Geometric Rotation

7 Key excerpts on "Geometric Rotation"

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  Essential Physics

  • John Matolyak, Ajawad Haija(Authors)
  • 2013(Publication Date)
  • CRC Press

    (Publisher)

  139 © 2010 Taylor & Francis Group, LLC Rotational Motion Translational motion and circular motion, uniform and nonuniform of an object, were discussed in Chapters 3 and 4. In this chapter, rotational motion of a point or extended object will be introduced. A force acting on an extended object creates a torque that rotates it about a fixed axis. A solid object of finite physical size is known as a rigid body. The axis of rotation could be about the center of mass of the rigid body or about other points where it is pivoted. In this chapter, Sections 8.2 through 8.7 discuss the kinematics of a rigid body, and Section 8.8 reviews the dynamics of the rigid body. In addition to drawing a parallel between Newton’s laws applied to point-like objects in linear motions and those applied to rotational motion of a rigid body, translational and rotational motions of a rigid body are discussed in detail. 8.1 ANGULAR KINEMATIC QUANTITIES Consider a point-like object moving in a circle of radius r (Figure 8.1). As the object moves from point P 1 at t 1 to point P 2 at t 2 , it sweeps through an arc Δ s that subtends an angle Δθ ( = θ 2 – θ 1 ) at the center, in a time interval Δ t = t 2 – t 1 . From geometry Δ s = r Δθ , (8.1) where Δθ is an angular displacement, measured in radians. To convert an angle expressed in degrees to radians, the following relation may be used: θ π θ rad deg deg 2 rad 360 =       , (8.2a) and from radians to degrees, the conversion is θ π θ de g r ad 360 2 . =       (8.2b) Dividing Equation 8.1 by Δ t gives the average linear velocity v ; that is, ∆ ∆ ∆ ∆ s t r t = θ or v r . = ω (8.3) For an object experiencing a constant linear acceleration, v v v 2 1 2 = + . 8 140 Essential Physics © 2010 Taylor & Francis Group, LLC Here, v 1 and v 2 are the instantaneous tangential, or linear, velocities of the object at points P 1 and P 2 , respectively.

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  Introductory Physics for the Life Sciences: Mechanics (Volume One)

  • David V. Guerra(Author)
  • 2023(Publication Date)
  • CRC Press

    (Publisher)

  13 Rotational Motion

  DOI: 10.1201/9781003308065-13

  13.1 Introduction

  Rotational motion is the motion of an object spinning around an axis that passes through the object itself. This is to be contrasted with translational motion, which is the motion of an object moving through space in a straight or curved path without rotation. As demonstrated in Figure 13.1 , a block sliding down an incline moves with only linear motion defined by a displacement, a velocity, and an acceleration. A disk rotating about a fixed axis moves with only rotational motion, but a ball rolling down an incline experiences both rotational and linear motion.

  FIGURE 13.1 Examples of different types of motion.

  As demonstrated in Chapter 12 , when an object is traveling in a circular path, the concepts of translation kinematics are commonly applied to the analysis, but sometimes the concepts of rotational motion, period, frequency, and angular frequency can be applied. So, circular motion provides a transition between the language of translational and rotational motion, which is formalized in this chapter. In addition, rotational dynamics will formalize the connection between the net rotational force, or net torque, on an object and the angular acceleration of the object.

  • Chapter question: There are bacteria that employ a rotating flagellum, tails that look a bit like a corkscrew, to propel themselves forward. In normal situations, these propulsion systems work well to move these bacteria forward through water. On the other hand, when a drop of water containing these bacteria is placed on a microscope slide, the bacteria begin to move in approximately circular paths at fairly constant speeds (Figure 13.2 ).
    FIGURE 13.2

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  Physics

  • John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  CHAPTER 8Rotational Kinematics

  The figure shows the front view of a turbine jet engine on a commercial aircraft. The rotating fan blades collect air into the engine before it is compressed, mixed with fuel, and ignited to produce thrust. The rotational motion of the blades can be described using the concepts of angular displacement, angular velocity, and angular acceleration within the framework of rotational kinematics.

  LEARNING OBJECTIVES

  After reading this module, you should be able to…

  • 8.1  Define angular displacement.
  • 8.2  Define angular velocity and angular acceleration.
  • 8.3  Solve rotational kinematics problems.
  • 8.4  Relate angular and tangential variables.
  • 8.5  Distinguish between centripetal and tangential accelerations.
  • 8.6  Analyze rolling motion.
  • 8.7  Use the right-hand rule to determine the direction of angular vectors.

