Rotational & RigidBody Mechanics. Lectures 3+4


 Shannon Holland
 1 years ago
 Views:
Transcription
1 Rotational & RigidBody Mechanics Lectures 3+4
2 Rotational Motion So far: point objects moving through a trajectory. Next: moving actual dimensional objects and rotating them. 2
3 Circular Motion  Definitions C is the center of rotation. P a point on to the object. r is the distance vector P C r = r the distance between C and P. Object rotates P travels along a circular path after t, P covered distance s, and angle θ. Unitlength axis of rotation: u. In this example, the Z axis going out from the screen. Rotation: counterclockwise (righthand rule). C θ r P s P 3
4 Angular Displacement This angle θ represents the rotation of the object: θ = s/r where s is the arc length and r the radius unit is radian (rad) 1 radian=angle for arc length 1 at a distance 1. ω C θ r P v s P 4
5 Angular Velocity Angular velocity: the rate of change of the angular displacement: ω = θ t = θ t + t θ(t) t unit is rad/s. The angular velocity vector is collinear with the rotation axis: ω = ωu 5
6 Angular Acceleration Angular acceleration: the rate of change of the angular velocity: α = ω ω t + t ω(t) = t t Exactly like acceleration is to velocity in a trajectory. unit is rad/s 2 6
7 Equations of Motion Deferring velocity from position and acceleration: ω(t + t) = ω(t) + α t ω = ω(t + t) + ω(t) 2 θ = 1 2 ω(t + t) + ω(t) t θ = ω(t) t α t2 ω(t + t) 2 = ω(t) 2 + 2α θ 7
8 Kinetics of Rotational Motion The centripetal force creates curved motion. Orthogonal to the velocity of the object. Object is in orbit. Constant force circular rotation with constant tangential velocity. Why? 8
9 Tangential and Angular Velocities Every point on a rigid body moves with the same angular velocity. Different points on a rigid body can have different tangential velocities. Different radii V(t) C P(t + t) P(t) T(t) 9
10 Tangential Velocity Angular velocity vector: direction of rotation axis Tangential velocity vector: direction of movement direction Relation: v = ω r ω Or: r v ω = r 2 C θ r P v s P Note that v = ω r (abs. values), due to s = θ r. Only the tangential part of any velocity matters for rotation! 10
11 Tangential Acceleration Tangential acceleration defined the same way a = α r where α is the angular acceleration. Similarly to the velocity equation v = ω r 11
12 Centripetal Acceleration The centripetal acceleration, orthogonal to the velocity (=towards the axis of rotation), drives the rotational movement: What is the centrifugal force? a n = v t 2 r = rω2 12
13 RigidBody Kinematics Rigid bodies have dimensions, as opposed to single points. Relative distances between all points are invariant. Movement can be decomposed into two components: Linear trajectory of any single points Relative rotation around the point Free body movement: around the center of mass (COM).
14 Mass The measure of the amount of matter in the volume of an object: m = ρ dv V where ρ is the density at each location in the object volume V. Equivalently: a measure of resistance to motion or change in motion. 14
15 Mass For a 3D object, mass is the integral over its volume: m = ρ(x, y, z) dx dy dz For uniform density objects (simple rigid bodies): m = ρ V where ρ is the density of the object and V is its volume. 15
16 Center of Mass The center of mass (COM) is the point at which all the mass can be considered to be concentrated obtained from the first moment, i.e. mass times distance. point of balance of the object if uniform density, COM is also the centroid COM 16
17 Center of Mass Coordinate of the COM: COM = 1 m V ρ p p dv where p is the position at each location in V. For a set of bodies: n COM = 1 m i=1 m i p i where m i is the mass of each COM p i in every body. 17
18 Center of Mass Example for a body made of two spheres in 1D x 2 x 1 x COM m 1 COM m 2 x COM = m 1x 1 + m 2 x 2 m 1 + m 2 18
19 Center of Mass Quite easy to determine for primitive shapes But what about complex surface based models? 19
20 Angular Momentum Remember linear momentum: p = m v. Rotational motion also produces angular momentum of a any point on the object about the center of mass (or any relative point): L = V r p dv unit is N m s Angular momentum is a conserved quantity! Just like the linear momentum. Remember: measured around a point. 20
21 Angular Momentum Plugging in angular velocity: r p = r v dm = r ω r dm Integrating, we get: L = r ω r dm M Note: The angular momentum and the angular velocity are not generally collinear! 21
22 Defining: r = x y z and ω = Moment of Inertia ω x ω y ω z. Remember that the angular velocity is constant throughout the body. We get: L = y 2 + z 2 ω x xyω y xzω z yxω x + z 2 + x 2 ω y yzω z zxω x zyω y + (x 2 + y 2 )ω z dm = I xx I xy I xz I yx I yy I yz I zx I zy I zz ω x ω y ω z.
