PHYS 705: Classical Mechanics. Non-inertial Reference Frames Vectors in Rotating Frames
|
|
- Gilbert Rose
- 5 years ago
- Views:
Transcription
1 1 PHYS 705: Classical Mechanics Non-inertial Reference Frames Vectors in Rotating Frames
2 2 Infinitesimal Rotations From our previous discussion, we have established that any orientation of a rigid body (with one point fixed) can be represented by a single rotation about a particular axis. And this rotation can be parameterized by three parameters (3 Euler angles) Question: Can we represent these three parameters by a vector? Suppose A and B are two such vectors representing two such rotations Then, for them to be vectors, they must commute, i.e. AB BA However, from our previous discussion, finite rotations are not commutative as an operation.
3 3 Infinitesimal Rotations Recall that if we represent the rotation A by a matrix M A, then the addition of two rotations A and B will be given by a matrix multiplication, i.e., AB M M And, matrix multiplications are not commutative : A B MAMB MBMA Thus, finite rotations CANNOT be represented as vectors! However, as we will demonstrate next. Infinitesimal rotations can be represented by vectors!
4 4 Infinitesimal Rotations Let consider a general infinitesimal transformation given by the following, x x x x ' i i ij j ij ij j so that the unprimed axes is changed to by an infinitesimal operator. In matrix notations, we have, ' ε ij x i x I ε x Since is assumed to be small, we will show that the sequence of infinitesimal transformation is not important. x ' i
5 5 Infinitesimal Rotations - Let say we have two infinitesimal transformations:, Op Op Switching the order of the operations, Op Op 2 1 I ε1 I ε Iε Iε2 I ε1iiε2 ε1ε2 Iε1ε2 O Iε Iε1 I ε2iiε1ε2ε1 Iε2 ε1o -By neglecting higher order terms, these two expressions are the same since matrix addition is commutative. Infinitesimal transformation is commutative. 2
6 6 Infinitesimal Rotations - Now, let consider an inverse of an infinitesimal transformation, A Iε It is given simply by: CHECK: - Then, let see what property will have if we want this infinitesimal transformation to be orthogonal. 1 A Iε 1 AA I ε I ε I (to first order) ε - For orthogonal transformations, we need to have A A T 1 - This then gives, A T I ε A I T 1 ε ε T ε ε is an anti-symmetric matrix
7 7 Infinitesimal Rotations - Further note that, det A det Iε 1 (to first order) An infinitesimal orthogonal transformation corresponds to a proper rotation. ε - Since is anti-symmetric, we can write it generally in component form as, 0 ε d d d d 2 1 d d d, d, d (The three quantities can be identified as three independent parameters specifying the infinitesimal rotation. )
8 8 Infinitesimal Rotations - As an example, we can explicitly write out the infinitesimal Euler rotation, A cos d cos d cos dsin dsin d cos d sin d cos d cos dsin d sin dsin d sin d cos d cos dsin dcos d sin d sin d cos d cos dcos d sin d cos d sin dsin d sin d cos d cos d with cos d cos d cos d 1 sin d d sin d d sin d d This gives
9 9 Infinitesimal Rotations - As an example, we can explicitly write out the infinitesimal Euler rotation, A 1 d d 0 cos d cos d cos dsin dsin d cos d sin d cos d cos dsin dsin dx sinddx sin, d sin d cos Ad cos sin cos d sin d sin cos d cos dcos d sin d cos d d d 1 d Iε cos dx 1, sin dsin d sin d cos d and keeping cos d 0 d 1 only linear terms And, dω d,0, d d T Note: 1. For notational purpose, we write as a differential but it is NOT an actual differential of a vector. dω dω 2. But, as you will see stands for a vector of differential changes
10 10 Infinitesimal Rotations - We will now show that these three quantities also form a particular kind of vector. - Under an infinitesimal rotation, the change in the coordinates can be expressed as, - Writing this out explicitly in components, we have, dr r' r Iε rr εr dx1 0 d3 d2x1 dr dx εr d 0 d x dx 3 d2 d1 0 x 3 - The result looks like the cross product between two vectors: dr rdω with x, x, x d, d, d dr dx x d x d dx x d x d dx x d x d T T r dω d1, d2, d3
11 11 d as a Pseudovector - Under an orthogonal transformation B, the anti-symmetric matrix transforms according to the similarity transform: ε' 1 B εb ε - Recall that a regular vector (called polar vector) r, it must satisfy the transformation rule: x ' i bx ij j b bb ij ik jk where is orthogonal, i.e, ij - One can also show that dω must transform as, d' det B b d (*) i ij j (not shown here) - Vectors transforming according to (*) is called an axial vector (pseudovector)
