Angular Kinematic Equations
Angular kinematic equations are used to describe the rotational motion of an object. They relate angular displacement, angular velocity, and angular acceleration. The three main equations are: ωf = ωi + αt, θ = ωit + 0.5αt^2, and ωf^2 = ωi^2 + 2αθ, where ω represents angular velocity, θ is angular displacement, α is angular acceleration, t is time, and the subscripts i and f denote initial and final values.
7 Key excerpts on "Angular Kinematic Equations"
eBook - ePubInstant Notes in Sport and Exercise Biomechanics
Second Edition
- Paul Grimshaw, Michael Cole, Adrian Burden, Neil Fowler(Authors)
- 2019(Publication Date)
- Garland Science(Publisher)
...SECTION C Kinematics and kinetics of angular motion C1 ANGULAR MOTION Paul Grimshaw Angular motion is rotatory movement about an imaginary or real axis of rotation and where all parts on a body (note that the term body need not necessarily be a human body) or segment move through the same angle. Angular kinematics describes quantities of angular motion using such terms as angular displacement, angular velocity and angular acceleration. Figure C1.1 identifies two examples of angular motion in more detail. Angular distance or displacement (scalar or vector quantity) is usually expressed in the units of degrees (where a complete circle is 360 degrees). Similarly, angular velocity (depicted by the Greek letter omega (ω)) and angular acceleration (depicted by the Greek letter alpha (α)) are often expressed as degrees per second (°/s) and degrees per second squared (°/s 2 or degrees/second 2) respectively. However, it is more convenient within human motion to use the unit radian due to the large amount of angular displacement involved. The value for 1 radian represents an angle of approximately 57.3° and there are 2π (where pi is approximately 3.142) radians in 360 degrees (one circle). As with the terms used to describe linear motion, within angular motion there exists both scalar and vector quantities. However, it is often possible and more easily understandable to describe angular movement using such definitions as clockwise or anti-clockwise rotation. Again, positive and negative signs can be used to denote the different directions (e.g. clockwise rotation may be assigned a negative sign and anti-clockwise rotation a positive sign which is the common convention used within biomechanics). Figure C1.1 Angular motion in sport Figure C1.2 Angular motion during kicking a ball Considering Figure C1.2 it is possible to see the actions employed by the leg when kicking a soccer ball...
eBook - ePub- W. Bolton(Author)
- 2015(Publication Date)
- Routledge(Publisher)
...Chapter 5 Angular motion 5.1 Introduction This chapter is concerned with describing angular motion, deriving and using the equations for such motion and relating linear motion of points on the circumference of rotating objects with their angular motion. The term torque is introduced. 5.1.1 Basic terms The following are basic terms used to describe angular motion. Angular displacement The angular displacement is the angle swept out by the rotation and is measured in radians. Thus, in Figure 5.1, the radial line rotates through an angular displacement of θ in moving from OA to OB. One complete rotation through 360° is an angular displacement of 2 π rad; one quarter of a revolution is 90° or π /2 rad. As 2 π rad 5 360°, then 1 rad 5 360°/2 π or about 57°. Figure 5.1 Angular motion 2 Angular velocity Angular velocity ω is the rate at which angular displacement occurs, the unit being rad/s. 3 Average angular velocity The average angular velocity over some time interval is the change in angular displacement during that time divided by the time...
eBook - ePubBiomechanics of Human Motion
Applications in the Martial Arts, Second Edition
- Emeric Arus, Ph.D.(Authors)
- 2017(Publication Date)
- CRC Press(Publisher)
...CHAPTER 7 Kinematics in Angular Motion In this chapter, the author describes almost the same physical properties, characteristics regarding to rotational (angular) motion. Besides distance, displacement, velocity, and acceleration, which have been described in Chapter 6, other physical properties such as inertia, and different variables including tangential, centrifugal, and centripetal accelerations related to angular motion will be described in this chapter. The reader should know the following (information) facts related to angular motion. Any object/mass that is going through a uniform circular motion with a constant speed will be accelerated by the force driving it because a rotary motion is related to acceleration, velocity, and direction. How should be this understood? The object/mass basically has nothing to do with speed, but has to do with velocity. Since the velocity is a vector that has both magnitude and direction, a change in either the magnitude or the direction constitutes a change in the velocity. So, the conclusion of the above statement is that an object moving in a circle at constant speed is indeed accelerating. It is accelerating because the direction of the velocity vector is changing. If the magnitude of the velocity is changing, then acceleration occurs too. Putting this in simple words, the rotary motion constitutes a change in direction even if that direction is a perfect circle, and if the rotation is going on and on, it still constitutes a change of direction. 7.1 DISTANCE AND DISPLACEMENT When a body rotates in a 2-D plane, the rotation is characterized by the fact that the rotating body (its external point) moves around the perimeter of a circle or a cylinder at the same distance from the center of rotation...
