Physics 111: Mechanics Lecture 10
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Rotational Motion (The Dynamics of a Rigid Body)
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Robert Katz Publications Research Papers in Physics and Astronomy 1-1958 Physics, Chapter 11: Rotational Motion (The Dynamics of a Rigid Body) Henry Semat City College of New York Robert Katz University of Nebraska-Lincoln, [email protected] Follow this and additional works at: https://digitalcommons.unl.edu/physicskatz Part of the Physics Commons Semat, Henry and Katz, Robert, "Physics, Chapter 11: Rotational Motion (The Dynamics of a Rigid Body)" (1958). Robert Katz Publications. 141. https://digitalcommons.unl.edu/physicskatz/141 This Article is brought to you for free and open access by the Research Papers in Physics and Astronomy at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Robert Katz Publications by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. 11 Rotational Motion (The Dynamics of a Rigid Body) 11-1 Motion about a Fixed Axis The motion of the flywheel of an engine and of a pulley on its axle are examples of an important type of motion of a rigid body, that of the motion of rotation about a fixed axis. Consider the motion of a uniform disk rotat ing about a fixed axis passing through its center of gravity C perpendicular to the face of the disk, as shown in Figure 11-1. The motion of this disk may be de scribed in terms of the motions of each of its individual particles, but a better way to describe the motion is in terms of the angle through which the disk rotates. -
Kinematics Study of Motion
Kinematics Study of motion Kinematics is the branch of physics that describes the motion of objects, but it is not interested in its causes. Itziar Izurieta (2018 october) Index: 1. What is motion? ............................................................................................ 1 1.1. Relativity of motion ................................................................................................................................ 1 1.2.Frame of reference: Cartesian coordinate system ....................................................................................................................................................................... 1 1.3. Position and trajectory .......................................................................................................................... 2 1.4.Travelled distance and displacement ....................................................................................................................................................................... 3 2. Quantities of motion: Speed and velocity .............................................. 4 2.1. Average and instantaneous speed ............................................................ 4 2.2. Average and instantaneous velocity ........................................................ 7 3. Uniform linear motion ................................................................................. 9 3.1. Distance-time graph .................................................................................. 10 3.2. Velocity-time -
PHYSICS of ARTIFICIAL GRAVITY Angie Bukley1, William Paloski,2 and Gilles Clément1,3
Chapter 2 PHYSICS OF ARTIFICIAL GRAVITY Angie Bukley1, William Paloski,2 and Gilles Clément1,3 1 Ohio University, Athens, Ohio, USA 2 NASA Johnson Space Center, Houston, Texas, USA 3 Centre National de la Recherche Scientifique, Toulouse, France This chapter discusses potential technologies for achieving artificial gravity in a space vehicle. We begin with a series of definitions and a general description of the rotational dynamics behind the forces ultimately exerted on the human body during centrifugation, such as gravity level, gravity gradient, and Coriolis force. Human factors considerations and comfort limits associated with a rotating environment are then discussed. Finally, engineering options for designing space vehicles with artificial gravity are presented. Figure 2-01. Artist's concept of one of NASA early (1962) concepts for a manned space station with artificial gravity: a self- inflating 22-m-diameter rotating hexagon. Photo courtesy of NASA. 1 ARTIFICIAL GRAVITY: WHAT IS IT? 1.1 Definition Artificial gravity is defined in this book as the simulation of gravitational forces aboard a space vehicle in free fall (in orbit) or in transit to another planet. Throughout this book, the term artificial gravity is reserved for a spinning spacecraft or a centrifuge within the spacecraft such that a gravity-like force results. One should understand that artificial gravity is not gravity at all. Rather, it is an inertial force that is indistinguishable from normal gravity experience on Earth in terms of its action on any mass. A centrifugal force proportional to the mass that is being accelerated centripetally in a rotating device is experienced rather than a gravitational pull. -
Introduction to Robotics Lecture Note 5: Velocity of a Rigid Body
