DOCSLIB.ORG

S Radians) (In R S Length Arc Θ = ≡ R

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
S Radians) (In R S Length Arc Θ = ≡ R Rotational Motion • Rotation (rigid body) versus translation (point particle) • Rotation concepts and variables • Rotational kinematic quantities Angular position and displacement Angular velocity Angular acceleration • Rotation kinematics formulas for constant angular acceleration 1 “Radian” “radian” : more convenient unit for angle than degree Definition: 360o 180o • 2π radians = 360 degree 1 radian = = = 57.3o 2π π s arc length ≡ s = r θ (in radians) θ ≡ rad r θ (in rad) s =×2π rr =θ 2π r s θ’ θ Example: r = 10 cm, θ = 100 radians Æ s = 1000 cm = 10 m. 1 Rigid body Rigid body: A “rigid” object, for which the position of each point relative to all other points in the body does not change. Example: Solid: Rigid body Liquid: Not rigid body Rigid body can still have translational and rotational motion. Angular position of rotating rigid body • By convention, θ is measured CCW from the x-axis • It keeps increasing past 2π, can be y negative, etc. • Each point of the body moves around θ the axis in a circle with some specific Reference x line rotates radius with body rotation axis “o” rigid body fixed to body parallel to z-axis 2 Angular displacement of rotating rigid body y θ Reference x line rota tes with body Angular displacement: rotation axis “o” rigid body • Net change in the angular coordinate fixed to body parallel to z-axis Δθ ≡ θfinal − θinital (an angle in rad.) Arc length: Δs • Measures distance covered by a point as it moves Reference line rotating with body y through Δθ (constant r) Δs = r Δθ Δs ≡ rΔθ (a distance along a circular arc) θf r r θo x Rigid body rotation: angular & tangential velocity Angular velocity ω: For any point, r is the perpendicular • Rate of change of the angular displacement distance to the rotation axis Δθ Δθ dθ ωave ≡ ωinst ≡ Lim ≡ Δt Δt → 0 Δt dt vT • Units: radians/sec. Positive in Counter-Clock-Wise sense • Frequency f = # of complete revolutions/unit time r • f = 1/T T = period (time for 1 complete revolution θ = ωΔτ x ω = 2πf = 2π/T f = ω/2π Tangential velocity vT: • Rate at which a point sweeps out arc length along circular path ΔΔs θ Æ Δ=Δs r θ Æ = r vrT = ω ΔΔtt 3 iClicker Quiz 1.1. The period of a rotating wheel is 12.57 seconds. The radius of the wheel is 3 meters. It’s angular speed is closest to: A. 79 rpm B. 0.5 rad/s C. 2.0 rad/s D. .08 rev/s E. 6.28 rev/s 1.2. A point on the rim of the same wheel has a tangential speed closest to: A. 12.57 rev/s B. 0.8 rev/s C. 0.24 m/s Δs ≡ rΔθ D. 1.5 m/s E. 6.28 m/s vT = ωr ω = 2πf = 2π/T Rigid body rotation: angular acceleration Angular acceleration α: Δω Δωωd • Rate of change of the angular velocity ααave ≡≡ inst Lim = ΔtdtΔ→t 0 Δt • Units: rad/s2 • CCW considered positive • for CONSTANT α: ωf = ω0 + αΔt 4 1D and Angular Kinematics Equations (Same mathematical forms) 1D motion with Angular motion with constant acceleration a constant angular acceleration α x(t), v(t), a(t) variables θ(t), ω(t), α(t) dx dv dθ dω v = a = Definitions ω = α = dt dt dt dt vf (t) = v0 + at Kinematic ωf (t) = ω0 + αt Equations x (t) = x + v t + 1 at 2 θ (t) = θ + ω t + 1 αt2 f 0 0 2 f 0 0 2 v2 (t) = v2 + 2a[x − x ] 2 2 f 0 f 0 ωf (t) = ω0 + 2α[θf − θ0 ] Rotational variables are vectors, having direction The angular displacement, speed, and acceleration ( θ, ω, α ) are vectors with direction. The directions are given by the right-hand rule: Fingers of right hand curl along the angular direction (See Fig.) Then, the direction of thumb is thdihe direct ion o fhf the angu lar quantity. 