any point of co-ordinate + z, can be associated with a point of co-ordinate - z, hence:
|
|
- Calvin Little
- 5 years ago
- Views:
Transcription
1 AMA GEMERY A3 implifications of matrices A3 olid ( ) has a plane of symmetry (,x,y) any point of co-ordinate + z, can be associated with a point of co-ordinate - z, hence: x y yzdm ( ) ( ) D E xzdm ( ) INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 3
2 AMA GEMERY A3 solid ( ) has an axis of symmetry z he axis of symmetry can be viewed as the intersection of two planes of symmetry and the simplification above can be extended, i e, yxdm ( ) yzdm ( ) ( ) x y xydm xzdm ( ) ( ) and the matrix of inertia in becomes diagonal when expressed in ( R) (,x,y,z) INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 4
3 AMA GEMERY A33 solid ( ) has an axis of revolution axis of revolution axis of symmetry + x and y play the same role A he matrix of inertia in reads, when (,z ) is the axis of revolution: [ I ] o,s A I (,) B A, i A C / R A I (,) B B, i INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 5
4 AMA GEMERY IMRAN ALICAIN **** Being ( ), a solid of revolution, show that the matrix of inertia in is the same in ( R ) and ( *) R,,,!"²+!"² ( ) z, z * θ x x * θ y * y INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 6
5 AMA GEMERY A4 Change of basis,%&,%',%' *,%& ( Note that ( is orthonormal: * + (,,,-,,- /,,,,-,,- /,,,, / 3, 3 4, 4 3, 4, 4 -, 3 Remark: It is possible to find a basis R* where the extra-diagonal terms (product of inertia) are nil, it is the principal basis of inertia (A34) INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 7
6 AMA GEMERY A5 Examples of Inertia Matrices Fill the matrices using Binet s notation and according to the symmetry properties of each solid z y, x,5 x z,,5 y INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 8
7 AMA GEMERY,,5,,5 INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 9
8 B KINEIC B KINEIC urpose: Kinetics relates Kinematics to Mass Geometry B Kinetic or Momentum wrench: B Definition: From the velocity field, let us define the following vector field (elemental vectors), dk ( ) ( ) dm where dm is the elemental mass associated with Consider a solid ( ) so that the following global quantities can be defined INA-LYN / CAN nd er Cycle/ Mechanics, nd semester
9 B KINEIC a) um: σ dk ( ) ( ) ( ) ( ) dm which is known as the linear momentum of solid ( ) b) Moment at C C C dk dm ( ) ( ) C ( ) ( ) which is the angular momentum at C INA-LYN / CAN nd er Cycle/ Mechanics, nd semester
10 B KINEIC c) Momentum wrench { K }: he moment at one point L and at another point M are related by the equation: proof: L M + LM σ ( L) ( ) ( L) LM ( ) ( ) L ( ) ( LM + M) ( ) ( ) dm + ( L) LM σ + ( M ) ( ) M ( ) dm dm LM ( ) ( ) dm + ( M ) his expression corresponds to the shifting (change of point) formula for the moment field of a wrench INA-LYN / CAN nd er Cycle/ Mechanics, nd semester
11 B KINEIC B Expression of the linear momentum: By definition σ ( ) ( ) dm hence by introducing the velocity at the centre of mass G: σ σ ( G) + G + G d G d d dm dm dm + M M σ ( G) ( ) G dm INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 3
12 B KINEIC B3 Expression of the angular momentum: B3 Angular momentum of a solid in its centre of inertia: ( G) ( ) G ( G) ( G) ( ) G ( G) ( ) ( ) ( ) ( G) ( G) G dm + G dm + ( ) ( ) G ( G) + L [ I ] ( G, ) G G + G G G, G dm dm dm ( G ) [ I ] G, INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 4
13 B KINEIC B3 Angular momentum of a solid in a point C By using the change of point formula, one gets: G, ( C ) [ I ] { } + CG M ( G ) C G + CG M ( G) KENIG' formula INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 5
14 B KINEIC B33 Angular momentum in a point s fixed in a frame ( R ): tarting from the general definition, s s dm s ( ) ( ) one obtains: s s s ( s) ( ) ( ) ( s) ( ) ( s) + Hence: ( ) + ( ) ( ) ( ) dm + dm dm s { } ( ) [ I ], dm INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 6