eBook - ePub- John Bird(Author)
- 2019(Publication Date)
- Routledge(Publisher)
...Chapter 23 Linear and angular motion Why it is important to understand: Linear and angular motion This chapter commences by defining linear and angular velocity and also linear and angular acceleration. It then derives the well-known relationships, under uniform acceleration, for displacement, velocity and acceleration, in terms of time and other parameters. The chapter then uses elementary vector analysis, similar to that used for forces in chapter 20, to determine relative velocities. This chapter deals with the basics of kinematics. A study of linear and angular motion is important for the design of moving vehicles. At the end of this chapter, you should be able to: appreciate that 2π radians corresponds to 360° define linear and angular velocity perform calculations on linear and angular velocity using v = ωτ and ω = 2πn define linear and angular acceleration perform calculations on linear and angular acceleration using v 2 = v 1 + at, ω 2 = ω 1 + at and a = τα select appropriate equations of motion when performing simple calculations appreciate the difference between scalar and vector quantities use vectors to determine relative velocities, by drawing and by calculation 23.1 Introduction This chapter commences by defining linear and angular velocity and also linear and angular acceleration. It then derives the well-known relationships, under uniform acceleration, for displacement, velocity and acceleration, in terms of time and other parameters. The chapter then uses elementary vector analysis, similar to that used for forces in chapter 20, to determine relative velocities. This chapter deals with the basics of kinematics. 23.2 The radian The unit of angular displacement is the radian, where one radian is the angle subtended at the centre of a circle by an arc equal in length to the radius, as shown in Figure 23.1. The relationship between angle in radians θ, arc length s and radius of a circle τ is: s = r θ (1) Science and Mathematics for Engineering...
eBook - ePub- Ronald J. Anderson(Author)
- 2020(Publication Date)
- Wiley(Publisher)
...This results in an angular velocity vector, (1.4) where the right handed set of unit vectors,, is defined in Figure 1.2. Note that it is essential that right handed coordinate systems be used for dynamic analysis because of the extensive use of the cross product and the directions of vectors arising from it. If there is a right handed coordinate system, with respective unit vectors, then the cross products are such that, Figure 1.2 Even 2D problems are 3D. Using this definition of the angular velocity, the motion of the tip of vector, resulting from the angular change in time, can be determined from the cross product which, by the rules of the vector cross product, has magnitude, and a direction that, according to the right hand rule 2 used for cross products, is perpendicular to both and and, in fact, lies in the direction of. Combining these two terms to get and substituting into Equation 1.3 results in, (1.5) The time derivative of any vector,, can therefore be written as, (1.6) It is important to understand that the angular velocity vector,, is the angular velocity of the coordinate system in which the vector,, is expressed. There is a danger that the rate of change of direction terms will be included twice if the angular velocity of the vector with respect to the coordinate system in which it measured is used instead. The example presented in Section 1.3 shows a number of different ways to arrive at the derivative of a vector which rotates in a plane. 1.2 Performing Kinematic Analysis Before proceeding with examples of kinematic analyses we state here the steps that are necessary in achieving a successful result. This first step in any dynamic analysis is vitally important. The goal is to derive expressions for the absolute velocities and accelerations of the centers of mass of the bodies making up the system being analyzed. In addition, expressions for the absolute angular velocities and angular accelerations of the bodies will be required...
eBook - ePub- William Bolton(Author)
- 2012(Publication Date)
- Routledge(Publisher)
...4 Linear and angular motion 4.1 Introduction This chapter is concerned with the behaviour of dynamic mechanical systems when there is uniform acceleration. The terms and basic equations associated with linear motion with uniform acceleration and angular motion with uniform angular acceleration, Newton’s laws of motion, moment of inertia and the effects of friction are revised and applied to the solution of mechanical system problems. The terms scalar quantity and vector quantity are used in this chapter, so as a point of revision: Scalar quantities are those that only need to have their size to be given in order for their effects to be determined, e.g. mass. Vector quantities are those that need to have both their size and direction to be given in order for their effects to be determined, e.g. force where we need to know the direction as well as the size to determine its effect. 4.2 Linear motion The following are basic terms used in the description of linear motion, i.e. motion that occurs in a straight line path rather than rotation which we will consider later in this chapter: 1 Distance and displacement The term distance tends to be used for distances measured along the path of an object, whatever form the path takes; the term displacement, however, tends to be used for the distance travelled in a particular straight line direction (Figure 4.1). For example, if an object moves in a circular path the distance travelled is the circumference of the path whereas the displacement might be zero if it ends up at the same point it started from...
eBook - ePub- Alan Hendrickson(Author)
- 2012(Publication Date)
- Routledge(Publisher)
...The crossing point of these axes is called the origin. Any point on the plane defined by these axes can be specified by two numbers, x and y, each of which relates to the distance between the origin and the point in a direction parallel to the corresponding axis (graph on the left of Figure 7.1). The other coordinate system, polar coordinates, describes the position of a point by specifying the length of a line between an origin and the point, and the angle between this line and a given reference direction (shown on the right of Figure 7.1). Polar coordinates are more useful than Cartesian when purely circular rotation occurs because only the angular coordinate changes with time, the length of the radial line remains constant. Given this, polar coordinates will be assumed for all the rotational motion chapters. Figure 7.1 Cartesian and polar coordinates Angular Displacement Since the rotational quantities of angular displacement, angular speed, and angular acceleration are analogous to their linear counterparts, much of the information in this chapter will appear familiar, albeit with different variables. For instance, angular displacement, denoted with the Greek characters “delta theta,” Δθ, represents the change of angular position over the time interval between t 1 and t 2 : Δ θ = θ 2 − θ 1 Where Δθ = the change of angular position (radians, degrees) θ 1, θ 2 = angular positions at different times (radians, degrees) In everyday use angles are expressed in degrees, and everyone is familiar with the fact that one full revolution or turn amounts to a rotation of 360°. Angle measurement in radians however is what must be used in most of the rotational motion formulas presented here. An angular measurement in radians is, by definition, the result of a ratio between arc length and radial distance (left side of Figure 7.2)...