23 Momentum and Inertia The inertia tensor only depends on the geometry of the object and the relative point (often, COM): I xx = y 2 + z 2 dm I xy = I yx = xy dm I yy = z 2 + x 2 dm I xz = I zx = xz dm I zz = x 2 + y 2 dm I yz = I zy = yz dm 23
24 The Inertia Tensor Finally the inertia can be expressed as the matrix y 2 + z 2 dm xy dm xz dm I = xy dm z 2 + x 2 dm yz dm xz dm yz dm x 2 + y 2 dm The diagonal elements are called the (principal) moment of inertia The offdiagonal elements are called products of inertia 24
25 The Inertia Tensor Equivalently, we separate mass elements to density and volume elements: I = V ρ x, y, z y 2 + z 2 xy xz xy z 2 + x 2 yz dx dy dz xz yz x 2 + y 2 The diagonal elements: distances to the respective principal axes. The nondiagonal elements: products of the perpendicular distances to the respective planes. 25
26 Moment of Inertia The moment of inertia of a rigid body is a measure of how much the mass of the body is spread out. A measure of the ability to resist change in rotational motion Defined with respect to a specific rotation axis u. Through the central rotation origin point. We have that: I u = V r u 2 dm 26
27 Moment and Tensor We have: r u 2 = u r 2 = u T I(q)u for any point q. (Remember: r is distance to origin). Thus: I u = M u T I(q)u dm=u T Iu The scalar angular momentum around the axis is then L u = I u ω. Reducible to a planar problem (axis is the new z axis).
28 Moment of Inertia For a mass point: I = m r u 2 r m For a collection of mass points: I = i m i r i 2 r 2 m 2 r 1 m 1 r 3 m 3 For a continuous mass distribution on the plane: I = M r 2 u dm dm r 28
29 Inertia of Primitive Shapes For primitive shapes, the inertia can be expressed with the parameters of the shape Illustration on a solid sphere Calculating inertia by integration of thin discs along one axis (e.g. z). Surface equation: x 2 + y 2 + z 2 = R 2 29
30 Inertia of Primitive Shapes Distance to axis of rotation is the radius of the disc at the cross section along z: r 2 = x 2 + y 2 = R 2 z 2. Summing moments of inertia of small cylinders of inertia I Z = r2 m We get: di Z = 1 2 r2 dm = 1 2 r2 ρdv = 1 2 r2 ρπr 2 dz 2 along the zaxis: I Z = 1 ρπ R 2 R r 4 dz = 1 ρπ R 2 R R 2 z 2 2 dz = 1 ρπ 2 R4 z 2R 2 z z 5 R 5 R as m = ρ = ρπ R πr 3, we finally obtain: I Z = 2 5 mr2. 30
31 Inertia of Primitive Shapes Solid sphere, radius r and mass m: I = 2 5 mr mr mr2 y Hollow sphere, radius r and mass m: 31 I = 2 3 mr mr mr2 z x
32 Inertia of Primitive Shapes Solid ellipsoid, semiaxes a, b, c and mass m: I = 1 5 m(b2 +c 2 ) m(a2 +c 2 ) m(a2 +b 2 ) z y x Solid box, width w, height h, depth d and mass m: 32 h w d I = 1 12 m(h2 +d 2 ) m(w2 +d 2 ) m(w2 +h 2 )
33 Inertia of Primitive Shapes Solid cylinder, radius r, height h and mass m: h I = 1 12 m(3r2 +h 2 ) m(3r2 +h 2 ) Hollow cylinder, radius r, height h and mass m: 1 2 mr2 h 33 I = 1 12 m(6r2 +h 2 ) m(6r2 +h 2 ) mr 2
34 ParallelAxis Theorem The object does not necessarily rotate around the center of mass. Some point can be fixed! parallel axis theorem: I v = I COM + md 2 Where: I v : inertia around axis u. I COM inertia about a parallel axis through the COM. d is the distance between the axes. 34
35 ParallelAxis Theorem More generally, for point displacements: d x, d y, d z I xx = y 2 + z 2 dm + md x 2 I xy = xy dm + md x d y I yy = z 2 + x 2 dm + md y 2 I xz = xz dm + md x d z I zz = x 2 + y 2 dm + md z 2 I yz = yz dm + md y d z 35
36 PerpendicularAxis Theorem For a planar 2D object, the moment of inertia about an axis perpendicular to the plane is the sum of the moments of inertia of two perpendicular axes through the same point in the plane: z y z y x x I z = I x + I y for any planar object 36 I z = 2I x = 2I y for symmetrical objects
37 Reference Frame The inertia tensor is coordinate dependent. If R changes bases from body to world coordinate, the inertia tensor in world space is: I world = R I body R T 37
38 RigidBody Dynamics
39 Torque The torque is a force F applied at a distance r from a (held) center of mass. Tangential part causes tangential acceleration: F t = m a t Multiplying by the distance, the torque is: τ = F t r = m a t r We know that a t = r α So we have τ = m r α r = m r 2 α unit is N m rotates an object about its axis of rotation through the center of mass. 39
40 Torque A force in general is not applied in the direction of the tangent. The torque τ is then defined as: τ = r F The direction of the torque is perpendicular to both F and r. 40
41 Newton s Second Law The law F = m a has an equivalent with inertia tensor and torque: τ = I α Force linear acceleration Torque angular acceleration 41
42 Rotational Kinetic Energy Translating energy formulas to rotational motion. The rotational kinetic energy is defined as: E Kr = 1 2 ωt I ω 42
43 Conservation of Mechanical Energy Adding rotational kinetic energy E Kt t + t + E P t + t + E Kr t + t = E Kt t + E P t + E Kr (t) + E O E Kt is the translational kinetic energy. E P is the potential energy. E Kr is the rotational kinetic energy. E O the lost energies (surface friction, air resistance etc.). 43
44 Torque and Angular Momentum Remember, in the linear case: F = d p dt ( p is the linear momentum). Similarly with torque and angular momentum: dl d r d p = p + r dt dt dt = v m v + r F = 0 + τ Force derivative of linear momentum. Torque derivative of angular momentum.