15 B KINEIC / : B34 Angular momentum relative to the instantaneous axis of rotation s n ( I ) n I ( ) Let s calculate ( ) dm n ( ) s / I ( ) ( I ) n I ( I ) dm n + I ( I) dm n ( ) ( ) I n L n +, ( ) I t ( I ) n n [ I ], { } t ( I ) n n [ I ] n w, as, w n nn I nn ω INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 7
16 B KINEIC B Dynamic wrench: B Definition: From the acceleration field and the mass of a material system, two vectors can be set up: a) um: b) Moment in C Σ δ ( ) J dm ( ) C dm C J ( ) ( ) INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 8
17 B KINEIC c)change of point formula: δ δ δ δ ( L) L J ( ) ( ) ( L) LM J ( ) ( ) dm dm + ( L) LM Σ + δ ( M ) ( L) δ ( M ) + LM Σ ( ) ( LM + M) J ( ) ( ) M J ( ) dm dm δ ( L) δ ( M ) + LM Σ d) consequence: δ L G + LG Σ δ INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 9
18 B KINEIC B Expression of the sum: Σ Σ J ( ) ( ) ( ) M J ( G) dm d dm d ( ) d ( ) dm σ M ( G) d Σ M J ( G) INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 3
19 B KINEIC B3 Expression of the moment: d d d ( C) C ( ) dm ( C ( ) ) dm ( ) ( ) d d d C C dm C ( ) dm d d d + d C C dm + C J dm ( ) ( ) ( ) + δ ( ) d C C dm C ( ) C δ C C dm d δ ( C) ( C) + ( C) σ δ d ( C) ( C) + ( C) σ INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 3
20 B KINEIC implifications: a) if C G b) or if C is fixed in ( R ) c) or if ( C) // ( G) (translation for example) d then δ ( C ) ( C ) In these three cases, the dynamic wrench { } wrench { K } D is the time derivative of the momentum INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 3
21 B KINEIC B3 Kinetic Energy: B3 Definition for a material point of mass dm, d ( ) dm by extension, it comes for a solid ( ), ( ) dm Caution: Kinetic energy is a CALAR but it depends on a frame of reference INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 33
22 B KINEIC INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 34 B3 Expression for a solid tarting from the general definition, kinetic energy reads: [ ] dm G G + [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ],,,,, G L G M dm G G G M G dm G G M G dm G G M G dm G G M dm G dm G [ ] [ ], G t I G M +
23 B KINEIC INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 35 B33 Expression for a solid with one point s fixed in the frame of reference R : imilarly, it comes: [ ] dm s s +, with s [ ] [ ] [ ] [ ],,,,, L dm dm dm dm dm [ ], t I if s fixed in the frame of reference R
24 B KINEIC B34 imple applications Rotation of a solid around a fixed axis ranslation of a solid along a fixed axis { } E F G ψɺ y ( s ), s { } E F G ( s ) yɺ y, INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 36
25 B KINEIC B35 Examples B35 Gyroscope - Using Binet s notations, give the simplest form of [I],, [I], Calculate the dynamic moments in with respect to (R ) for,, U 3 Determine the total kinetic energy with respect to (R ) INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 37
26 B KINEIC B35 Euler's pendulum: x z z z θ l G x, mass of solid : m mass of solid : m - Determine the momentum wrench with respect to (R ) for, in - Calculate the total kinetic energy with respect to (R ) INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 38
27 B KINEIC B353 Gyroscopic pendulum and are massless, 3 mass m 3 Calculate for 3 in : - the angular momentum, - the dynamic moment, - the kinetic energy INA-LYN / CAN nd er Cycle/ Mechanics, nd semester 39
Lecture 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 informationPhysics 312, Winter 2007, Practice Final
Physics 312, Winter 2007, Practice Final Time: Two hours Answer one of Question 1 or Question 2 plus one of Question 3 or Question 4 plus one of Question 5 or Question 6. Each question carries equal weight.