ECE5463: Introduction to Robotics Lecture Note 5: Velocity of a Rigid Body Prof. Wei Zhang Department of Electrical and Computer Engineering Ohio State University Columbus, Ohio, USA Spring 2018 Lecture 5 (ECE5463 Sp18) Wei Zhang(OSU) 1 / 24 Outline Introduction • Rotational Velocity • Change of Reference Frame for Twist (Adjoint Map) • Rigid Body Velocity • Outline Lecture 5 (ECE5463 Sp18) Wei Zhang(OSU) 2 / 24 Introduction For a moving particle with coordinate p(t) 3 at time t, its (linear) velocity • R is simply p_(t) 2 A moving rigid body consists of infinitely many particles, all of which may • have different velocities. What is the velocity of the rigid body? Let T (t) represent the configuration of a moving rigid body at time t.A • point p on the rigid body with (homogeneous) coordinate p~b(t) and p~s(t) in body and space frames: p~ (t) p~ ; p~ (t) = T (t)~p b ≡ b s b Introduction Lecture 5 (ECE5463 Sp18) Wei Zhang(OSU) 3 / 24 Introduction Velocity of p is d p~ (t) = T_ (t)p • dt s b T_ (t) is not a good representation of the velocity of rigid body • - There can be 12 nonzero entries for T_ . - May change over time even when the body is under a constant velocity motion (constant rotation + constant linear motion) Our goal is to find effective ways to represent the rigid body velocity. • Introduction Lecture 5 (ECE5463 Sp18) Wei Zhang(OSU) 4 / 24 Outline Introduction • Rotational Velocity • Change of Reference Frame for Twist (Adjoint Map) • Rigid Body Velocity • Rotational Velocity Lecture 5 (ECE5463 Sp18) Wei Zhang(OSU) 5 / 24 Illustrating -
Linear and Angular Velocity Examples
Linear and Angular Velocity Examples Example 1 Determine the angular displacement in radians of 6.5 revolutions. Round to the nearest tenth. Each revolution equals 2 radians. For 6.5 revolutions, the number of radians is 6.5 2 or 13 . 13 radians equals about 40.8 radians. Example 2 Determine the angular velocity if 4.8 revolutions are completed in 4 seconds. Round to the nearest tenth. The angular displacement is 4.8 2 or 9.6 radians. = t 9.6 = = 9.6 , t = 4 4 7.539822369 Use a calculator. The angular velocity is about 7.5 radians per second. Example 3 AMUSEMENT PARK Jack climbs on a horse that is 12 feet from the center of a merry-go-round. 1 The merry-go-round makes 3 rotations per minute. Determine Jack’s angular velocity in radians 4 per second. Round to the nearest hundredth. 1 The merry-go-round makes 3 or 3.25 revolutions per minute. Convert revolutions per minute to radians 4 per second. 3.25 revolutions 1 minute 2 radians 0.3403392041 radian per second 1 minute 60 seconds 1 revolution Jack has an angular velocity of about 0.34 radian per second. Example 4 Determine the linear velocity of a point rotating at an angular velocity of 12 radians per second at a distance of 8 centimeters from the center of the rotating object. Round to the nearest tenth. v = r v = (8)(12 ) r = 8, = 12 v 301.5928947 Use a calculator. The linear velocity is about 301.6 centimeters per second. -
Chapter 5 ANGULAR MOMENTUM and ROTATIONS
Chapter 5 ANGULAR MOMENTUM AND ROTATIONS In classical mechanics the total angular momentum L~ of an isolated system about any …xed point is conserved. The existence of a conserved vector L~ associated with such a system is itself a consequence of the fact that the associated Hamiltonian (or Lagrangian) is invariant under rotations, i.e., if the coordinates and momenta of the entire system are rotated “rigidly” about some point, the energy of the system is unchanged and, more importantly, is the same function of the dynamical variables as it was before the rotation. Such a circumstance would not apply, e.g., to a system lying in an externally imposed gravitational …eld pointing in some speci…c direction. Thus, the invariance of an isolated system under rotations ultimately arises from the fact that, in the absence of external …elds of this sort, space is isotropic; it behaves the same way in all directions. Not surprisingly, therefore, in quantum mechanics the individual Cartesian com- ponents Li of the total angular momentum operator L~ of an isolated system are also constants of the motion. The di¤erent components of L~ are not, however, compatible quantum observables. Indeed, as we will see the operators representing the components of angular momentum along di¤erent directions do not generally commute with one an- other. Thus, the vector operator L~ is not, strictly speaking, an observable, since it does not have a complete basis of eigenstates (which would have to be simultaneous eigenstates of all of its non-commuting components). This lack of commutivity often seems, at …rst encounter, as somewhat of a nuisance but, in fact, it intimately re‡ects the underlying structure of the three dimensional space in which we are immersed, and has its source in the fact that rotations in three dimensions about di¤erent axes do not commute with one another. -
The Spin, the Nutation and the Precession of the Earth's Axis Revisited