45 Impulse We may apply offcenter forces for a very short amount of time. Such angular impulse results in a change in angular momentum, i.e. in angular velocity: τ t = L 45
46 Rigid Body Forces A force can be applied anywhere on the object, producing also a rotational motion. F COM α a 46
47 Position of An Object Remember: the object moves linearly as the COM moves. Rotation: the movement for all points relatively to the COM. Total motion: sum of the two motions. F 47
48 Particle System Most simple instance of a physics system Each object (body) is a particle Each particle have forces acting upon it Constant, e.g. gravity Position dependent, e.g. force fields Velocity dependent, e.g. drag forces Event based, e.g. collision forces Restrictive, e.g. joint constraint So net force is a function F p o, v, a, m, t, Discretization: e.g., V f q dm becomes a sum: i=1 n f q i m i 48
49 Particle System Use the equations of motion to find the position of each particle at each frame. At the start of each frame: Sum up all of the forces for each particle. From these forces compute the acceleration. Integrate into velocity and position. Rigid body: all particles receive the same rotation and translation. 49
50 Complex Objects When an object consists of multiple primitive shapes: Calculate the individual inertia of each shape. Use parallel axis theorem to transform to inertia about an axis through the COM of the object. Add the inertia matrices together. 50
51 Motion Constraints A rigid body may not be free to move on its own. We wish to constrain its movement: wheels on a chair human body parts trigger of a gun opening door actually almost anything you can think of in a game... 51
52 Degree of freedom To describe how a body can move in space, specify its degrees of freedom (DOF): Translational Rotational 52
53 Kinematic pair A kinematic pair is a connection between two bodies that imposes constraints on their relative movement Lower pair, constraint on a point, line or plane: Revolute pair, or hinged joint: 1 rotational DOF. Prismatic joint, or slider: 1 translational DOF. Screw pair: 1 coordinated rotation/translation DOF. Cylindrical pair: 1 translational + 1 rotational DOF. Spherical pair, or ballandsocket joint: 3 rotational DOF. Planar pair: 3 translational DOF. Higher pair, constraint on a curve or surface. 53
Lecture II: RigidBody Physics
RigidBody Motion Previously: Point dimensionless objects moving through a trajectory. Today: Objects with dimensions, moving as one piece. 2 RigidBody Kinematics Objects as sets of points. Relative distances
More informationGame Physics. Game and Media Technology Master Program  Utrecht University. Dr. Nicolas Pronost
Game and Media Technology Master Program  Utrecht University Dr. Nicolas Pronost Rigid body physics Particle system Most simple instance of a physics system Each object (body) is a particle Each particle
More informationArtificial Intelligence & Neuro Cognitive Systems Fakultät für Informatik. Robot Dynamics. Dr.Ing. John Nassour J.
Artificial Intelligence & Neuro Cognitive Systems Fakultät für Informatik Robot Dynamics Dr.Ing. John Nassour 25.1.218 J.Nassour 1 Introduction Dynamics concerns the motion of bodies Includes Kinematics
More informationGame Physics. Game and Media Technology Master Program  Utrecht University. Dr. Nicolas Pronost
Game and Media Technology Master Program  Utrecht University Dr. Nicolas Pronost Essential physics for game developers Introduction The primary issues Let s move virtual objects Kinematics: description
More informationCP1 REVISION LECTURE 3 INTRODUCTION TO CLASSICAL MECHANICS. Prof. N. Harnew University of Oxford TT 2017
CP1 REVISION LECTURE 3 INTRODUCTION TO CLASSICAL MECHANICS Prof. N. Harnew University of Oxford TT 2017 1 OUTLINE : CP1 REVISION LECTURE 3 : INTRODUCTION TO CLASSICAL MECHANICS 1. Angular velocity and
More informationGeneral Physics I. Lecture 10: Rolling Motion and Angular Momentum.