5 Example: A grindstone is rotating with constant angular acceleration about a fixed axis in space. Initial conditions 2 at t = 0: α = 0.35 rad/s ω0 = - 4.6 rad/s When is Δθ = 0 again in addition to t=0? Positive directions: right hand rule Example: Wheel rotating and accelerating At t = 0, a wheel rotating about a fixed axis at a constant angular acceleration has an angular velocity of 2.0 rad/s. Two seconds later it has turned through 5.0 complete revolutions. Find the angular acceleration of this wheel? ωf (t) = ω0 + αt θ (t) = θ + ω t + 1 αt2 f 0 0 2 2 2 ωf (t) = ω0 + 2α[θf − θ0 ] 6 Rigid body rotation: radial and tangential acceleration Centripetal (radial) acceleration ac or ar • Radial acceleration component, points toward rotation axis v2 T 2 Fma= arr == ωω (use v = r ) rr a r T T vT r ac ω,α Tangential acceleration aT: x • Tangential acceleration component • Proportional to angular acceleration α and also to radius r • Units: length / time 2 aT = rα Fmatangential = T Rotation variables: angular vs. linear s = rΔθ vrT = ω arT = α v 2 a = T =rω 2 r r 7 A ladybug sits at the outer edge of a merry-go-round, and a gentleman bug sits halfway between her and the axis of rotation. The merry-go-round makes a complete revolution once each second. The gentleman bug’s angular velocity is A. half the ladybug’s. B. the same as the ladybug’s. C. twice the ladybug’s. D. impossible to determine A ladybug sits at the outer edge of a merry-go-round, and a gentleman bug sits halfway between her and the axis of rotation. The merry-go-round makes a complete revolution once each second. The gentleman bug’s velocity is A. half the ladybug’s. B. the same as the ladybug’s. C. twice the ladybug’s. D. impossible to determine 8 Rotational Dynamics We want something like “F=ma” for rotational motion….. • Moment of inertia – rotational analog of mass • Torque – rotational analog of force 17 Something like mass for rotational motion: Moment of Inertia, I G L Kinetic energy of ladybug and gentlemanbug 11 1 1 Kmvmvmr=+22 =()ωω 2 + mr () 2 22LL GG 2 L L 2 G G 1122 22 1 2 2 2 1 2 =+mrLLω mr GGωωω =+=( mr LL mr GG ) I 22 2 2 22 I =+mrL LGG mr 1 Generally, Imrmrmr=+++222... Kinetic energy: K = Iω 2 11 2 2 33 2 9 Example: Find moment of inertia for a crossed dumbbell •Four identical balls as shown: m m = 2 kg •Connected by massless rods: 2 d d length d = 1 m. mmdd A BC d Rotational inertia I depends on axis chosen m A) Choose rotation axis perpendicular to figure through point “A” B) Now c hoose ax is perpen dicul ar t o fi gure th rough poi nt “B” C) Let rotation axis pass through points “B” and “C” Calculation of Moment of inertia for continuous mass distributions requires “Integration, a kind of calculus”. We will just use the result. 10 Moments of Inertia of Various Rigid Objects Now we want to define “torque, τ”, so that “τ = I α”. F Newton’s Law along tangential direction θ F T m FTT= ma= m r α r Multiplying “r”, so that we have “I” on right side rp axis 2 rFT == m r α Iα So, let’s define torque as τ ≡ rFT Then we got τ = Iα Since FFT = sinθ and rrp = sinθ τ ==rFTp rFsinθ = r F (could be positive or negative) 11 F τ ==rFTp rFsinθ = r F θ F T m If r = 0, torque is zero. r rp axis If theta = 0 or 180 degree, the torque is zero. For multiple forces τ net =+++τττ123... τ net = Iα m1 m2 m1=100 kg adult, m2=10 kg baby. Distance to fulcrum point is 1 m and 11 m respectively. The seesaw s tar ts at hor izon tal positi on f rom rest. Which direction will it rotates? (a) Counter-Clockwise (b) Clockwise (c) No rotation (d) Not enough information Example: Find the net torque, moment of inertia, and initial angular acceleration. Choose axis of rotation through fulcrum point. 