12 12 d as a Pseudovector - Recall that if B is a an orthogonal transformation, det B 1 det B = + 1, then B is called proper, (rotation only, no inversion) det B = -1, then B is called improper (with inversion) - So, we have the following distinction between a polar and axial vector under a improper orthogonal transformation B: V' BV if V is a polar vector V' BV if V is a axial vector (or pseudovector) - Common examples of axial vectors in physics involves the cross product of two polar vectors such as angular velocity, angular momentum,
13 13 Geometric View of an Infinitesimal Rotations dω We just saw that an inf. rotation leads to an inf. change for : Here is the same situation in the active view: dr rdω (**This was in the passive view) dr r dω r sin r d r dr dr Here, we are rotating the vector (active view) instead of rotating the coordinates (passive view) thus the reverse order of the cross prod. magnitude: direction: rsind RHR r dr dωr (We will take this active view from now on.)
14 14 Geometric View of an Infinitesimal Rotations Dividing dt from both sides of the equation, we have, dr d dt Ω v r ω dt r d ω Ω dt where is the instantaneous angular velocity ω where lies along the axis of infinitesimal rotation in the time tt, interval in a direction known as the instantaneous axis of rotation. dt (the familiar definition for angular velocity)
15 15 Rate of Change of a Vector under Rotation x z z ' r R x ' P y ' r ' y Looking at infinitesimal rotation more generally: We are interested in describing vectors in the body (primed) frame as measured in the fixed (unprimed) frame when the rigid body frame is being rotated about a fixed axis. - For simplicity, we also assume (for now) that a point P described by in the body frame does not move wrt the body frame. R ( is assumed to be fixed for now.) r ' Note: Our discussion can be for any vector instead of the specific position vector r ' in the body frame. G
16 16 Rate of Change of a Vector in under Rotation - However, as viewed from the fixed frame, will be moving (due to ) dω r ' - From our discussion on rotation before, viewed in the fixed frame, dr' dωr' fixed - Or, dividing by dt, we have, dr' dω r ' ω r ' dt dt fixed r ' - More generally, if we now allow to move wrt the body frame as well, dr' dr' ωr' dt dt fixed body
17 17 Rate of Change of a Vector in under Rotation - Although our discussion was for a given position vector in the body frame, the discussion applies equally well to ANY vector in the body frame. dg dg ωg dt dt fixed body r ' G - So, it is useful to abstract out this operation for any vectors, d d ω dt dt fixed body
18 18 Rate of Change of a Vector in under Rotation This relation can also be formally derived. The components of a vector measured in the body frame and in the fixed frame are related by an orthogonal transformation G or G( body) AG( fixed) G a G ' i ij j A a ij where is the Euler rotation matrix Reversing the expression between the fixed and body frames, 1 T G a G' a G' a G' i ij j ij j ji j
19 19 Rate of Change of a Vector in under Rotation As the body moves infinitesimally in time, the components of in the body dg ' j frame will change by and the instantaneous rotation matrix will be given by the infinitesimal rotation matrix, G ' i AIε a ji ji ji So, the components in the fixed frame is then changed by, ' ' G dg G dg i i ji ji j j Keeping terms up to to 1 st order, we have, G dg G' dg' G' i i j j ji j
20 20 Rate of Change of a Vector in under Rotation Without loss of generality, we pick our fixed frame to be instantaneously coincident with the body frame at time t so that we choose G i G' j at time t BUT, the differential of G in the body and fixed frame are in general not the same and is related by, dg dg ' G ' i j ji j Using the anti-symmetric property of, i.e., dg dg ' G ' i i ij j ε ji ij
21 21 Rate of Change of a Vector in under Rotation dg dg ' G ' i j ij j Then, recall that we have ε 0 d d d d 2 1 d d 0 And using the Levi Civita tensor The last term can be rewritten as, ijk 1, ijk is of forward permutation 1, ijk is of backward permutation 0, otherwise G' d G' d G' ij j ijk k j ikj k j
22 22 Rate of Change of a Vector in under Rotation So, we have, dg dg ' d G ' i i ikj k j or dg dg dωg fixed body And, finally dg dg ωg dt dt fixed body And the operator equation again, d d ω dt dt fixed body
23 23 Acceleration in a Rotating Frame - Let use the operator equation to calculate the velocity and acceleration of a point P in the body frame as measured in the fixed frame z z ' r r ' P y ' unprimed wrt fixed frame x R x ' y primed wrt body frame (Here, we also allow R to move as well.)