The spin, the nutation and the precession of the Earth’s axis revisited The spin, the nutation and the precession of the Earth’s axis revisited: A (numerical) mechanics perspective W. H. M¨uller [email protected] Abstract Mechanical models describing the motion of the Earth’s axis, i.e., its spin, nutation and its precession, have been presented for more than 400 years. Newton himself treated the problem of the precession of the Earth, a.k.a. the precession of the equinoxes, in Liber III, Propositio XXXIX of his Principia [1]. He decomposes the duration of the full precession into a part due to the Sun and another part due to the Moon, to predict a total duration of 26,918 years. This agrees fairly well with the experimentally observed value. However, Newton does not really provide a concise rational derivation of his result. This task was left to Chandrasekhar in Chapter 26 of his annotations to Newton’s book [2] starting from Euler’s equations for the gyroscope and calculating the torques due to the Sun and to the Moon on a tilted spheroidal Earth. These differential equations can be solved approximately in an analytic fashion, yielding Newton’s result. However, they can also be treated numerically by using a Runge-Kutta approach allowing for a study of their general non-linear behavior. This paper will show how and explore the intricacies of the numerical solution. When solving the Euler equations for the aforementioned case numerically it shows that besides the precessional movement of the Earth’s axis there is also a nu- tation present. -
Rotational Motion of Electric Machines
Rotational Motion of Electric Machines • An electric machine rotates about a fixed axis, called the shaft, so its rotation is restricted to one angular dimension. • Relative to a given end of the machine’s shaft, the direction of counterclockwise (CCW) rotation is often assumed to be positive. • Therefore, for rotation about a fixed shaft, all the concepts are scalars. 17 Angular Position, Velocity and Acceleration • Angular position – The angle at which an object is oriented, measured from some arbitrary reference point – Unit: rad or deg – Analogy of the linear concept • Angular acceleration =d/dt of distance along a line. – The rate of change in angular • Angular velocity =d/dt velocity with respect to time – The rate of change in angular – Unit: rad/s2 position with respect to time • and >0 if the rotation is CCW – Unit: rad/s or r/min (revolutions • >0 if the absolute angular per minute or rpm for short) velocity is increasing in the CCW – Analogy of the concept of direction or decreasing in the velocity on a straight line. CW direction 18 Moment of Inertia (or Inertia) • Inertia depends on the mass and shape of the object (unit: kgm2) • A complex shape can be broken up into 2 or more of simple shapes Definition Two useful formulas mL2 m J J() RRRR22 12 3 1212 m 22 JRR()12 2 19 Torque and Change in Speed • Torque is equal to the product of the force and the perpendicular distance between the axis of rotation and the point of application of the force. T=Fr (Nm) T=0 T T=Fr • Newton’s Law of Rotation: Describes the relationship between the total torque applied to an object and its resulting angular acceleration. -
Unit 1: Motion
Macomb Intermediate School District High School Science Power Standards Document Physics The Michigan High School Science Content Expectations establish what every student is expected to know and be able to do by the end of high school. They also outline the parameters for receiving high school credit as dictated by state law. To aid teachers and administrators in meeting these expectations the Macomb ISD has undertaken the task of identifying those content expectations which can be considered power standards. The critical characteristics1 for selecting a power standard are: • Endurance – knowledge and skills of value beyond a single test date. • Leverage - knowledge and skills that will be of value in multiple disciplines. • Readiness - knowledge and skills necessary for the next level of learning. The selection of power standards is not intended to relieve teachers of the responsibility for teaching all content expectations. Rather, it gives the school district a common focus and acts as a safety net of standards that all students must learn prior to leaving their current level. The following document utilizes the unit design including the big ideas and real world contexts, as developed in the science companion documents for the Michigan High School Science Content Expectations. 1 Dr. Douglas Reeves, Center for Performance Assessment Unit 1: Motion Big Ideas The motion of an object may be described using a) motion diagrams, b) data, c) graphs, and d) mathematical functions. Conceptual Understandings A comparison can be made of the motion of a person attempting to walk at a constant velocity down a sidewalk to the motion of a person attempting to walk in a straight line with a constant acceleration. -
Physics B Topics Overview ∑
