Physics
Linear Motion
Linear motion refers to the movement of an object in a straight line, with constant velocity or acceleration. It is characterized by the absence of rotation or angular movement. In linear motion, the object's position, velocity, and acceleration can be described using equations that involve time and initial conditions.
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11 Key excerpts on "Linear Motion"
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 - PDFEngineering Mechanics
Problems and Solutions
- Arshad Noor Siddiquee, Zahid A. Khan, Pankul Goel(Authors)
- 2018(Publication Date)
- Cambridge University Press(Publisher)
1 2 3 4 B A Fig. 10.1 500 Engineering Mechanics The average acceleration is defined as the ratio of average velocity to the time internal Δ t and expressed as a v t avg avg = ∆ The instantaneous acceleration is defined as the ratio of instantaneous velocity to the time interval Δ t and expressed as lim . t inst velocity v t dv dt → 0 ∆ or 10.3 RectiLinear Motion During motion, a particle can have three types of motion i.e., translational, rotational or general plane motion. In this chapter, we will study translational motion. Rotational motion and general plane motion are discussed in chapter 15. In translational motion, the particle travels along a straight line path without rotation; it is also known as rectiLinear Motion. In this type of motion, all particles of a body travel along a straight line only i.e., bears only one dimensional motion. The rectiLinear Motion may lie along X-axis or Y-axis. The examples of rectiLinear Motion along X-axis are motion of vehicles on a straight track (road without speed breaker), motion of piston in a horizontal cylinder. However, the motion of lift in a building and falling of a stone vertically are examples of rectiLinear Motion along Y-axis. 10.4 RectiLinear Motion in Horizontal Direction (X-axis) A particle moving in a straight line can have either uniform or variable acceleration. Generally the particles consist of variable acceleration in actual practice. The motion of a body under variable acceleration can be analyzed by using differential and integral equations of motion. The motion under uniform acceleration is analyzed by using equations of motion as discussed in next section (10.4.2). 10.4.1 Motion with variable acceleration Differential equations of motion: These equations are used to analyze the motion of a body having variable acceleration. The equations for four parameters of motion i.e., displacement, time, velocity and acceleration are given by v dx dt a dv dt d x dt v dv dx = = or or 2 2
eBook - PDF- Richard C. Hill, Kirstie Plantenberg(Authors)
- 2013(Publication Date)
- SDC Publications(Publisher)
14 2.2.5) Constant acceleration equations ....................................................................... 15 2.2.6) General notes .................................................................................................... 16 2.3) ERRATIC RECTILinear Motion......................................................................... 25 2.4) SOLVING RECTILINEAR PROBLEMS GRAPHICALLY ....................................... 32 CHAPTER 2 REVIEW PROBLEMS ............................................................................... 37 CHAPTER 2 PROBLEMS .............................................................................................. 41 CHAPTER 2 COMPUTER PROBLEMS ......................................................................... 48 CHAPTER 2 DESIGN PROBLEMS ................................................................................ 49 CHAPTER 2 ACTIVITIES ............................................................................................... 49 Conceptual Dynamics Kinematics: Chapter 2 – Kinematics of Particles - RectiLinear Motion 2 - 2 CHAPTER SUMMARY In this chapter, we will study kinematics of particles. Kinematics involves the study of a body's motion without regard to the forces that generate that motion. In particular, kinematics involves studying the relationship between displacement, velocity and acceleration. This chapter will focus on analyzing simple one-dimensional motion. The next chapter will move on to more complex two-dimensional motion. The treatment of particles precedes rigid bodies because they are simpler to analyze. A particle may be treated as a point; it has mass but no size. Therefore, we will only need to consider translational motion and not worry about rotation. Conceptual Dynamics Kinematics: Chapter 2 – Kinematics of Particles - RectiLinear Motion 2 - 3 2.1) RECTILinear Motion 2.1.1) RECTILinear Motion The motion of a real object with size and mass is very complex.