  8.1 Rotational Motion and Angular Displacement

  In the simplest kind of rotation, points on a rigid object move on circular paths. In Figure 8.1 , for example, we see the circular paths for points A, B, and C on a spinning skater. The centers of all such circular paths define a line, called the axis of rotation .

  Figure 8.1

  When a rigid object rotates, points on the object, such as A, B, or C, move on circular paths. The centers of the circles form a line that is the axis of rotation.

  The angle through which a rigid object rotates about a fixed axis is called the angular displacement . Figure 8.2 shows how the angular displacement is measured for a rotating compact disc (CD). Here, the axis of rotation passes through the center of the disc and is perpendicular to its surface. On the surface of the CD we draw a radial line, which is a line that intersects the axis of rotation perpendicularly. As the CD turns, we observe the angle through which this line moves relative to a convenient reference line that does not rotate. The radial line moves from its initial orientation at angle θ0 to a final orientation at angle θ (Greek letter theta). In the process, the line sweeps out the angle θ − θ0 . As with other differences that we have encountered (Δx = x − x0 , Δv = v − v0 , Δt = t − t0 ), it is customary to denote the difference between the final and initial angles by the notation Δθ (read as “delta theta”): Δθ = θ − θ0 . The angle Δθ

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  Physics for Students of Science and Engineering

  • A. L. Stanford, J. M. Tanner(Authors)
  • 2014(Publication Date)
  • Academic Press

    (Publisher)

  rigid body . Rigid bodies are an idealization—they don’t exist in any state—but actual bodies can flex. We use the notion of a rigid body to facilitate the analysis of the motion of actual physical bodies. In the analysis of many physical bodies, the rigid-body assumption is an excellent approximation. In this chapter we will assume that all extended bodies are rigid bodies.

  Let us now consider the description of the purely rotational motion of a rigid body turning about an axis fixed in space.

  Rotational Kinematics

  Consider the circular disk of Figure 7.1(a) . It is a rigid body attached to a rod lying along an axis passing through the center of the disk normal to the face of the disk. An axis of rotation of a body is a line in space about which the particles within the body maintain a constant distance and, therefore, move in a circular path about the axis. Because the disk is a rigid body, the rotational motion of the disk may be described by the motion of an arbitrary particle within the disk. Look, then, at the particle P on the face of the disk shown in Figure 7.1(b) . We will now define the variables that specify the rotational motion of the disk.

  Figure 7.1 A rigid circular disk with a fixed axis of rotation passing through its center. The axis is perpendicular to the face of the disk.

  The angle θ, measured counterclockwise from a fixed reference line to a radial line through the point P , as shown in Figure 7.1 (b) , is the angular position of the particle at point P . The length s of the arc that lies at a distance r from the axis is related to θby

  s = r θ

  (7-1)

  (7-1)

  when θis measured in radians . The rate at which s changes with respect to time is obtained by differentiating Equation (7-1) with respect to time. Recognizing that r is a constant, we obtain

  d s

  d t

  = r

  d θ

  d t

  (7-2)

  (7-2)

  In Equation (7-2) , ds /dt is

  vt

  , the tangential component of the instantaneous velocity of the particle, or

  v t

  d s

  d t

  (7-3)

  (7-3)

  We define dθ/dt , the instantaneous time rate of change of angular position, to be the instantaneous angular velocity ω of P (and, therefore, of the disk), or

  ω =

  d θ

  d t

  (7-4)

  (7-4)

  The unit of angular velocity is radians per second (rad/s). Using the definitions of Equations (7-3) and (7-4) , we may write Equation (7-2)

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  3D Math Primer for Graphics and Game Development

  • Fletcher Dunn, Ian Parberry(Authors)
  • 2011(Publication Date)
  • A K Peters/CRC Press

    (Publisher)

  Section 2.4.1 discussed that it’s impossible to describe the position of an object in absolute terms—we must always do so within the context of a specific reference frame. When we investigated the relationship between “points” and “vectors,” we noticed that specifying a position is actually the same as specifying an amount of translation from some other given reference point (usually the origin of some coordinate system).

  In the same way, orientation cannot be described in absolute terms. Just as a position is given by a translation from some known point, an orientation is given by a rotation from some known reference orientation (often called the “identity” or “home” orientation). The amount of rotation is known as an angular displacement . In other words, describing an orientation is mathematically equivalent to describing an angular displacement.

