Lecture II: Rigid-Body Physics
|
|
- Marvin Dorsey
- 5 years ago
- Views:
Transcription
1
2 Rigid-Body Motion Previously: Point dimensionless objects moving through a trajectory. Today: Objects with dimensions, moving as one piece. 2
3 Rigid-Body Kinematics Objects as sets of points. Relative distances between all points are invariant to rigid movement. Free body movement: around the center of mass (COM). Movement has two components: Linear trajectory of a central point. Relative rotation around the point. 3
4 Mass The measure of the amount of matter in the volume of an object: m = # ρ dv ( ρ : the density of each point the object volume V. dv: the volume element. Equivalently: a measure of resistance to motion or change in motion. 4
5 Mass For a 3D object, mass is the integral over its volume: m = # # # ρ(x, y, z) dx dy dz For uniform density (ρ constant): m = ρ V 5
6 Center of Mass The center of mass (COM) is the average point of the object, weighted by density: COM = 1 m p: point coordinates. # ρ 4 p dv ( Point of balance for the object. Uniform density: COM ó centroid. COM 6
7 Center of Mass of System Sets of bodies have a mutual center of mass: m 8 : mass of each body. 9 COM = 1 m 7 m 8 p 8 8:; p 8 : location of individual COM. m= 9 m 8 8:;. Example: two spheres in 1D x 2 x =>? = m ;x ; + m A x A m ; + m A x 1 m 1 x COM COM m 2 7
8 Center of Mass Quite easy to determine for primitive shapes What about complex surface based models? 8
9 Rotational Motion P a point on to the object. C is the center of rotation. Distance vector r = P C. u C θ r P v s P r = r : the distance. Object rotates ó P travels along a circular path. Unit-length axis of rotation: u. Here, u=z ( out from the screen). Rotation: counterclockwise. right-hand rule. 9
10 Angular Displacement Point p covers linear distance s. θ is the angular displacement of the object: u C θ r P v s P θ = s/r s: arc length. Unit is radian (rad) 1 radian = angle for arc length 1 at a distance 1. 10
11 Angular Velocity Angular speed: the rate of change of the angular displacement: unit is UVW XYZ. ω = dθ dt The angular velocity vector is collinear with the rotation axis: ω = ωu 11
12 Angular Acceleration Angular acceleration: the rate of change of the angular velocity: α = dω dt Paralleling definition of linear acceleration. Unit is rad/s A 12
13 Tangential and Angular Velocities Every point moves with the same angular velocity. Direction of vector: u. Tangential velocity vector: Or: v = ω r ω = r v r A u C θ r P v s P ω = ^_ (abs. values) U due to s = θ 4 r. V(t) C P(t + t) P(t) Only the tangential part matters! T(t) 13
14 Dynamics The centripetal force creates curved motion. In the direction of (negative) r Object is in orbit. Constant force ó circular rotation with constant tangential velocity. Why? 14
15 Tangential & Centripetal Accelerations Tangential acceleration α holds: a = α r cf. velocity equation v = ω r. The centripetal acceleration drives the rotational movement: a 9 = ^b U r = ωa r. What is the centrifugal force? 15
16 Angular Momentum Linear motion è linear momentum: p = mv. Rotational motion è angular momentum about any fixed relative point (to which r is measured): L = # r p ρdv unit is N 4 m 4 s ρdv = dm (mass element) Angular momentum is conserved! Just like the linear momentum. Caveat: conserved w.r.t. the same point. ( 16
17 Angular Momentum Plugging in angular velocity: r p dv = r v dm = r ω r dm Integrating, we get: L = # r ω r dm ( Note: The angular momentum and the angular velocity are not generally collinear! 17
18 Define: r = x y z Moment of Inertia and ω = ω f ω g ω h. For a single rotating body: the angular velocity is constant. We get: L = # r ω r dm ( = # y A + z A ω f xyω g xzω h yxω f + z A + x A ω g yzω h zxω f zyω g + (x A + y A )ω h dm = I ff I fg I fh I gf I gg I gh I hf I hg I hh ω f ω g ω h. Note: replacing integral with a (constant) matrix operating on a vector! 18
19 Momentum and Inertia r = x y z The inertia tensor I only depends on the geometry of the object and the relative fixed point (often, COM): I ff = # y A + z A dm I fg = I gf = # xy dm I gg = # z A + x A dm I fh = I hf = # xz dm I hh = # x A + y A dm I gh = I hg = # yz dm 19
20 The Inertia Tensor Compact form: # y A + z A dm # xy dm # xz dm I = # xy dm # z A + x A dm # yz dm # xz dm # yz dm # x A + y A dm The diagonal elements are called the (principal) moment of inertia. The off-diagonal elements are called products of inertia. 20