12 Example: second law for rotation PP10606-49*: When she is launched from a springboard, a diver's angular speed about her center of mass changes from zero to 6.20 rad/s in 220 ms. Her rotational inertia about her center of mass is constant at 12.0 kg·m2. During the launch, what are the magnitudes of (a) her average angular acceleration and (b) the average external torque on her from the board? G G τnet = Itot α 5 N tangential force is applied at the edge of a uniform disk of radius 2 m and mass of 8 kg. Find angular acceleration. F=5 N Axis of rotation 1 Formula: I = MR2 2 13 5 N tangential force is applied at 1 m from the center of a uniform disk of radius 2 m and mass of 8 kg. Find angular acceleration. FF5=5N N Axis of rotation 1 Formula: I = MR2 2 5 N force is applied at 1 m from the center of a uniform disk of radius 2 m and mass of 8 kg. Find torque. 45 degree F=5 N Axis of rotation 14 Torque on extended object by gravitational force Æ Assume that the total gravitational force effectively acts at the center of mass.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Chapter 24 Physical Pendulum
    Chapter 24 Physical Pendulum 24.1 Introduction ........................................................................................................... 1 24.1.1 Simple Pendulum: Torque Approach .......................................................... 1 24.2 Physical Pendulum ................................................................................................ 2 24.3 Worked Examples ................................................................................................. 4 Example 24.1 Oscillating Rod .................................................................................. 4 Example 24.3 Torsional Oscillator .......................................................................... 7 Example 24.4 Compound Physical Pendulum ........................................................ 9 Appendix 24A Higher-Order Corrections to the Period for Larger Amplitudes of a Simple Pendulum ..................................................................................................... 12 Chapter 24 Physical Pendulum …. I had along with me….the Descriptions, with some Drawings of the principal Parts of the Pendulum-Clock which I had made, and as also of them of my then intended Timekeeper for the Longitude at Sea.1 John Harrison 24.1 Introduction We have already used Newton’s Second Law or Conservation of Energy to analyze systems like the spring-object system that oscillate. We shall now use torque and the rotational equation of motion to study oscillating systems like pendulums and torsional springs. 24.1.1 Simple
    [Show full text]
  • Rotation: Kinematics
    Rotation: Kinematics Rotation refers to the turning of an object about a fixed axis and is very commonly encountered in day to day life. The motion of a fan, the turning of a door knob and the opening of a bottle cap are a few examples of rotation. Rotation is also commonly observed as a component of more complex motions that result as a combination of both rotation and translation. The motion of a wheel of a moving bicycle, the motion of a blade of a moving helicopter and the motion of a curveball are a few examples of combined rotation and translation. This module focuses on the kinematics of pure rotation. This module begins by defining the angular variables and then proceeds to describe the relation between these variables and the variables of linear motion. Angular Variables Angular Position Consider an object rotating about a fixed axis. Let us define a coordinate system such that the axis of rotation passes through the origin and is perpendicular to the x- and y- axes. Fig. 1 shows a view of the x-y plane. If a reference line is drawn through the origin and a fixed point in the object, the angle between this line and a fixed direction is used to define the angular position of the object. In Fig. 1, the angular position is measured from the positive x-direction. Fig. 1: Rotating object - Angular position. Also, from geometry, the angular position can be written as the ratio of the length of the path traveled by any point on the reference line measured from the fixed direction to the radius of the path, ...Eq.