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 informationRigid bodies - general theory
Rigid bodies - general theory Kinetic Energy: based on FW-26 Consider a system on N particles with all their relative separations fixed: it has 3 translational and 3 rotational degrees of freedom. Motion
More informationSTATICS Chapter 1 Introductory Concepts
Contents Preface to Adapted Edition... (v) Preface to Third Edition... (vii) List of Symbols and Abbreviations... (xi) PART - I STATICS Chapter 1 Introductory Concepts 1-1 Scope of Mechanics... 1 1-2 Preview
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 informationRigid Body Dynamics: Kinematics and Kinetics. Rigid Body Dynamics K. Craig 1
Rigid Body Dynamics: Kinematics and Kinetics Rigid Body Dynamics K. Craig 1 Topics Introduction to Dynamics Basic Concepts Problem Solving Procedure Kinematics of a Rigid Body Essential Example Problem
More informationDynamics 12e. Copyright 2010 Pearson Education South Asia Pte Ltd. Chapter 20 3D Kinematics of a Rigid Body
Engineering Mechanics: Dynamics 12e Chapter 20 3D Kinematics of a Rigid Body Chapter Objectives Kinematics of a body subjected to rotation about a fixed axis and general plane motion. Relative-motion analysis
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 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 informationClassical Mechanics III (8.09) Fall 2014 Assignment 3
Classical Mechanics III (8.09) Fall 2014 Assignment 3 Massachusetts Institute of Technology Physics Department Due September 29, 2014 September 22, 2014 6:00pm Announcements This week we continue our discussion
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 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 informationVideo 2.1a Vijay Kumar and Ani Hsieh
Video 2.1a Vijay Kumar and Ani Hsieh Robo3x-1.3 1 Introduction to Lagrangian Mechanics Vijay Kumar and Ani Hsieh University of Pennsylvania Robo3x-1.3 2 Analytical Mechanics Aristotle Galileo Bernoulli
More informationExercise 1: Inertia moment of a simple pendulum
Exercise : Inertia moment of a simple pendulum A simple pendulum is represented in Figure. Three reference frames are introduced: R is the fixed/inertial RF, with origin in the rotation center and i along
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 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 informationLAWS OF GYROSCOPES / CARDANIC GYROSCOPE
LAWS OF GYROSCOPES / CARDANC GYROSCOPE PRNCPLE f the axis of rotation of the force-free gyroscope is displaced slightly, a nutation is produced. The relationship between precession frequency or nutation
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 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 informationPhysics A - PHY 2048C
Physics A - PHY 2048C and 11/15/2017 My Office Hours: Thursday 2:00-3:00 PM 212 Keen Building Warm-up Questions 1 Did you read Chapter 12 in the textbook on? 2 Must an object be rotating to have a moment
More information41514 Dynamics of Machinery
41514 Dynamics of Machinery Theory, Experiment, Phenomenology and Industrial Applications Ilmar Ferreira Santos 1. Recapitulation Mathematical Modeling & Steps 2. Example System of Particle 3. Example
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 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 informationRotational & 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 informationLecture II: Rigid-Body Physics
Rigid-Body Motion Previously: Point dimensionless objects moving through a trajectory. Today: Objects with dimensions, moving as one piece. 2 Rigid-Body Kinematics Objects as sets of points. Relative distances
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 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 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 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 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 informationPhysical Dynamics (PHY-304)
Physical Dynamics (PHY-304) Gabriele Travaglini March 31, 2012 1 Review of Newtonian Mechanics 1.1 One particle Lectures 1-2. Frame, velocity, acceleration, number of degrees of freedom, generalised coordinates.