Physics 106 Lecture 1 Introduction to Rotation SJ 7th Ed.: Chap. 10.1 to 3 • Course Introduction • Course Rules & Assignment • TiTopics Overv iew • Rotation (rigid body) versus translation (point particle) • Rotation concepts and variables • Rotational kinematic quantities Angular position and displacement Angular velocity & acceleration • Rotation kinematics formulas for constant angular acceleration • Analogy with linear kinematics 1 Physics B Topics Overview PHYSICS A motion of point bodies COVERED: kinematics - translation dynamics ∑Fext = ma conservation laws: energy & momentum motion of “Rigid Bodies” (extended, finite size) PHYSICS B rotation + translation, more complex motions possible COVERS: rigid bodies: fixed size & shape, orientation matters kinematics of rotation dynamics ∑Fext = macm and ∑ Τext = Iα rotational modifications to energy conservation conservation laws: energy & angular momentum TOPICS: 3 weeks: rotation: ▪ angular versions of kinematics & second law ▪ angular momentum ▪ equilibrium 2 weeks: gravitation, oscillations, fluids 1 Angular variables: language for describing rotation Measure angles in radians simple rotation formulas Definition: 360o 180o • 2π radians = full circle 1 radian = = = 57.3o • 1 radian = angle that cuts off arc length s = radius r 2π π s arc length ≡ s = r θ (in radians) θ ≡ rad r r s θ’ θ Example: r = 10 cm, θ = 100 radians Æ s = 1000 cm = 10 m. Rigid body rotation: angular displacement and arc length Angular displacement is the angle an object (rigid body) rotates through during some time interval..... ...also the angle that a reference line fixed in a body sweeps out A rigid body rotates about some rotation axis – a line located somewhere in or near it, pointing in some y direction in space • One polar coordinate θ specifies position of the whole body about this rotation axis. -
Hydraulics Manual Glossary G - 3
Glossary G - 1 GLOSSARY OF HIGHWAY-RELATED DRAINAGE TERMS (Reprinted from the 1999 edition of the American Association of State Highway and Transportation Officials Model Drainage Manual) G.1 Introduction This Glossary is divided into three parts: · Introduction, · Glossary, and · References. It is not intended that all the terms in this Glossary be rigorously accurate or complete. Realistically, this is impossible. Depending on the circumstance, a particular term may have several meanings; this can never change. The primary purpose of this Glossary is to define the terms found in the Highway Drainage Guidelines and Model Drainage Manual in a manner that makes them easier to interpret and understand. A lesser purpose is to provide a compendium of terms that will be useful for both the novice as well as the more experienced hydraulics engineer. This Glossary may also help those who are unfamiliar with highway drainage design to become more understanding and appreciative of this complex science as well as facilitate communication between the highway hydraulics engineer and others. Where readily available, the source of a definition has been referenced. For clarity or format purposes, cited definitions may have some additional verbiage contained in double brackets [ ]. Conversely, three “dots” (...) are used to indicate where some parts of a cited definition were eliminated. Also, as might be expected, different sources were found to use different hyphenation and terminology practices for the same words. Insignificant changes in this regard were made to some cited references and elsewhere to gain uniformity for the terms contained in this Glossary: as an example, “groundwater” vice “ground-water” or “ground water,” and “cross section area” vice “cross-sectional area.” Cited definitions were taken primarily from two sources: W.B. -
Exploring Artificial Gravity 59
Exploring Artificial gravity 59 Most science fiction stories require some form of artificial gravity to keep spaceship passengers operating in a normal earth-like environment. As it turns out, weightlessness is a very bad condition for astronauts to work in on long-term flights. It causes bones to lose about 1% of their mass every month. A 30 year old traveler to Mars will come back with the bones of a 60 year old! The only known way to create artificial gravity it to supply a force on an astronaut that produces the same acceleration as on the surface of earth: 9.8 meters/sec2 or 32 feet/sec2. This can be done with bungee chords, body restraints or by spinning the spacecraft fast enough to create enough centrifugal acceleration. Centrifugal acceleration is what you feel when your car ‘takes a curve’ and you are shoved sideways into the car door, or what you feel on a roller coaster as it travels a sharp curve in the tracks. Mathematically we can calculate centrifugal acceleration using the formula: V2 A = ------- R Gravitron ride at an amusement park where V is in meters/sec, R is the radius of the turn in meters, and A is the acceleration in meters/sec2. Let’s see how this works for some common situations! Problem 1 - The Gravitron is a popular amusement park ride. The radius of the wall from the center is 7 meters, and at its maximum speed, it rotates at 24 rotations per minute. What is the speed of rotation in meters/sec, and what is the acceleration that you feel? Problem 2 - On a journey to Mars, one design is to have a section of the spacecraft rotate to simulate gravity.