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 - PDFFrom Atoms to Galaxies
A Conceptual Physics Approach to Scientific Awareness
- Sadri Hassani(Author)
- 2010(Publication Date)
- CRC Press(Publisher)
Chapter 6 Kinematics: Describing Motion Chapter 4 taught us some basic knowledge of motion along a straight line. This chapter generalizes the concept of motion and introduces some fundamental quantities necessary for its description. 6.1 Position, Displacement, and Distance What is motion? Vaguely speaking, it is the change in the state of an object. More precisely, it is the change in the position of an object with time. Still more precisely, we must speak of the motion of an object relative to an observer , although the “object” could be a person, and the “observer” a thing. To analyze the motion, draw an arrow from the “observer” O to the “object” A , and call the arrow the position vector . 1 The very definition of the position vector assumes Position vector. that both O and A are points. The position vector, denoted commonly by r , determines the instantaneous position of the object A [Figure 6.1(a)] relative to an observer O . The word “instantaneous” is important because the position of the point object A is, in general, constantly changing. If we were to take snapshots of A at various times, t 1 , t 2 , t 3 , etc., and label the corresponding points at which A is located by A 1 , A 2 , A 3 , etc, we would have a situation depicted in Figure 6.1(b), with position vectors r 1 , r 2 , r 3 , etc. In this figure, only three out of an infinitude of possible snapshots are shown. Bear in mind that every directed line segment from O to a point on the curve in Figure 6.1 is a possible position vector. What do you know? 6.1. Can you say that the observer—as defined in the descrip-tion of motion—does not move? If the point A does not change, i.e., if the position vector r does not vary with time, we say that the object is stationary relative to O . You have to make a clear distinction between the distance between O and A , which is the length of r , and the position vector, which is the directed line segment r .
eBook - PDF- W. C. Bolton(Author)
- 2013(Publication Date)
- Wiley-Blackwell(Publisher)
Chapter 10 Linear and angular motion 10.1 Linear Motion Consider a particle moving along a straight line so that at some time t its displacement along the line from some reference point is s . If in a time interval δ t the displacement increases by δ s then the average velocity of the particle during that time interval is defined as being δ s / δ t . As the time interval considered is reduced, so the average velocity approaches the velocity at an instant. Thus, in the limit, the instantaneous velocity v is v = d s d t [1] If the velocity of the particle changes by δ v in a time interval δ t then the average acceleration is defined as being δ v / δ t . As the time interval considered is reduced, so the average acceleration approaches the acceleration at an instant. Thus, in the limit, the instantaneous acceleration a is a = d v d t [2] It is sometimes useful to express acceleration as a = d v d t = d v d s × d s d t = d v d s × v Example An element in a machine accelerates from rest in a straight line such that its velocity v in m/s changes with time t in s according to v = 0 . 8 t 2 + 0 . 2 t . What is the acceleration and distance covered after 2 s? Using equation [2], a = d v d t = d d t ( 0 . 8 t 2 + 0 . 2 t) = 1 . 6 t + 0 . 2 Thus after 2 s the acceleration is 3.4 m/s 2 . The distance covered can be obtained by using equation [1]. v = d s d t = 0 . 8 t 2 + 0 . 2 t 254 Linear and angular motion 255 s 0 d s = t 0 ( 0 . 8 t 2 + 0 . 2 t) d t s = 0 . 8 3 t 3 + 0 . 2 2 t 2 Thus when t = 2 s then s = 2 .
