity in multiple dimensions. Displacement and velocity in two or three dimensions are straightforward extensions of the one-dimensional definitions. However, now they are vector quantities, so calculations with them have to follow the rules of vector algebra, not scalar algebra. Displacement Vector To describe motion in two and three dimensions, we must first establish a coordinate system and a convention for the axes. We generally use the coordinates x, y, and z to locate a particle at point P(x, y, z) in three dimensions. If the particle is moving, the variables x, y, and z are functions of time (t): (4.1) x = x(t) y = y(t) z = z(t). The position vector from the origin of the coordinate system to point P is r → (t). In unit vector notation, introduced in Coordinate Systems and Components of a Vector, r → (t) is (4.2) r → (t) = x(t) i ^ + y(t) j ^ + z(t) k ^ . Figure 4.2 shows the coordinate system and the vector to point P, where a particle could be located at a particular time t.
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
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