21 The Inertia Tensor Equivalently, we separate mass elements to density and volume elements: I = # ρ x, y, z ( y A + z A xy xz xy z A + x A yz dx dy dz xz yz x A + y A The diagonal elements: distances to the respective principal axes. The non-diagonal elements: products of the perpendicular distances to the respective planes. 21
22 Moment of Inertia The moment of inertia I j, with respect to a rotation axis u, measures how much the mass spreads out : I j = # r j A dm ( r j : perpendicular distance to axis. Through the central rotation origin point. Measures ability to resist change in rotational motion. The angular equivalent to mass! 22
23 Moment and Tensor We have: r j A = u r A = u k I(q)u for any point q. (Remember: r is distance to origin). Thus: I j = u k I(q)u dm=u k Iu? The scalar angular momentum around the axis is then L j = I j ω. Reducible to a planar problem (axis as Z axis). 23
24 For a mass point: I = m 4 r j A Moment of Inertia For a collection of mass points: I = m 8 r 8 A 8 r A r m m r ; ; m A For a continuous mass distribution on the plane: I = r A? j dm dm r o r m o 24
25 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 A + y A + z A = R A 25
26 Inertia of Primitive Shapes Distance to axis of rotation is the radius of the disc at the cross section along z: r A = x A + y A = R A z A. Summing moments of inertia of small cylinders of inertia I q along the z-axis: = Ub r A We get: v di q = 1 2 ra dm = 1 2 ra ρdv = 1 2 ra ρπr A dz v I q = ; ρπ A wv ru dz = ; ρπ A wv RA z A A dz = ; ρπ [R u z 2R A z o 3 A + z z 5] v wv = ρπ R z. As m = ρ 4 3 πr o, we finally obtain: I q = A z mra. 26
27 Inertia of Primitive Shapes Solid sphere, radius r and mass m: I = 2 5 mra mra mra Hollow sphere, radius r and mass m: I = 2 3 mra mra mra z y x 27
28 Inertia of Primitive Shapes Solid ellipsoid, semi-axes a, b, c and mass m: I = 1 5 m(ba +c A ) m(aa +c A ) m(aa +b A ) z y x Solid box, width w, height h, depth d and mass m: h w d I = 1 12 m(ha +d A ) m(wa +d A ) m(wa +h A ) 28
29 Inertia of Primitive Shapes Solid cylinder, radius r, height h and mass m: h I = 1 12 m(3ra +h A ) m(3ra +h A ) mra Hollow cylinder, radius r, height h and mass m: h I = 1 12 m(6ra +h A ) m(6ra +h A ) mr A 29
30 Parallel-Axis Theorem The object does not necessarily rotate around the center of mass. Some point can be fixed! parallel axis theorem: I^: inertia around axis u. I^ = I =>? + md A I =>? inertia about a parallel axis through the COM. d is the distance between the axes. 30
31 Parallel-Axis Theorem More generally, for point displacements: d f, d g, d h I ff = # y A + z A dm + md f A I fg = # xy dm + md f d g I gg = # z A + x A dm + md g A I fh = # xz dm + md f d h I hh = # x A + y A dm + md h A I gh = # yz dm + md g d h 31
32 Perpendicular-Axis 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 h = I f + I g for any planar object I h = 2I f = 2I g for symmetrical objects 32
33 Reference Frame The representation of the inertia tensor is coordinate dependent. The physical effect should be invariant to coordinates! If transformation R changes bases from body to world coordinate, the inertia tensor in world space is: I U W = R 4 I Wg 4 R k The moment of inertia is invariant! Why? 33
34 Torque A force F applied at a distance r from a (fixed) center of mass. Tangential part causes tangential acceleration: F = m 4 a The torque τ is defined as: τ = r F So we get τ = m r α r = m r A α unit is N m rotates an object about its axis of rotation. 34
35 Newton s Second Law The law F = m 4 a has an equivalent with the inertia tensor and torque: τ = Iα Force ó linear acceleration Torque ó angular acceleration 35
36 Torque and Angular Momentum Reminder: in the linear case: F = W (p is the linear WŒ momentum). Similarly with torque and angular momentum: dl dt = dr dp p + r dt dt = v mv + r F = 0 + τ Kinematics: dl dt = d(iω) = I dω = Iα = τ dt dt Force ó derivative of linear momentum. Torque ó derivative of angular momentum. 36
37 Rotational Kinetic Energy Translating energy formulas to rotational motion. The rotational kinetic energy is defined as: E ŽU = 1 2 ωk I ω 37
38 Conservation of Mechanical Energy Adding rotational kinetic energy E ŽŒ t + t + E t + t + E ŽU t + t = E ŽŒ t + E t + E ŽU (t) + E > E ŽŒ is the translational kinetic energy. E is the potential energy. E ŽU is the rotational kinetic energy. E > the lost energies (surface friction, air resistance etc.). 38
39 Impulse We may apply off-center 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 39
40 Rigid Body Forces A force can be applied anywhere on the object, producing also a rotational motion. F COM α a 40 40
41 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 41 41
42 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