eBook - PDFEngineering Mechanics
Dynamics
- L. G. Kraige, J. N. Bolton(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
The direction of n lies in the local plane of the curve. † These last two mea- surements are called path variables. The motion of particles (or rigid bodies) can be described by using coordinates measured from fixed reference axes (absolute-motion analysis) or by using coordinates measured from moving reference axes (relative-motion analysis). Both descriptions will be developed and applied in the articles which follow. With this conceptual picture of the description of particle motion in mind, we restrict our attention in the first part of this chapter to the case of plane motion where all movement occurs in or can be represented as occurring in a single plane. A large proportion of the motions of machines and structures in engineering can be represented as plane motion. Later, in Chapter 7, an introduction to three- dimensional motion is presented. We begin our discussion of plane motion with rectiLinear Motion, which is motion along a straight line, and follow it with a de- scription of motion along a plane curve. 2/2 RectiLinear Motion Consider a particle P moving along a straight line, Fig. 2∕ 2. The position of P at any instant of time t can be specified by its distance s measured from some convenient reference point O fixed on the line. At time t + Δ t the particle has moved to P′ and its coordi- nate becomes s + Δs. The change in the position coordinate during the interval Δ t is called the displacement Δs of the particle. The displacement would be negative if the particle moved in the nega- tive s-direction. B P R Path x y r y x z n t z A FIGURE 2/1 O s − s + s P s Pʹ Δ FIGURE 2/2 *Often called Cartesian coordinates, named after René Descartes (1596–1650), a French mathematician who was one of the inventors of analytic geometry.
- Paul Anthony Russell(Author)
- 2021(Publication Date)
- Reeds(Publisher)
Kinematics is about the study of motion in the real world without considering the size or shape of machines or objects or the forces that cause the motion. Linear Motion SPEED is the rate at which a body moves through space; however, speed does not define the direction of a moving body and is, therefore, expressed as the distance travelled in a given time. The units of speed are metres per second (ms −1 ), kilometres per hour (km/h), etc. Speed may also change during a journey; for example, if a car covers180 km in 3 h, it is very improbable that it has been moving at a constant speed of 60 km/h, but the average is 60 km/h. Therefore, speed is described as a scalar quantity, as only one fact about the body is defined. VELOCITY, however, indicates speed in a specified direction. Velocity represents two facts about a moving body – its speed and also its direction; consequently, velocity is a vector quantity and hence can be illustrated by drawing an arrow (vextor) to scale, the length of which represents the speed of the body and the arrowhead on it represents its direction. See Figure 2.1. RESULTANT VELOCITY is obtained from vector diagrams of velocities in the same manner as with vector diagrams of forces. Resultant velocity is an outcome of vector addition. KINEMATICS 2 N S W E 2 m/s due east 18 km/h south-west ▲ Figure 2.1 Vectors showing velocity Example 2.1. A ship travelling due north at 16 knots runs into a 4-knot current moving south-east. Find the resultant speed and direction of the ship. ( ) ( ) ( ) cos cos ac ab bc ab bc b 2 2 2 2 2 2 16 4 2 16 4 45 256 16 = + - × × × = + - × × × ° = + - = = = ° = × = 90 51 181 49 13 47 4 13 47 45 4 0 7071 13 47 . . . sin . sin sin . . ac a a 0 2100 12 7 . ∴ = ° ′ a ∴ = = ° ′ resultant speed knots resultant direction east of 13 47 12 7 . north A CHANGE OF VELOCITY will take place if speed changes, or if direction changes, or if both speed and direction change.
eBook - PDF- R. Douglas Gregory(Author)
- 2006(Publication Date)
- Cambridge University Press(Publisher)
1 . 18 Find the unit tangent vector, the unit normal vector and the curvature of the parabola x = ap 2 , y = 2ap, z = 0 at the point with parameter p. Chapter Two Velocity, acceleration and scalar angular velocity KEY FEATURES The key concepts in this chapter are the velocity and acceleration of a particle and the angular velocity of a rigid body in planar motion. Kinematics is the study of the motion of material bodies without regard to the forces that cause their motion. The subject does not seek to answer the question of why bod- ies move as they do; that is the province of dynamics. It merely provides a geometrical description of the possible motions. The basic building block for bodies in mechanics is the particle, an idealised body that occupies only a single point of space. The impor- tant kinematical quantities in the motion of a particle are its velocity and acceleration. We begin with the simple case of straight line particle motion, where velocity and accel- eration are scalars, and then progress to three-dimensional motion, where velocity and acceleration are vectors. The other important idealisation that we consider is the rigid body, which we regard as a collection of particles linked by a light rigid framework. The important kinematical quantity in the motion of a rigid body is its angular velocity. In this chapter, we con- sider only those rigid body motions that are essentially two-dimensional, so that angular velocity is a scalar quantity. The general three-dimensional case is treated in Chapter 16. 2.1 STRAIGHT LINE MOTION OF A PARTICLE Consider a particle P moving along the x -axis so that its displacement x from the origin O is a known function of the time t . Then the mean velocity of P over the time O P x v FIGURE 2.1 The particle P moves in a straight line and has displacement x and velocity v at time t .