More informationSPRING SEMESTER AE 262 DYNAMICS. (02) Dr. Yavuz YAMAN
2012-2013 SPRING SEMESTER AE 262 DYNAMICS INSTRUCTOR (01) Dr. Yavuz YAMAN (02) Dr. Yavuz YAMAN TEXTBOOK Vector Mechanics for Engineers DYNAMICS F.P. Beer, E.R. Johnston Jr. and W.E. Clausen Eighth Edition
More informationLecture PowerPoints. Chapter 8 Physics: Principles with Applications, 6 th edition Giancoli
Lecture PowerPoints Chapter 8 Physics: Principles with Applications, 6 th edition Giancoli 2005 Pearson Prentice Hall This work is protected by United States copyright laws and is provided solely for the
More informationM2A2 Problem Sheet 3 - Hamiltonian Mechanics
MA Problem Sheet 3 - Hamiltonian Mechanics. The particle in a cone. A particle slides under gravity, inside a smooth circular cone with a vertical axis, z = k x + y. Write down its Lagrangian in a) Cartesian,
More informationLaws of gyroscopes / cardanic gyroscope
Principle If the axis of rotation of the force-free gyroscope is displaced slightly, a nutation is produced. The relationship between precession frequency or nutation frequency and gyrofrequency is examined
More informationDynamics. Basilio Bona. Semester 1, DAUIN Politecnico di Torino. B. Bona (DAUIN) Dynamics Semester 1, / 18
Dynamics Basilio Bona DAUIN Politecnico di Torino Semester 1, 2016-17 B. Bona (DAUIN) Dynamics Semester 1, 2016-17 1 / 18 Dynamics Dynamics studies the relations between the 3D space generalized forces
More informationAPM1612. Tutorial letter 203/1/2018. Mechanics 2. Semester 1. Department of Mathematical Sciences APM1612/203/1/2018
APM6/03//08 Tutorial letter 03//08 Mechanics APM6 Semester Department of Mathematical Sciences IMPORTANT INFORMATION: This tutorial letter contains solutions to assignment 3, Sem. BARCODE Define tomorrow.
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 informationChapter 8. Rotational Equilibrium and Rotational Dynamics
Chapter 8 Rotational Equilibrium and Rotational Dynamics Force vs. Torque Forces cause accelerations Torques cause angular accelerations Force and torque are related Torque The door is free to rotate about
More informationSOLUTIONS, PROBLEM SET 11
SOLUTIONS, PROBLEM SET 11 1 In this problem we investigate the Lagrangian formulation of dynamics in a rotating frame. Consider a frame of reference which we will consider to be inertial. Suppose that
More informationPhysical Dynamics (SPA5304) Lecture Plan 2018
Physical Dynamics (SPA5304) Lecture Plan 2018 The numbers on the left margin are approximate lecture numbers. Items in gray are not covered this year 1 Advanced Review of Newtonian Mechanics 1.1 One Particle
More informationLecture Outline Chapter 11. Physics, 4 th Edition James S. Walker. Copyright 2010 Pearson Education, Inc.