  We say “mathematically equivalent” because in this book, we make a subtle distinction between “orientation” and terms such as “angular displacement” and “rotation.” It is helpful to think of an “angular displacement” as an operator that accepts an input and produces an output. A particular direction of transformation is implied; for example, the angular displacement from the old orientation to the new orientation, or from upright space to object space. An example of an angular displacement is, “Rotate 90º about the z -axis.” It’s an action that we can perform on a vector.

  However, we frequently encounter state variables and other situations in which this operator framework of input/output is not helpful and a parent/child relationship is more natural. We tend to use the word “orientation” in those situations. An example of an orientation is, “Standing upright and facing east.” It describes a state of affairs.

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  Analytical Mechanics

  • Nivaldo A. Lemos(Author)
  • 2018(Publication Date)
  • Cambridge University Press

    (Publisher)

  3 Kinematics of Rotational Motion But though the professed aim of all scientific work is to unravel the secrets of nature, it has another effect, not less valuable, on the mind of the worker. It leaves him in possession of methods which nothing but scientific work could have led him to invent. James Clerk Maxwell, The Theory of Molecules The dynamics of rigid bodies is a chapter of classical mechanics that deserves to be highlighted not only owing to its intrinsic physical interest but also because it involves important mathematical techniques. Before, however, embarking on the study of dynamics, it is necessary to formulate efficacious methods to describe the motion of rigid bodies. A considerable space will be dedicated to the study of rotational kinematics in the perspective that several of the mathematical tools to be developed are of great generality, finding wide application in other domains of theoretical physics. 3.1 Orthogonal Transformations A rigid body has, in general, six degrees of freedom. Obviously, three of them correspond to translations of the body as a whole, whereas the other three degrees of freedom describe the orientations of the body relative to a system of axes fixed in space. A simple way to specify the orientation of the rigid body consists in setting up a Cartesian system of axes fixed in the body, which move along with it, and consider the angles that these axes make with axes parallel to those that remain fixed in space, represented by dashed lines in Fig. 3.1. Direction Cosines Let  be a Cartesian coordinate system (x 1 , x 2 , x 3 ) with corresponding unit vectors ˆ e 1 , ˆ e 2 , ˆ e 3 representing axes fixed in space, and let   be a Cartesian coordinate system (x  1 , x  2 , x  3 ) with unit vectors ˆ e  1 , ˆ e  2 , ˆ e  3 whose axes remain attached to the rigid body, as in Fig.

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  Reading Physics

  A Guide to Understanding Basic Classical Mechanics without Mathematical Expressions

  • Jae Jun Kim, Ph.D.(Authors)
  • 2023(Publication Date)
  • Universal Publishers

    (Publisher)

  So, let us go over some important quantities. Which one can we think of easily? Yes, displacement is one of them. For displacement in linear motion, what is a corresponding quantity going to be in the rotational motion? Is it going to be the same sort of displacement in the rotational motion? No, it is not, but we have a rather different convention to describe a rotational motion. So, what can we think of? Yes, it is “angle” that is going to describe rotational motion. Angle is a corresponding quantity in rotational motion. If it is displacement in the linear motion, then it’s angle in the rotational motion. The reason that the physical unit for the two motions are different is because the distance in rotational motion is maintained as a constant, so that can be pulled out as a constant. Remember: we care about what is changing as a function of time when studying classical mechanics.

  We care about what is changing as a function of time.

  After that, understanding velocity and acceleration follow naturally. Instead of linear velocity, we have angular velocity. Instead of linear acceleration, we have angular acceleration. You changed the word “linear” to “angular,” and the list goes on for all the quantities that we have covered.

  How about the dynamics part? The story is a bit different when it comes down to the work done to or by the system.

  For all the quantities we deal with in the rotational motion, replace the position by angle, velocity by angular velocity, acceleration by angular acceleration, mass by moment of inertia, but time stays the same as it is.

  Is that not simple? You just need to follow the rules, and you are going to get the corresponding quantities that you need when describing a rotational motion.

  Remember: for all the physical quantities that we utilize to describe linear motion associated with an object, we have corresponding physical quantities in rotational motion. It is important to remember that. They just happen to look different, but they’re not that different in terms of analyzing motion.

  Problems:

  Rotational motion could be thought of as a combination of the linear motion. That means that the kinematics equation for rotational motion can be derived from the equations for linear motions. Think about why this could be so and write a paragraph on it.

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