General Physics I Lecture 10: Rolling Motion and Angular Momentum Prof. WAN, Xin (万歆) 万歆 ) xinwan@zju.edu.cn http://zimp.zju.edu.cn/~xinwan/ Outline Rolling motion of a rigid object: centerofmass motion
More informationPLANAR KINETIC EQUATIONS OF MOTION (Section 17.2)
PLANAR KINETIC EQUATIONS OF MOTION (Section 17.2) We will limit our study of planar kinetics to rigid bodies that are symmetric with respect to a fixed reference plane. As discussed in Chapter 16, when
More informationChapter 8 Lecture Notes
Chapter 8 Lecture Notes Physics 2414  Strauss Formulas: v = l / t = r θ / t = rω a T = v / t = r ω / t =rα a C = v 2 /r = ω 2 r ω = ω 0 + αt θ = ω 0 t +(1/2)αt 2 θ = (1/2)(ω 0 +ω)t ω 2 = ω 0 2 +2αθ τ
More informationHandout 6: Rotational motion and moment of inertia. Angular velocity and angular acceleration
1 Handout 6: Rotational motion and moment of inertia Angular velocity and angular acceleration In Figure 1, a particle b is rotating about an axis along a circular path with radius r. The radius sweeps
More informationHandout 6: Rotational motion and moment of inertia. Angular velocity and angular acceleration
1 Handout 6: Rotational motion and moment of inertia Angular velocity and angular acceleration In Figure 1, a particle b is rotating about an axis along a circular path with radius r. The radius sweeps
More informationTwoDimensional Rotational Kinematics
TwoDimensional Rotational Kinematics Rigid Bodies A rigid body is an extended object in which the distance between any two points in the object is constant in time. Springs or human bodies are nonrigid
More informationCircular Motion, Pt 2: Angular Dynamics. Mr. Velazquez AP/Honors Physics
Circular Motion, Pt 2: Angular Dynamics Mr. Velazquez AP/Honors Physics Formulas: Angular Kinematics (θ must be in radians): s = rθ Arc Length 360 = 2π rads = 1 rev ω = θ t = v t r Angular Velocity α av
More information= o + t = ot + ½ t 2 = o + 2
Chapters 89 Rotational Kinematics and Dynamics Rotational motion Rotational motion refers to the motion of an object or system that spins about an axis. The axis of rotation is the line about which the
More informationCIRCULAR MOTION AND ROTATION
1. UNIFORM CIRCULAR MOTION So far we have learned a great deal about linear motion. This section addresses rotational motion. The simplest kind of rotational motion is an object moving in a perfect circle
More informationRotation. Rotational Variables
Rotation Rigid Bodies Rotation variables Constant angular acceleration Rotational KE Rotational Inertia Rotational Variables Rotation of a rigid body About a fixed rotation axis. Rigid Body an object that
More informationPLANAR KINETICS OF A RIGID BODY FORCE AND ACCELERATION
PLANAR KINETICS OF A RIGID BODY FORCE AND ACCELERATION I. Moment of Inertia: Since a body has a definite size and shape, an applied nonconcurrent force system may cause the body to both translate and rotate.
More informationLesson Rigid Body Dynamics
Lesson 8 Rigid Body Dynamics Lesson 8 Outline Problem definition and motivations Dynamics of rigid bodies The equation of unconstrained motion (ODE) User and time control Demos / tools / libs Rigid Body
More informationChapter 10. Rotation of a Rigid Object about a Fixed Axis
Chapter 10 Rotation of a Rigid Object about a Fixed Axis Angular Position Axis of rotation is the center of the disc Choose a fixed reference line. Point P is at a fixed distance r from the origin. A small
More informationTorque and Rotation Lecture 7
Torque and Rotation Lecture 7 ˆ In this lecture we finally move beyond a simple particle in our mechanical analysis of motion. ˆ Now we consider the socalled rigid body. Essentially, a particle with extension
More informationHandout 7: Torque, angular momentum, rotational kinetic energy and rolling motion. Torque and angular momentum
Handout 7: Torque, angular momentum, rotational kinetic energy and rolling motion Torque and angular momentum In Figure, in order to turn a rod about a fixed hinge at one end, a force F is applied at a
More informationGeneral Definition of Torque, final. Lever Arm. General Definition of Torque 7/29/2010. Units of Chapter 10
Units of Chapter 10 Determining Moments of Inertia Rotational Kinetic Energy Rotational Plus Translational Motion; Rolling Why Does a Rolling Sphere Slow Down? General Definition of Torque, final Taking
More informationLecture PowerPoints. Chapter 10 Physics for Scientists and Engineers, with Modern Physics, 4 th edition Giancoli
Lecture PowerPoints Chapter 10 Physics for Scientists and Engineers, with Modern Physics, 4 th edition Giancoli 2009 Pearson Education, Inc. This work is protected by United States copyright laws and is
More informationRigid body simulation. Once we consider an object with spatial extent, particle system simulation is no longer sufficient
Rigid body dynamics Rigid body simulation Once we consider an object with spatial extent, particle system simulation is no longer sufficient Rigid body simulation Unconstrained system no contact Constrained
More informationPart 8: Rigid Body Dynamics
Document that contains homework problems. Comment out the solutions when printing off for students. Part 8: Rigid Body Dynamics Problem 1. Inertia review Find the moment of inertia for a thin uniform rod
More informationTranslational vs Rotational. m x. Connection Δ = = = = = = Δ = = = = = = Δ =Δ = = = = = 2 / 1/2. Work
Translational vs Rotational / / 1/ Δ m x v dx dt a dv dt F ma p mv KE mv Work Fd / / 1/ θ ω θ α ω τ α ω ω τθ Δ I d dt d dt I L I KE I Work / θ ω α τ Δ Δ c t s r v r a v r a r Fr L pr Connection Translational
More information1.1. Rotational Kinematics Description Of Motion Of A Rotating Body
PHY 19 PHYSICS III 1. Moment Of Inertia 1.1. Rotational Kinematics Description Of Motion Of A Rotating Body 1.1.1. Linear Kinematics Consider the case of linear kinematics; it concerns the description
More informationRotation. EMU Physics Department. Ali ÖVGÜN.
Rotation Ali ÖVGÜN EMU Physics Department www.aovgun.com Rotational Motion Angular Position and Radians Angular Velocity Angular Acceleration Rigid Object under Constant Angular Acceleration Angular and
More informationWe define angular displacement, θ, and angular velocity, ω. What's a radian?