eBook - PDFMechanics
Lectures on Theoretical Physics
- Arnold Sommerfeld(Author)
- 2013(Publication Date)
- Academic Press(Publisher)
Like the latter it is a directed magnitude, a vector. We write 4 (1) p=rav and formulate the first law of motion in its final form : (2) p=constant in the absence of forces. We shall put the law of inertia thus formulated at the head of our mechanics. It is the result of an evolution extending over many centuries, and is by no means as self-evident as it appears to us today. The philosopher Kant, for instance, says in his paper, Thoughts on the True Estimation of Living Forces/' written in 1747, long after Newton: There exist two kinds of motions; those which have ceased after a certain time, and those which persist. The motions that in Kant's opinion cease by themselves are, according to modern ideas — and those of Newton — motions which are attenuated by frictional forces and finally destroyed. The expression quantity of motion is unfortunately chosen in that it does not take into account the vector character of my. Thus a better term would be the word impulse, which conveys the idea of a push of a certain magnitude in a definite direction that the given mv is able to impart, by collision, to some body initially at rest. Since the term impulse is, however, used in a somewhat different sense in mechanics, we shall have to retain the name quantity of motion, or, in modern language, momentum for the vector p. Instead of the law of inertia and Newton's first law of motion we can then speak of the law of conservation of momentum. We shall now discuss Newton's second law, the real law of motion: The change in motion is proportional to the force acting and takes place in the direction of the straight line along which the force acts. 4 We assume that the reader is familiar with the elements of vector algebra. Since, however, vector operations originated in close association with mechanics (includ-ing the mechanics of fluids), we shall often have occasion to explain vector concepts simultaneously with mechanical concepts.
eBook - PDF- Robert Resnick, David Halliday, Kenneth S. Krane(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
65 MOTION IN TWO AND THREE DIMENSIONS I n this chapter we consider an extension of the con- cepts presented in Chapters 2 and 3. In those chapters we introduced kinematics and dynamics in terms of vectors, but we considered only applications in one dimension. In this chapter we broaden the discussion to include two- and three-dimensional applications. Keeping track of the separate x, y, and z components of the motion is greatly simplified if we rely on vectors to describe the particle’s position, velocity, and accel- eration, as well as the forces that may act on the particle. To illustrate the vector techniques, we discuss two examples: a projectile launched with both horizontal and vertical velocity components in the Earth’s grav- ity, and an object moving in a circular path. 4-1 MOTION IN THREE DIMENSIONS WITH CONSTANT ACCELERATION In Section 2-5 we developed a procedure for analyzing the position, velocity, and acceleration of a particle that moves in one dimension with constant acceleration. Knowing the acceleration, we can find the velocity at all times according to Eq. 2-26 and the position at all times from Eq. 2-28 Now we consider the possibility that the particle moves in three dimensions with constant acceleration. That is, as the particle moves, the acceleration does not vary in either magnitude or direction. Equivalently, we can represent the acceleration as a vector with three components each of which is constant. In general the parti- cle moves in a curved path. As is the case in one-dimen- sional motion, we would like to know the particle’s velocity (a vector with components and its position (a vector with components x, y, z ) at all times. We can obtain the general equations for motion with constant by setting and a z constant. a y constant, a x constant, a B r B v x , v y , v z ) v B (a x , a y , a z ), a B (x x 0 v 0x t 1 2 a x t 2 ).
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