Rotational & Rigid-Body Mechanics. Lectures 3+4
Rotational & Rigid-Body Mechanics Lectures 3+4 Rotational Motion So far: point objects moving through a trajectory. Next: moving actual dimensional objects and rotating them. 2 Circular Motion - Definitions
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 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 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 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 informationTwo-Dimensional Rotational Kinematics
Two-Dimensional 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 non-rigid
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 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 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 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 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 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 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 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 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 information= o + t = ot + ½ t 2 = o + 2
Chapters 8-9 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 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 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 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 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 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 informationRotational kinematics
Rotational kinematics Suppose you cut a circle out of a piece of paper and then several pieces of string which are just as long as the radius of the paper circle. If you then begin to lay these pieces
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 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 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 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 informationAngular Displacement. θ i. 1rev = 360 = 2π rads. = "angular displacement" Δθ = θ f. π = circumference. diameter
Rotational Motion Angular Displacement π = circumference diameter π = circumference 2 radius circumference = 2πr Arc length s = rθ, (where θ in radians) θ 1rev = 360 = 2π rads Δθ = θ f θ i = "angular displacement"
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 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 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 informationDEVIL PHYSICS BADDEST CLASS ON CAMPUS IB PHYSICS
DEVIL PHYSICS BADDEST CLASS ON CAMPUS IB PHYSICS OPTION B-1A: ROTATIONAL DYNAMICS Essential Idea: The basic laws of mechanics have an extension when equivalent principles are applied to rotation. Actual
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 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 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 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 informationAngular velocity and angular acceleration CHAPTER 9 ROTATION. Angular velocity and angular acceleration. ! equations of rotational motion
Angular velocity and angular acceleration CHAPTER 9 ROTATION! r i ds i dθ θ i Angular velocity and angular acceleration! equations of rotational motion Torque and Moment of Inertia! Newton s nd Law for
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 informationMotion Of An Extended Object. Physics 201, Lecture 17. Translational Motion And Rotational Motion. Motion of Rigid Object: Translation + Rotation
Physics 01, Lecture 17 Today s Topics q Rotation of Rigid Object About A Fixed Axis (Chap. 10.1-10.4) n Motion of Extend Object n Rotational Kinematics: n Angular Velocity n Angular Acceleration q Kinetic
More informationChapter 8- Rotational Motion
Chapter 8- Rotational Motion Assignment 8 Textbook (Giancoli, 6 th edition), Chapter 7-8: Due on Thursday, November 13, 2008 - Problem 28 - page 189 of the textbook - Problem 40 - page 190 of the textbook
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 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 informationRotation Angular Momentum
Rotation Angular Momentum Lana Sheridan De Anza College Nov 28, 2017 Last time rolling motion Overview Definition of angular momentum relation to Newton s 2nd law angular impulse angular momentum of rigid
More informationRotational Motion and Torque
Rotational Motion and Torque Introduction to Angular Quantities Sections 8- to 8-2 Introduction Rotational motion deals with spinning objects, or objects rotating around some point. Rotational motion is
More informationChapter 10 Rotational Kinematics and Energy. Copyright 2010 Pearson Education, Inc.