We define angular displacement, θ, and angular velocity, ω Units: θ = rad ω = rad/s What's a radian? Radian is the ratio between the length of an arc and its radius note: counterclockwise is + clockwise
More information14. Rotational Kinematics and Moment of Inertia
14. Rotational Kinematics and Moment of nertia A) Overview n this unit we will introduce rotational motion. n particular, we will introduce the angular kinematic variables that are used to describe the
More informationChap10. Rotation of a Rigid Object about a Fixed Axis
Chap10. Rotation of a Rigid Object about a Fixed Axis Level : AP Physics Teacher : Kim 10.1 Angular Displacement, Velocity, and Acceleration  A rigid object rotating about a fixed axis through O perpendicular
More informationRotational Motion and Torque
Rotational Motion and Torque Introduction to Angular Quantities Sections 8 to 82 Introduction Rotational motion deals with spinning objects, or objects rotating around some point. Rotational motion is
More informationGeneral Physics I. Lecture 8: Rotation of a Rigid Object About a Fixed Axis. Prof. WAN, Xin ( 万歆 )
General Physics I Lecture 8: Rotation of a Rigid Object About a Fixed Axis Prof. WAN, Xin ( 万歆 ) xinwan@zju.edu.cn http://zimp.zju.edu.cn/~xinwan/ New Territory Object In the past, point particle (no rotation,
More informationDynamics. 1 Copyright c 2015 Roderic Grupen
Dynamics The branch of physics that treats the action of force on bodies in motion or at rest; kinetics, kinematics, and statics, collectively. Websters dictionary Outline Conservation of Momentum Inertia
More informationRotational Kinetic Energy
Lecture 17, Chapter 10: Rotational Energy and Angular Momentum 1 Rotational Kinetic Energy Consider a rigid body rotating with an angular velocity ω about an axis. Clearly every point in the rigid body
More informationRotational Kinematics and Dynamics. UCVTS AIT Physics
Rotational Kinematics and Dynamics UCVTS AIT Physics Angular Position Axis of rotation is the center of the disc Choose a fixed reference line Point P is at a fixed distance r from the origin Angular Position,
More informationDEVIL PHYSICS BADDEST CLASS ON CAMPUS IB PHYSICS
DEVIL PHYSICS BADDEST CLASS ON CAMPUS IB PHYSICS OPTION B1A: ROTATIONAL DYNAMICS Essential Idea: The basic laws of mechanics have an extension when equivalent principles are applied to rotation. Actual
More informationLecture 13 REVIEW. Physics 106 Spring What should we know? What should we know? Newton s Laws
Lecture 13 REVIEW Physics 106 Spring 2006 http://web.njit.edu/~sirenko/ What should we know? Vectors addition, subtraction, scalar and vector multiplication Trigonometric functions sinθ, cos θ, tan θ,
More informationKinematics (special case) Dynamics gravity, tension, elastic, normal, friction. Energy: kinetic, potential gravity, spring + work (friction)
Kinematics (special case) a = constant 1D motion 2D projectile Uniform circular Dynamics gravity, tension, elastic, normal, friction Motion with a = constant Newton s Laws F = m a F 12 = F 21 Time & Position
More informationPhysics. TOPIC : Rotational motion. 1. A shell (at rest) explodes in to smalll fragment. The C.M. of mass of fragment will move with:
TOPIC : Rotational motion Date : Marks : 120 mks Time : ½ hr 1. A shell (at rest) explodes in to smalll fragment. The C.M. of mass of fragment will move with: a) zero velocity b) constantt velocity c)
More informationPhysics of Rotation. Physics 109, Introduction To Physics Fall 2017
Physics of Rotation Physics 109, Introduction To Physics Fall 017 Outline Next two lab periods Rolling without slipping Angular Momentum Comparison with Translation New Rotational Terms Rotational and
More informationRotational Motion, Torque, Angular Acceleration, and Moment of Inertia. 8.01t Nov 3, 2004
Rotational Motion, Torque, Angular Acceleration, and Moment of Inertia 8.01t Nov 3, 2004 Rotation and Translation of Rigid Body Motion of a thrown object Translational Motion of the Center of Mass Total
More informationChapter 6: Momentum Analysis
61 Introduction 62Newton s Law and Conservation of Momentum 63 Choosing a Control Volume 64 Forces Acting on a Control Volume 65Linear Momentum Equation 66 Angular Momentum 67 The Second Law of
More informationPhys101 Lectures 19, 20 Rotational Motion
Phys101 Lectures 19, 20 Rotational Motion Key points: Angular and Linear Quantities Rotational Dynamics; Torque and Moment of Inertia Rotational Kinetic Energy Ref: 101,2,3,4,5,6,8,9. Page 1 Angular Quantities
More informationChapter 8 Lecture. Pearson Physics. Rotational Motion and Equilibrium. Prepared by Chris Chiaverina Pearson Education, Inc.