    [Show full text]
  • CHAPTER 9: Rotation of a Rigid Body About a Fixed Axis up Until Know
    Lecture 13: Reviews of Rotational Kinematics and Dynamics 1 CHAPTER 9: Rotation of a Rigid Body about a Fixed Axis Up until know we have always been looking at \point particles" or the motion of the center{of{mass of extended objects. In this chapter we begin the study of rotations of an extended object about a ¯xed axis. Such objects are called rigid bodies because when they rotate they maintain their overall shape. It is just their orientation in space which is changing. Angular Variables: , !, ® The variables used to described the motion of \point particles" are displacement, velocity, and acceleration. For a rotating rigid body, there are three completely analogous variables: angular displacement, angular velocity, and angular acceler- ation. These angular variables are very useful because the they can be assigned to every point on the rigid body as it rotates about a ¯xed axis. The ordi- nary displacement, velocity, and acceleration can be calculated at a given point from the angular displacement, angular velocity, and angular acceleration just by multiplying by the distance r of that point from the axis of rotation. Equations of motion with constant angular acceleration ®: For every equation which you have learned to describe the linear motion of a point particle, there is an exactly analogous equation to describe the rotational motion ((t) and !(t)) about a ¯xed axis. 1 2 1 2 Position with time: x(t) = x0 +v0t+ at corresponds to (t) = 0 +!0t+ ®t 2 2 Speed with time: v(t) = v0 + at corresponds to !(t) = !0 + ®t 2 2 2 2 Speed with distance: v (x) = v0+2a(x¡x0) corresponds to ! () = !0+2®(¡0) Relating linear kinematics with angular kinematics For a purely rotating body, all points on the body move in circles about the axis of rotation.
    [Show full text]
  • Angular, Or Rotational Motion
    Chapter 8 Angular, or Rotational Motion The radian There are 2 types of pure unmixed motion: n Translational - linear motion n Rotational - motion involving a rotation or revolution around a fixed chosen axis (an axis which does not move). We need a system that defines BOTH types of motion working together on a system. Rotational quantities are usually defined with units involving a radian measure. If we take the radius of a circle and LAY IT DOWN on the circumference, it will create an angle whose arc length is equal to R. In other words, one radian angle subtends an arc length Ds equal to the radius of the circle (R) Angular Quantities In purely rotational motion, all points on the object move in circles around the axis of rotation (“O”). The radius of the circle is r. Note that all points on a straight line drawn through the axis move through the same angle in the same time. s The angle θ is 1 radian if the arc length is equal to the radius. Therefore, The radian Half a radian would subtend an arc length, s, equal to half the radius, and 2 radians would subtend an arc length equal to two times the radius. A general Radian Angle (Dq) subtends an arc length (Ds) equal to R. The theta in this case represents ANGULAR DISPLACEMENT. Angular Velocity Since velocity is defined as the rate of change of displacement, ANGULAR VELOCITY is defined as the rate of change of ANGULAR DISPLACEMENT. Angular Acceleration Once again, following the same lines of logic… Since acceleration is defined as the rate of change of velocity, we can say the ANGULAR ACCELERATION is defined as the rate of change of the angular velocity.
    [Show full text]
  • Circular Motion
    Lecture 6 Circular Motion Pre-reading: KJF §6.1 and 6.2 Please log in to Socrative, room HMJPHYS1002 CIRCULAR MOTION KJF §6.1–6.4 Angular position If an object moves in a circle of radius r, then after travelling a distance s it has moved an angular displacement θ: s θ = r θ is measured in radians (2π radians = 360°) KJF §3.8 3 Tangential velocity If motion is uniform and object takes time t to execute motion, then it has tangential velocity of magnitude v given by s v = t Period of motion T = time to complete one revolution (units: s) Frequency f = number of revolutions per second (units: s–1 or Hz) 1 f = T 4 Angular velocity Define an angular velocity ω angular displacement θ ω = = time interval t Uniform circular motion is when ω is constant. Combining last 3 equations: v = rω 2π T = period ω KJF §6.1 5 Question You place a beetle on a uniformly rotating record (a) Is the beetle's tangential velocity different or the same at different radial positions? (b)Is the beetle's angular velocity different or the same at the different radial positions? Remember; all points on a rigid rotating object will experience the same angular velocity 6 Consider an object is moving in uniform circular motion – tangential speed is constant. Is the object accelerating? Velocity is a vector ∴ changing direction ⇒ acceleration ⇒ net force 7 The change in velocity Δv = v2 – v1 and Δv points towards the centre of the circle Angle between velocity vectors is θ so Δv = vθ and so ∆v vθ v2 a = = = ∆t rθ/v r KJF §3.8 8 Centripetal acceleration Acceleration points towards centre – centripetal acceleration ac v2 a = = ω2r c r Since the object is accelerating, there must be a force to keep it moving in a circle mv2 F = = mω2r c r This centripetal force may be provided by friction, tension in a string, gravity etc.