Chapter 10 Rotational Kinematics and Energy Copyright 010 Pearson Education, Inc. 10-1 Angular Position, Velocity, and Acceleration Copyright 010 Pearson Education, Inc. 10-1 Angular Position, Velocity,
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: center-of-mass motion
More informationConservation of Angular Momentum
Physics 101 Section 3 March 3 rd : Ch. 10 Announcements: Monday s Review Posted (in Plummer s section (4) Today start Ch. 10. Next Quiz will be next week Test# (Ch. 7-9) will be at 6 PM, March 3, Lockett-6
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 informationLab 9 - Rotational Dynamics
145 Name Date Partners Lab 9 - Rotational Dynamics OBJECTIVES To study angular motion including angular velocity and angular acceleration. To relate rotational inertia to angular motion. To determine kinetic
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 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 informationMoment of Inertia & Newton s Laws for Translation & Rotation
Moment of Inertia & Newton s Laws for Translation & Rotation In this training set, you will apply Newton s 2 nd Law for rotational motion: Στ = Σr i F i = Iα I is the moment of inertia of an object: I
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 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 informationPhysics 8, Fall 2011, equation sheet work in progress
1 year 3.16 10 7 s Physics 8, Fall 2011, equation sheet work in progress circumference of earth 40 10 6 m speed of light c = 2.9979 10 8 m/s mass of proton or neutron 1 amu ( atomic mass unit ) = 1 1.66
More informationPhysics 201. Professor P. Q. Hung. 311B, Physics Building. Physics 201 p. 1/1
Physics 201 p. 1/1 Physics 201 Professor P. Q. Hung 311B, Physics Building Physics 201 p. 2/1 Rotational Kinematics and Energy Rotational Kinetic Energy, Moment of Inertia All elements inside the rigid
More informationRolling without slipping Angular Momentum Conservation of Angular Momentum. Physics 201: Lecture 19, Pg 1
Physics 131: Lecture Today s Agenda Rolling without slipping Angular Momentum Conservation o Angular Momentum Physics 01: Lecture 19, Pg 1 Rolling Without Slipping Rolling is a combination o rotation and
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 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 informationRotation Basics. I. Angular Position A. Background
Rotation Basics I. Angular Position A. Background Consider a student who is riding on a merry-go-round. We can represent the student s location by using either Cartesian coordinates or by using cylindrical
More informationFundamentals Physics. Chapter 10 Rotation
Fundamentals Physics Tenth Edition Halliday Chapter 10 Rotation 10-1 Rotational Variables (1 of 15) Learning Objectives 10.01 Identify that if all parts of a body rotate around a fixed axis locked together,
More information16. Rotational Dynamics
6. Rotational Dynamics A Overview In this unit we will address examples that combine both translational and rotational motion. We will find that we will need both Newton s second law and the rotational
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 informationCEE 271: Applied Mechanics II, Dynamics Lecture 24: Ch.17, Sec.1-3
1 / 38 CEE 271: Applied Mechanics II, Dynamics Lecture 24: Ch.17, Sec.1-3 Prof. Albert S. Kim Civil and Environmental Engineering, University of Hawaii at Manoa Tuesday, Nov. 13, 2012 2 / 38 MOMENT OF
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 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. Rotational Motion. Rotational Motion
I. Rotational Kinematics II. Rotational Dynamics (Netwton s Law for Rotation) III. Angular Momentum Conservation 1. Remember how Newton s Laws for translational motion were studied: 1. Kinematics (x =
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 informationPhysics 8, Fall 2013, equation sheet work in progress
(Chapter 1: foundations) 1 year 3.16 10 7 s Physics 8, Fall 2013, equation sheet work in progress circumference of earth 40 10 6 m speed of light c = 2.9979 10 8 m/s mass of proton or neutron 1 amu ( atomic
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 so-called rigid body. Essentially, a particle with extension
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 informationPhysics 5A Final Review Solutions
Physics A Final Review Solutions Eric Reichwein Department of Physics University of California, Santa Cruz November 6, 0. A stone is dropped into the water from a tower 44.m above the ground. Another stone
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 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 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 MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised
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 10 Rotational Kinematics and Energy. Copyright 2010 Pearson Education, Inc.