Chapter 8 Lecture Pearson Physics Rotational Motion and Equilibrium Prepared by Chris Chiaverina Chapter Contents Describing Angular Motion Rolling Motion and the Moment of Inertia Torque Static Equilibrium
More informationAngular Displacement (θ)
Rotational Motion Angular Displacement, Velocity, Acceleration Rotation w/constant angular acceleration Linear vs. Angular Kinematics Rotational Energy Parallel Axis Thm. Angular Displacement (θ) Angular
More informationChapters 10 & 11: Rotational Dynamics Thursday March 8 th
Chapters 10 & 11: Rotational Dynamics Thursday March 8 th Review of rotational kinematics equations Review and more on rotational inertia Rolling motion as rotation and translation Rotational kinetic energy
More informationIn this section of notes, we look at the calculation of forces and torques for a manipulator in two settings:
Introduction Up to this point we have considered only the kinematics of a manipulator. That is, only the specification of motion without regard to the forces and torques required to cause motion In this
More informationUniform Circular Motion
Uniform Circular Motion Motion in a circle at constant angular speed. ω: angular velocity (rad/s) Rotation Angle The rotation angle is the ratio of arc length to radius of curvature. For a given angle,
More informationChapter 8 Rotational Motion
Chapter 8 Rotational Motion Assignment 8 Textbook (Giancoli, 6 th edition), Chapter 78: Due on Thursday, November 13, 2008  Problem 28  page 189 of the textbook  Problem 40  page 190 of the textbook
More informationRigid body dynamics. Basilio Bona. DAUIN  Politecnico di Torino. October 2013
Rigid body dynamics Basilio Bona DAUIN  Politecnico di Torino October 2013 Basilio Bona (DAUIN  Politecnico di Torino) Rigid body dynamics October 2013 1 / 16 Multiple pointmass bodies Each mass is
More informationAngular Motion, General Notes
Angular Motion, General Notes! When a rigid object rotates about a fixed axis in a given time interval, every portion on the object rotates through the same angle in a given time interval and has the same
More informationRotational Kinematics
Rotational Kinematics Rotational Coordinates Ridged objects require six numbers to describe their position and orientation: 3 coordinates 3 axes of rotation Rotational Coordinates Use an angle θ to describe
More information6. 3D Kinematics DE2EA 2.1: M4DE. Dr Connor Myant
DE2EA 2.1: M4DE Dr Connor Myant 6. 3D Kinematics Comments and corrections to connor.myant@imperial.ac.uk Lecture resources may be found on Blackboard and at http://connormyant.com Contents ThreeDimensional
More informationRotational Motion About a Fixed Axis
Rotational Motion About a Fixed Axis Vocabulary rigid body axis of rotation radian average angular velocity instantaneous angular average angular Instantaneous angular frequency velocity acceleration acceleration
More informationRotational Motion About a Fixed Axis
Rotational Motion About a Fixed Axis Vocabulary rigid body axis of rotation radian average angular velocity instantaneous angular average angular Instantaneous angular frequency velocity acceleration acceleration
More informationPhysics 121. March 18, Physics 121. March 18, Course Announcements. Course Information. Topics to be discussed today:
Physics 121. March 18, 2008. Physics 121. March 18, 2008. Course Information Topics to be discussed today: Variables used to describe rotational motion The equations of motion for rotational motion Course
More informationMotion in Space. MATH 311, Calculus III. J. Robert Buchanan. Fall Department of Mathematics. J. Robert Buchanan Motion in Space
Motion in Space MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Fall 2011 Background Suppose the position vector of a moving object is given by r(t) = f (t), g(t), h(t), Background
More informationInelastic Collisions. Experiment Number 8 Physics 109 Fall 2017
Inelastic Collisions Experiment Number 8 Physics 109 Fall 2017 Midterm Exam Scores 6 5 4 Number 3 2 1 0 049 5059 6069 7079 Score Range 8089 90100 Outline Ballistic Pendulum Physics of Rotation Angular
More informationGeneral Physics I. Lecture 8: Rotation of a Rigid Object About a Fixed Axis. Prof. WAN, Xin ( 万歆 )
General Physics I Lecture 8: Rotation of a Rigid Object About a Fixed Axis Prof. WAN, Xin ( 万歆 ) xinwan@zju.edu.cn http://zimp.zju.edu.cn/~xinwan/ New Territory Object In the past, point particle (no rotation,
More information9 Kinetics of 3D rigid bodies  rotating frames
9 Kinetics of 3D rigid bodies  rotating frames 9. Consider the two gears depicted in the figure. The gear B of radius R B is fixed to the ground, while the gear A of mass m A and radius R A turns freely
More informationChapter 8: Momentum, Impulse, & Collisions. Newton s second law in terms of momentum:
linear momentum: Chapter 8: Momentum, Impulse, & Collisions Newton s second law in terms of momentum: impulse: Under what SPECIFIC condition is linear momentum conserved? (The answer does not involve collisions.)