Lecture Outline Chapter 11 Physics, 4 th Edition James S. Walker Chapter 11 Rotational Dynamics and Static Equilibrium Units of Chapter 11 Torque Torque and Angular Acceleration Zero Torque and Static
More informationGyroscopic matrixes of the straight beams and the discs
Titre : Matrice gyroscopique des poutres droites et des di[...] Date : 29/05/2013 Page : 1/12 Gyroscopic matrixes of the straight beams and the discs Summarized: This document presents the formulation
More informationProblem 1. Mathematics of rotations
Problem 1. Mathematics of rotations (a) Show by algebraic means (i.e. no pictures) that the relationship between ω and is: φ, ψ, θ Feel free to use computer algebra. ω X = φ sin θ sin ψ + θ cos ψ (1) ω
More informationPhys 270 Final Exam. Figure 1: Question 1
Phys 270 Final Exam Time limit: 120 minutes Each question worths 10 points. Constants: g = 9.8m/s 2, G = 6.67 10 11 Nm 2 kg 2. 1. (a) Figure 1 shows an object with moment of inertia I and mass m oscillating
More informationChapter 8. Rotational Equilibrium and Rotational Dynamics
Chapter 8 Rotational Equilibrium and Rotational Dynamics 1 Force vs. Torque Forces cause accelerations Torques cause angular accelerations Force and torque are related 2 Torque The door is free to rotate
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 information3-D Kinetics of Rigid Bodies
3-D Kinetics of Rigid Bodies Angular Momentum Generalized Newton s second law for the motion for a 3-D mass system Moment eqn for 3-D motion will be different than that obtained for plane motion Consider
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 informationGeneral Physics (PHY 2130)
General Physics (PHY 130) Lecture 0 Rotational dynamics equilibrium nd Newton s Law for rotational motion rolling Exam II review http://www.physics.wayne.edu/~apetrov/phy130/ Lightning Review Last lecture:
More informationChapter 10.A. Rotation of Rigid Bodies
Chapter 10.A Rotation of Rigid Bodies P. Lam 7_23_2018 Learning Goals for Chapter 10.1 Understand the equations govern rotational kinematics, and know how to apply them. Understand the physical meanings
More informationIf the symmetry axes of a uniform symmetric body coincide with the coordinate axes, the products of inertia (Ixy etc.
Prof. O. B. Wright, Autumn 007 Mechanics Lecture 9 More on rigid bodies, coupled vibrations Principal axes of the inertia tensor If the symmetry axes of a uniform symmetric body coincide with the coordinate
More informationCHAPTER 8: ROTATIONAL OF RIGID BODY PHYSICS. 1. Define Torque
7 1. Define Torque 2. State the conditions for equilibrium of rigid body (Hint: 2 conditions) 3. Define angular displacement 4. Define average angular velocity 5. Define instantaneous angular velocity
More information1. The first thing you need to find is the mass of piece three. In order to find it you need to realize that the masses of the three pieces must be
1. The first thing you need to find is the mass of piece three. In order to find it you need to realize that the masses of the three pieces must be equal to the initial mass of the starting rocket. Now
More informationSet No - 1 I B. Tech I Semester Regular Examinations Jan./Feb ENGINEERING MECHANICS
3 Set No - 1 I B. Tech I Semester Regular Examinations Jan./Feb. 2015 ENGINEERING MECHANICS (Common to CE, ME, CSE, PCE, IT, Chem E, Aero E, AME, Min E, PE, Metal E) Time: 3 hours Question Paper Consists
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 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 information7. The gyroscope. 7.1 Introduction. 7.2 Theory. a) The gyroscope
K 7. The gyroscope 7.1 Introduction This experiment concerns a special type of motion of a gyroscope, called precession. From the angular frequency of the precession, the moment of inertia of the spinning
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 106b/196b Problem Set 9 Due Jan 19, 2007
Physics 06b/96b Problem Set 9 Due Jan 9, 2007 Version 3: January 8, 2007 This problem set focuses on dynamics in rotating coordinate systems (Section 5.2), with some additional early material on dynamics
More information31 ROTATIONAL KINEMATICS
31 ROTATIONAL KINEMATICS 1. Compare and contrast circular motion and rotation? Address the following Which involves an object and which involves a system? Does an object/system in circular motion have
More informationW13D1-1 Reading Quiz and Concept Questions
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Physics 8.01 Fall Term 2009 W13D1-1 Reading Quiz and Concept Questions A person spins a tennis ball on a string in a horizontal circle (so that
More informationSimple and Physical Pendulums Challenge Problem Solutions
Simple and Physical Pendulums Challenge Problem Solutions Problem 1 Solutions: For this problem, the answers to parts a) through d) will rely on an analysis of the pendulum motion. There are two conventional
More informationRotational Dynamics. A wrench floats weightlessly in space. It is subjected to two forces of equal and opposite magnitude: Will the wrench accelerate?