Chapter 10 Rotational Kinematics and Energy 10-1 Angular Position, Velocity, and Acceleration 10-1 Angular Position, Velocity, and Acceleration Degrees and revolutions: 10-1 Angular Position, Velocity,
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 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 informationChapter 10: Rotation. Chapter 10: Rotation
Chapter 10: Rotation Change in Syllabus: Only Chapter 10 problems (CH10: 04, 27, 67) are due on Thursday, Oct. 14. The Chapter 11 problems (Ch11: 06, 37, 50) will be due on Thursday, Oct. 21 in addition
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 informationAPC PHYSICS CHAPTER 11 Mr. Holl Rotation
APC PHYSICS CHAPTER 11 Mr. Holl Rotation Student Notes 11-1 Translation and Rotation All of the motion we have studied to this point was linear or translational. Rotational motion is the study of spinning
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 informationNotes on Torque. We ve seen that if we define torque as rfsinθ, and the N 2. i i
Notes on Torque We ve seen that if we define torque as rfsinθ, and the moment of inertia as N, we end up with an equation mr i= 1 that looks just like Newton s Second Law There is a crucial difference,
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: 10-1,2,3,4,5,6,8,9. Page 1 Angular Quantities
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 informationChapter 10: Rotation
Chapter 10: Rotation Review of translational motion (motion along a straight line) Position x Displacement x Velocity v = dx/dt Acceleration a = dv/dt Mass m Newton s second law F = ma Work W = Fdcosφ
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 informationr CM = ir im i i m i m i v i (2) P = i
Physics 121 Test 3 study guide Thisisintendedtobeastudyguideforyourthirdtest, whichcoverschapters 9, 10, 12, and 13. Note that chapter 10 was also covered in test 2 without section 10.7 (elastic collisions),
More informationRotational Motion Rotational Kinematics
Rotational Motion Rotational Kinematics Lana Sheridan De Anza College Nov 16, 2017 Last time 3D center of mass example systems of many particles deforming systems Overview rotation relating rotational
More informationRIGID BODY MOTION (Section 16.1)
RIGID BODY MOTION (Section 16.1) There are cases where an object cannot be treated as a particle. In these cases the size or shape of the body must be considered. Rotation of the body about its center
More informationLecture D20-2D Rigid Body Dynamics: Impulse and Momentum
J Peraire 1607 Dynamics Fall 004 Version 11 Lecture D0 - D Rigid Body Dynamics: Impulse and Momentum In lecture D9, we saw the principle of impulse and momentum applied to particle motion This principle
More informationTute M4 : ROTATIONAL MOTION 1
Tute M4 : ROTATIONAL MOTION 1 The equations dealing with rotational motion are identical to those of linear motion in their mathematical form. To convert equations for linear motion to those for rotational
More informationRotational Dynamics continued
Chapter 9 Rotational Dynamics continued 9.4 Newton s Second Law for Rotational Motion About a Fixed Axis ROTATIONAL ANALOG OF NEWTON S SECOND LAW FOR A RIGID BODY ROTATING ABOUT A FIXED AXIS I = ( mr 2
More informationRotation. I. Kinematics - Angular analogs
Rotation I. Kinematics - Angular analogs II. III. IV. Dynamics - Torque and Rotational Inertia Work and Energy Angular Momentum - Bodies and particles V. Elliptical Orbits The student will be able to:
More informationChapter 11. Angular Momentum
Chapter 11 Angular Momentum Angular Momentum Angular momentum plays a key role in rotational dynamics. There is a principle of conservation of angular momentum. In analogy to the principle of conservation
More informationMotion Part 4: Projectile Motion
Motion Part 4: Projectile Motion Last modified: 28/03/2017 CONTENTS Projectile Motion Uniform Motion Equations Projectile Motion Equations Trajectory How to Approach Problems Example 1 Example 2 Example
More information6. 3D Kinematics DE2-EA 2.1: M4DE. Dr Connor Myant
DE2-EA 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 Three-Dimensional
More information2007 Problem Topic Comment 1 Kinematics Position-time equation Kinematics 7 2 Kinematics Velocity-time graph Dynamics 6 3 Kinematics Average velocity
2007 Problem Topic Comment 1 Kinematics Position-time equation Kinematics 7 2 Kinematics Velocity-time graph Dynamics 6 3 Kinematics Average velocity Energy 7 4 Kinematics Free fall Collisions 3 5 Dynamics
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 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 information