More informationMoment of Inertia Race
Review Two points, A and B, are on a disk that rotates with a uniform speed about an axis. Point A is closer to the axis than point B. Which of the following is NOT true? 1. Point B has the greater tangential
More informationMoments of Inertia (7 pages; 23/3/18)
Moments of Inertia (7 pages; 3/3/8) () Suppose that an object rotates about a fixed axis AB with angular velocity θ. Considering the object to be made up of particles, suppose that particle i (with mass
More informationChapter 12: Rotation of Rigid Bodies. Center of Mass Moment of Inertia Torque Angular Momentum Rolling Statics
Chapter 1: Rotation of Rigid Bodies Center of Mass Moment of Inertia Torque Angular Momentum Rolling Statics Translational vs Rotational / / 1/ m x v dx dt a dv dt F ma p mv KE mv Work Fd P Fv / / 1/ I
More informationCHAPTER 10 ROTATION OF A RIGID OBJECT ABOUT A FIXED AXIS WENBIN JIAN ( 簡紋濱 ) DEPARTMENT OF ELECTROPHYSICS NATIONAL CHIAO TUNG UNIVERSITY
CHAPTER 10 ROTATION OF A RIGID OBJECT ABOUT A FIXED AXIS WENBIN JIAN ( 簡紋濱 ) DEPARTMENT OF ELECTROPHYSICS NATIONAL CHIAO TUNG UNIVERSITY OUTLINE 1. Angular Position, Velocity, and Acceleration 2. Rotational
More information1/30. Rigid Body Rotations. Dave Frank
. 1/3 Rigid Body Rotations Dave Frank A Point Particle and Fundamental Quantities z 2/3 m v ω r y x Angular Velocity v = dr dt = ω r Kinetic Energy K = 1 2 mv2 Momentum p = mv Rigid Bodies We treat a rigid
More informationPhysics 2A Chapter 10  Rotational Motion Fall 2018
Physics A Chapter 10  Rotational Motion Fall 018 These notes are five pages. A quick summary: The concepts of rotational motion are a direct mirror image of the same concepts in linear motion. Follow
More informationPSI AP Physics I Rotational Motion
PSI AP Physics I Rotational Motion MultipleChoice questions 1. Which of the following is the unit for angular displacement? A. meters B. seconds C. radians D. radians per second 2. An object moves from
More informationPSI AP Physics I Rotational Motion
PSI AP Physics I Rotational Motion MultipleChoice questions 1. Which of the following is the unit for angular displacement? A. meters B. seconds C. radians D. radians per second 2. An object moves from
More informationBig Idea 4: Interactions between systems can result in changes in those systems. Essential Knowledge 4.D.1: Torque, angular velocity, angular
Unit 7: Rotational Motion (angular kinematics, dynamics, momentum & energy) Name: Big Idea 3: The interactions of an object with other objects can be described by forces. Essential Knowledge 3.F.1: Only
More informationz F 3 = = = m 1 F 1 m 2 F 2 m 3  Linear Momentum dp dt F net = d P net = d p 1 dt d p n dt  Conservation of Linear Momentum Δ P = 0
F 1 m 2 F 2 x m 1 O z F 3 m 3 y Ma com = F net F F F net, x net, y net, z = = = Ma Ma Ma com, x com, y com, z p = mv  Linear Momentum F net = dp dt F net = d P dt = d p 1 dt +...+ d p n dt Δ P = 0  Conservation
More informationPhysics 1A Lecture 10B
Physics 1A Lecture 10B "Sometimes the world puts a spin on life. When our equilibrium returns to us, we understand more because we've seen the whole picture. Davis Barton Cross Products Another way to
More informationUniform Circular Motion:Circular motion is said to the uniform if the speed of the particle (along the circular path) remains constant.
Circular Motion: Uniform Circular Motion:Circular motion is said to the uniform if the speed of the particle (along the circular path) remains constant. Angular Displacement: Scalar form:?s = r?θ Vector
More informationAdvanced Higher Physics. Rotational Motion. Problems: Solutions
Advanced Higher Physics Rotational Motion Problems: Solutions AH Physics: Rotational Motion Problems Solutions Page 1 013 AH Physics: Rotational Motion Problems Solutions Page 013 TUTORIAL 1.0 Equations
More informationRotation. Kinematics Rigid Bodies Kinetic Energy. Torque Rolling. featuring moments of Inertia
Rotation Kinematics Rigid Bodies Kinetic Energy featuring moments of Inertia Torque Rolling Angular Motion We think about rotation in the same basic way we do about linear motion How far does it go? How
More informationSlide 1 / 37. Rotational Motion
Slide 1 / 37 Rotational Motion Slide 2 / 37 Angular Quantities An angle θ can be given by: where r is the radius and l is the arc length. This gives θ in radians. There are 360 in a circle or 2π radians.
More informationManipulator Dynamics 2. Instructor: Jacob Rosen Advanced Robotic  MAE 263D  Department of Mechanical & Aerospace Engineering  UCLA
Manipulator Dynamics 2 Forward Dynamics Problem Given: Joint torques and links geometry, mass, inertia, friction Compute: Angular acceleration of the links (solve differential equations) Solution Dynamic
More informationCircular motion. Aug. 22, 2017
Circular motion Aug. 22, 2017 Until now, we have been observers to Newtonian physics through inertial reference frames. From our discussion of Newton s laws, these are frames which obey Newton s first
More informationRotation Basics. I. Angular Position A. Background
Rotation Basics I. Angular Position A. Background Consider a student who is riding on a merrygoround. We can represent the student s location by using either Cartesian coordinates or by using cylindrical
More information2007 Problem Topic Comment 1 Kinematics Positiontime equation Kinematics 7 2 Kinematics Velocitytime graph Dynamics 6 3 Kinematics Average velocity
2007 Problem Topic Comment 1 Kinematics Positiontime equation Kinematics 7 2 Kinematics Velocitytime graph Dynamics 6 3 Kinematics Average velocity Energy 7 4 Kinematics Free fall Collisions 3 5 Dynamics
More informationRotational Motion. Every quantity that we have studied with translational motion has a rotational counterpart
Rotational Motion & Angular Momentum Rotational Motion Every quantity that we have studied with translational motion has a rotational counterpart TRANSLATIONAL ROTATIONAL Displacement x Angular Displacement
More informationRotational Motion. Chapter 4. P. J. Grandinetti. Sep. 1, Chem P. J. Grandinetti (Chem. 4300) Rotational Motion Sep.