    [Show full text]
  • Open JDHPHD.Pdf
    The Pennsylvania State University The Graduate School College of Health and Human Development A KINEMATIC AND KINETIC ANALYSIS OF POSTURAL PERTURBATION A Thesis in Exercise and Sport Science by John D. Henley Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy May 2002 We approve the thesis of John D. Henley. Date of Signature Peter R. Cavanagh Distinguished Professor of Locomotion Studies, Medicine, Orthopedics and Rehabilitation, and Biobehavioral Health Thesis Advisor Chair of Committee R. Scott Kretchmar Professor of Exercise and Sport Science Interim Head of the Department of Kinesiology Karl M. Newell Professor of Kinesiology and Biobehavioral Health H. Joseph Sommer III Professor of Mechanical Engineering Vladimir Zatsiorsky Professor of Kinesiology iii Abstract This study was designed to quantify postural reaction to a sudden toe-up rotational perturbation by twenty-two healthy young adults (eleven males, eleven females, average age of 26 years, height of 1.72 m, and weight of 649.62 N) with no physical or physiological conditions that would compromise their ability to react to the perturbation. The goal was to achieve an understanding of the mechanical requirements necessary for maintenance of balance and stance. The human body was modeled as a system of five rigid segments. The act of reacting to a movement of the ground that suddenly dorsiflexed the ankle was monitored by a video analysis system operating at 60 Hz. Special care was taken to maximize marker resolution by covering the action with several cameras each zoomed in on part of the body. Segmental kinematics and kinetics were calculated after numerically filtering at 6 Hz and differentiating the digitized location of retroreflective markers attached to the subjects’ skin.
    [Show full text]
  • Goals Rotational Quantities As Vectors Angular Momentum Q Math: Cross
    Physics 106 Week 5 Torque and Angular Momentum as Vectors SJ 7thEd.: Chap 11.2 to 3 • Rotational quantities as vectors • Cross product • Torque expressed as a vector • Angular momentum defined • Angular momentum as a vector • Newton’s second law in vector form 1 Goals Rotational quantities as vectors Math: Cross Product Angular momentum 1 So far: simple (planar) geometries Rotational quantities Δθ, ω, α, τ, etc… represented by positive or negative numbers Rotation axis was specified simply as CCW or CW Problems were 2 dimensional with a perpendicular rotation axis Now: 3D geometries rotation represented in full vector form Angular displacement, angular velocity, angular acceleration as vectors, having direction The angular displacement, speed, and acceleration ( θ, ω, α ) are vectors with direction. The directions are given by the right-hand rule: Fingers of right hand curl along the angular direction (See Fig.) Then, the direction of thumb is thdihe direct ion o fhf the angu lar quantity. 2 Example: triad of unit vectors showing rotation in x- y plane +z K ω = ωkˆ +y +x ω A disk rotates at 3 rad/s in xy plane as shown above. What is anggyular velocity vector? Torque as a vector ? Torque “vector” is defined from position vector and force vector, using “cross product”. 3 Math: Cross Product Cross Product (Vector Product) two vectors Æ a third vector normal to the plane they define measures the component of one vector normal to the other θ = smaller angle between the vectors G G G G G G G G G G c ≡ a ×b = ab sin(θ) a ×b = −b × a c = a ×b G cross product of any parallel vectors = zero b cross product is a maximum for perpendicular vectors cross products of Cartesian unit vectors: θ ˆ ˆ ˆ ˆ ˆ ˆ G i × i = j × j = k ×k = 0 i a kˆ = ˆi × ˆj = −ˆj × ˆi ˆj = kˆ × ˆi = −ˆi ×kˆ j k ˆi = ˆj ×kˆ = −kˆ × ˆj Direction is defined by right hand rule.