Rotational Dynamics A wrench floats weightlessly in space. It is subjected to two forces of equal and opposite magnitude: Will the wrench accelerate? A. yes B. no C. kind of? Rotational Dynamics 10.1-3
More informationAutonomous Underwater Vehicles: Equations of Motion
Autonomous Underwater Vehicles: Equations of Motion Monique Chyba - November 18, 2015 Departments of Mathematics, University of Hawai i at Mānoa Elective in Robotics 2015/2016 - Control of Unmanned Vehicles
More information12. Rigid Body Dynamics I
University of Rhode Island DigitalCommons@URI Classical Dynamics Physics Course Materials 015 1. Rigid Body Dynamics I Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative Commons License
More informationPHYSICS. Chapter 12 Lecture FOR SCIENTISTS AND ENGINEERS A STRATEGIC APPROACH 4/E RANDALL D. KNIGHT Pearson Education, Inc.
PHYSICS FOR SCIENTISTS AND ENGINEERS A STRATEGIC APPROACH 4/E Chapter 12 Lecture RANDALL D. KNIGHT Chapter 12 Rotation of a Rigid Body IN THIS CHAPTER, you will learn to understand and apply the physics
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 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 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 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 informationFlight Dynamics and Control
Flight Dynamics and Control Lecture 1: Introduction G. Dimitriadis University of Liege Reference material Lecture Notes Flight Dynamics Principles, M.V. Cook, Arnold, 1997 Fundamentals of Airplane Flight
More informationDYNAMICS OF SERIAL ROBOTIC MANIPULATORS
DYNAMICS OF SERIAL ROBOTIC MANIPULATORS NOMENCLATURE AND BASIC DEFINITION We consider here a mechanical system composed of r rigid bodies and denote: M i 6x6 inertia dyads of the ith body. Wi 6 x 6 angular-velocity
More informationIII. Work and Energy
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 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 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 informationChapter 8 Rotational Motion and Equilibrium
Chapter 8 Rotational Motion and Equilibrium 8.1 Rigid Bodies, Translations, and Rotations A rigid body is an object or a system of particles in which the distances between particles are fixed (remain constant).
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 informationPhysics Waves & Oscillations. Mechanics Lesson: Circular Motion. Mechanics Lesson: Circular Motion 1/18/2016. Spring 2016 Semester Matthew Jones
Physics 42200 Waves & Oscillations Lecture 5 French, Chapter 3 Spring 2016 Semester Matthew Jones Mechanics Lesson: Circular Motion Linear motion: Mass: Position: Velocity: / Momentum: Acceleration: /
More informationDept of ECE, SCMS Cochin
B B2B109 Pages: 3 Reg. No. Name: APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY SECOND SEMESTER B.TECH DEGREE EXAMINATION, MAY 2017 Course Code: BE 100 Course Name: ENGINEERING MECHANICS Max. Marks: 100 Duration:
More informationLecture 9 Kinetics of rigid bodies: Impulse and Momentum
Lecture 9 Kinetics of rigid bodies: Impulse and Momentum Momentum of 2-D Rigid Bodies Recall that in lecture 5, we discussed the use of momentum of particles. Given that a particle has a, and is travelling
More informationRotation. PHYS 101 Previous Exam Problems CHAPTER
PHYS 101 Previous Exam Problems CHAPTER 10 Rotation Rotational kinematics Rotational inertia (moment of inertia) Kinetic energy Torque Newton s 2 nd law Work, power & energy conservation 1. Assume that
More informationPhysics Mechanics. Lecture 32 Oscillations II
Physics 170 - Mechanics Lecture 32 Oscillations II Gravitational Potential Energy A plot of the gravitational potential energy U g looks like this: Energy Conservation Total mechanical energy of an object