Rotational Motion Chapter 4 P. J. Grandinetti Chem. 4300 Sep. 1, 2017 P. J. Grandinetti (Chem. 4300) Rotational Motion Sep. 1, 2017 1 / 76 Angular Momentum The angular momentum of a particle with respect
More information8.012 Physics I: Classical Mechanics Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 8.012 Physics I: Classical Mechanics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MASSACHUSETTS INSTITUTE
More informationLecture Outline Chapter 10. Physics, 4 th Edition James S. Walker. Copyright 2010 Pearson Education, Inc.
Lecture Outline Chapter 10 Physics, 4 th Edition James S. Walker Chapter 10 Rotational Kinematics and Energy Units of Chapter 10 Angular Position, Velocity, and Acceleration Rotational Kinematics Connections
More informationRotation of Rigid Objects
Notes 12 Rotation and Extended Objects Page 1 Rotation of Rigid Objects Real objects have "extent". The mass is spread out over discrete or continuous positions. THERE IS A DISTRIBUTION OF MASS TO "AN
More informationDYNAMICS OF RIGID BODIES
DYNAMICS OF RIGID BODIES Measuring angles in radian Define the value of an angle θ in radian as θ = s r, or arc length s = rθ a pure number, without dimension independent of radius r of the circle one
More informationRelating Translational and Rotational Variables
Relating Translational and Rotational Variables Rotational position and distance moved s = θ r (only radian units) Rotational and translational speed d s v = dt v = ω r = ds dt = d θ dt r Relating period
More informationLecture 6 Physics 106 Spring 2006
Lecture 6 Physics 106 Spring 2006 Angular Momentum Rolling Angular Momentum: Definition: Angular Momentum for rotation System of particles: Torque: l = r m v sinφ l = I ω [kg m 2 /s] http://web.njit.edu/~sirenko/
More informationChapter 10. Rotation
Chapter 10 Rotation Rotation Rotational Kinematics: Angular velocity and Angular Acceleration Rotational Kinetic Energy Moment of Inertia Newton s nd Law for Rotation Applications MFMcGrawPHY 45 Chap_10HaRotationRevised
More information1. Which of the following is the unit for angular displacement? A. Meters B. Seconds C. Radians D. Radian per second E. Inches
AP Physics B Practice Questions: Rotational Motion MultipleChoice Questions 1. Which of the following is the unit for angular displacement? A. Meters B. Seconds C. Radians D. Radian per second E. Inches
More informationUniform circular motion (UCM) is the motion of an object in a perfect circle with a constant or uniform speed.
Uniform circular motion (UCM) is the motion of an object in a perfect circle with a constant or uniform speed. 1. Distance around a circle? circumference 2. Distance from one side of circle to the opposite
More informationMET 327 APPLIED ENGINEERING II (DYNAMICS) 1D Dynamic System Equation of Motion (EOM)
Handout #1 by Hejie Lin MET 327 APPLIED ENGINEERING II (DYNAMICS) 1. Introduction to Statics and Dynamics 1.1 Statics vs. Dynamics 1 Ch 9 Moment of Inertia A dynamic system is characterized with mass (M),
More informationPhysics 1A. Lecture 10B
Physics 1A Lecture 10B Review of Last Lecture Rotational motion is independent of translational motion A free object rotates around its center of mass Objects can rotate around different axes Natural unit
More informationPhys 7221 Homework # 8
Phys 71 Homework # 8 Gabriela González November 15, 6 Derivation 56: Torque free symmetric top In a torque free, symmetric top, with I x = I y = I, the angular velocity vector ω in body coordinates with
More informationClassical Mechanics. Luis Anchordoqui
1 Rigid Body Motion Inertia Tensor Rotational Kinetic Energy Principal Axes of Rotation Steiner s Theorem Euler s Equations for a Rigid Body Eulerian Angles Review of Fundamental Equations 2 Rigid body
More informationparticle p = m v F ext = d P = M d v cm dt
Lecture 11: Momentum and Collisions; Introduction to Rotation 1 REVIEW: (Chapter 8) LINEAR MOMENTUM and COLLISIONS The first new physical quantity introduced in Chapter 8 is Linear Momentum Linear Momentum
More informationAdvanced Higher Physics. Rotational Motion
Wallace Hall Academy Physics Department Advanced Higher Physics Rotational Motion Solutions AH Physics: Rotational Motion Problems Solutions Page 1 013 TUTORIAL 1.0 Equations of motion 1. (a) v = ds, ds
More informationω = 0 a = 0 = α P = constant L = constant dt = 0 = d Equilibrium when: τ i = 0 τ net τ i Static Equilibrium when: F z = 0 F net = F i = ma = d P
Equilibrium when: F net = F i τ net = τ i a = 0 = α dp = 0 = d L = ma = d P = 0 = I α = d L = 0 P = constant L = constant F x = 0 τ i = 0 F y = 0 F z = 0 Static Equilibrium when: P = 0 L = 0 v com = 0
More informationChapter 6: Momentum Analysis of Flow Systems
Chapter 6: Momentum Analysis of Flow Systems Introduction Fluid flow problems can be analyzed using one of three basic approaches: differential, experimental, and integral (or control volume). In Chap.
More informationTranslational and Rotational Dynamics!
Translational and Rotational Dynamics Robert Stengel Robotics and Intelligent Systems MAE 345, Princeton University, 217 Copyright 217 by Robert Stengel. All rights reserved. For educational use only.
More information