    [Show full text]
  • Physics, Chapter 6: Circular Motion and Gravitation
    University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Robert Katz Publications Research Papers in Physics and Astronomy 1-1958 Physics, Chapter 6: Circular Motion and Gravitation 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 6: Circular Motion and Gravitation" (1958). Robert Katz Publications. 146. https://digitalcommons.unl.edu/physicskatz/146 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. 6 Circular Motion and Gravitation 6-1 Circular Motion Our earlier discussion of the kinematics of a particle was developed prin­ cipally from the point of view of being able to describe that motion easily within a rectangular coordinate system. Thus the most complex case with which we dealt was that of a projectile motion, in which the acceleration was constant and was directed along one of the coordinate axes. A more convenient framework within which to discuss rotational and circular mo­ tions is provided by a set of polar coordinates. In the present discussion we will restrict ourselves to motion +y in which the polar coordinate r is constant, or fixed; that is, //---- the particle is constrained to /' // move in a circular path. I I I I 6-2 Angular Displacement I I I When a particle is constrained I I \ I +x to move in a circular path, it is \ I \ I convenient to superimpose a \ I \ I , I coordinate system on the mo­ , I tion so that the x-y plane is in '....
    [Show full text]
  • Chapter 20 Rigid Body: Translation and Rotational Motion Kinematics for Fixed Axis Rotation
    Chapter 20 Rigid Body: Translation and Rotational Motion Kinematics for Fixed Axis Rotation 20.1 Introduction ........................................................................................................... 1 20.2 Constrained Motion: Translation and Rotation ................................................ 1 20.2.1 Rolling without slipping ................................................................................ 5 Example 20.1 Bicycle Wheel Rolling Without Slipping ........................................ 6 Example 20.2 Cylinder Rolling Without Slipping Down an Inclined Plane ........ 9 Example 20.3 Falling Yo-Yo .................................................................................. 10 Example 20.4 Unwinding Drum ............................................................................ 11 20.3 Angular Momentum for a System of Particles Undergoing Translational and Rotational .................................................................................................................... 12 Example 20.5 Earth’s Motion Around the Sun .................................................... 15 20.4 Kinetic Energy of a System of Particles ............................................................ 17 20.5 Rotational Kinetic Energy for a Rigid Body Undergoing Fixed Axis Rotation ...................................................................................................................................... 18 Appendix 20A Chasles’s Theorem: Rotation and Translation of a Rigid Body ... 19 Chapter 20 Rigid
    [Show full text]
  • Estimation of Rocking and Torsion Associated with Surface Waves Extracted from Recorded Motions
    Soil Dynamics and Earthquake Engineering 80 (2016) 225–240 Contents lists available at ScienceDirect Soil Dynamics and Earthquake Engineering journal homepage: www.elsevier.com/locate/soildyn Estimation of rocking and torsion associated with surface waves extracted from recorded motions Kristel C. Meza-Fajardo a,n, Apostolos S. Papageorgiou b a Department of Civil Engineering, Universidad Nacional Autónoma de Honduras, Tegucigalpa, Honduras b Department of Civil Engineering, University of Patras, GR-26500 Patras, Greece article info abstract Article history: By exploiting the capability of identifying and extracting surface waves existing in a seismic signal, we Received 1 April 2015 can proceed to estimate the angular displacement (rotation about the horizontal axis normal to the Received in revised form direction of propagation of the wave; rocking) associated with Rayleigh waves as well as the angular 23 October 2015 displacement (rotation about the vertical axis; torsion) associated with Love waves. Accepted 29 October 2015 For a harmonic Rayleigh (Love) wave, rocking (torsion) would be proportional to the harmonic ver- Available online 21 November 2015 tical (transverse horizontal) velocity component and inversely proportional to the phase velocity cor- Keywords: responding to the particular frequency of the harmonic wave (a fact that was originally exploited by Ground motion variability Newmark (1969) [15] to estimate torsional excitation). Evidently, a reliable estimate of the phase velocity Wave identification (as a function of frequency) is necessary. As pointed out by Stockwell (2007) [17], because of its abso- Time–frequency analysis lutely referenced phase information, the S-Transform can be employed in a cross-spectrum analysis in a Angular displacements Phase velocity spectrum local manner.
    [Show full text]