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 informationPhys 7221 Homework # 8
Phys 71 Homework # 8 Gabriela González November 15, 6 Derivation 5-6: 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 informationPhysics Fall Mechanics, Thermodynamics, Waves, Fluids. Lecture 20: Rotational Motion. Slide 20-1
Physics 1501 Fall 2008 Mechanics, Thermodynamics, Waves, Fluids Lecture 20: Rotational Motion Slide 20-1 Recap: center of mass, linear momentum A composite system behaves as though its mass is concentrated
More informationScalar product Work Kinetic energy Work energy theorem Potential energy Conservation of energy Power Collisions
BLOOM PUBLIC SCHOOL Vasant Kunj, New Delhi Lesson Plan Class: XI Subject: Physics Month: August No of Periods: 11 Chapter No. 6: Work, energy and power TTT: 5 WT: 6 Chapter : Work, energy and power Scalar
More informationChapter 9. Rotational Dynamics
Chapter 9 Rotational Dynamics In pure translational motion, all points on an object travel on parallel paths. The most general motion is a combination of translation and rotation. 1) Torque Produces angular
More informationCourse syllabus Engineering Mechanics - Dynamics
Course syllabus Engineering Mechanics - Dynamics COURSE DETAILS Type of study programme Study programme Course title Course code ECTS (Number of credits allocated) Course status Year of study Course Web
More informationExam 3 December 1, 2010
Exam 3 Instructions: You have 60 minutes to complete this exam. This is a closed-book, closed-notes exam. You are allowed to use a calculator during the exam. All work must be shown to receive credit.
More information12. Foundations of Statics Mechanics of Manipulation
12. Foundations of Statics Mechanics of Manipulation Matt Mason matt.mason@cs.cmu.edu http://www.cs.cmu.edu/~mason Carnegie Mellon Lecture 12. Mechanics of Manipulation p.1 Lecture 12. Foundations of statics.
More informationAP Pd 3 Rotational Dynamics.notebook. May 08, 2014
1 Rotational Dynamics Why do objects spin? Objects can travel in different ways: Translation all points on the body travel in parallel paths Rotation all points on the body move around a fixed point An
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 informationDYNAMICS OF PARALLEL MANIPULATOR
DYNAMICS OF PARALLEL MANIPULATOR The 6nx6n matrices of manipulator mass M and manipulator angular velocity W are introduced below: M = diag M 1, M 2,, M n W = diag (W 1, W 2,, W n ) From this definitions
More informationExperiment 08: Physical Pendulum. 8.01t Nov 10, 2004
Experiment 08: Physical Pendulum 8.01t Nov 10, 2004 Goals Investigate the oscillation of a real (physical) pendulum and compare to an ideal (point mass) pendulum. Angular frequency calculation: Practice
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 informationEngineering Mechanics
2019 MPROVEMENT Mechanical Engineering Engineering Mechanics Answer Key of Objective & Conventional Questions 1 System of forces, Centoriod, MOI 1. (c) 2. (b) 3. (a) 4. (c) 5. (b) 6. (c) 7. (b) 8. (b)
More informationPhysics 121, March 25, Rotational Motion and Angular Momentum. Department of Physics and Astronomy, University of Rochester
Physics 121, March 25, 2008. Rotational Motion and Angular Momentum. Physics 121. March 25, 2008. Course Information Topics to be discussed today: Review of Rotational Motion Rolling Motion Angular Momentum
More informationConstrained motion and generalized coordinates
Constrained motion and generalized coordinates based on FW-13 Often, the motion of particles is restricted by constraints, and we want to: work only with independent degrees of freedom (coordinates) k
More information