24 24 Acceleration in a Rotating Frame - So, for the velocity vector, we have (as before), dr dr dr' dt dt dt fixed fixed fixed dr dr dr ' dt dt dt fixed fixed body ωr' dr' dr' ωr' dt dt fixed body vel of P rel to vel of rotating vel of P rel to extra piece due to fixed axes frame origin rotating frame rotation of the frame v v v' ωr' f R
25 25 Acceleration in a Rotating Frame - Differentiate in the fixed frame again, we have dv dvr dv' dr' ω ω r' dt dt dt dt fixed fixed fixed fixed Both v and r are vectors in the body frame but the differentiation is measured in the fixed frame so that we need to apply the operator equation again to expand them.
26 26 Acceleration in a Rotating Frame dv dvr dv ' dr ' ω ω r ' dt dt dt dt fixed fixd e fixed fixed dv dvr d ' d ' ' ' ' dt f dt v f dt r ω v ω ω r body dt ω r body a a a 2 ωv' ω ωr' ω r' f R r acc of P rel to acc of P rel to fixed axes rotating frame acc of rotating frame origin vel of P in the rotating frame pos of P in the rotating frame
27 27 Acceleration in a Rotating Frame - To gain some physical insights into the meanings of the various terms, 1. We assume that the body frame is rotating with a constant angular frequency around a fixed axis and the origin of the body frame is not moving as well, i.e., F ma ω 0 and a 0 f Fma 2 m ωv' mω ωr' r R 2. Newton 2 nd Law applies in inertia frames only. That is why we have calculated a f explicitly in the fixed frame and this gives, where F is the net force acting on m as observed in the fixed frame.
28 28 Acceleration in a Rotating Frame - Rearranging the terms, we can rewrite the equation into the following form, F2 m ωv' mω ωr' mar - Defining F F2 ωv' ωωr' eff m m - We can re-interpret the Newton 2 nd law as an equation applying in the rotating frame, F eff ma r This is F = ma in the rotating frame!
29 29 Acceleration in a Rotating Frame Feff F2 m ωv' mω ωr' Notes: - F is the actual physical forces in the inertia frame - The extra two terms corresponding to apparent forces (a penalty you pay) for NOT being in an inertial frame. mω ωr' The familiar centrifugal force 2 m ωv' The coriolis force - These two fictitious forces are not associated with any real forces acting on objects in an inertial frame and they are not visible to an inertial observer!
30 30 Example of Fictitious Forces in a Rotating Frame Just standing on a merry-go-round ω r ' ωr' ω ωr ' - For simplicity, let saying you are standing still (remain stationary relative v ' 0 to the rotating frame) so that there is no coriolis force! - Everything is perpendicular here. The fictitious centrifugal force is mv mωωr m r outward r ' 2 2 ' ' ( ) - Note: this fictitious centrifugal force ~ r so that it gets larger as one moves out and vanishes at the origin.
31 31 Example of Fictitious Forces in a Rotating Frame Now, we allow v ' 0 Consider standing in the middle and throwing a rock directly at a fixed target fixed outside the merry-go-around ω View in fixed frame In the fixed inertial frame, the rock experiences no net force and it will go out toward the target along the straight path as shown. - However, as the rocks moves outward, the thrower will rotate. Target fixed in fixed frame To the thrower in the rotating frame, the rock will look like veering off View in rotating frame to right.
32 32 Example of Fictitious Forces in a Rotating Frame So, within the rotating frame, there is a fictitious Coriolis force v ' ω View in rotating frame 2 ωv' Rule of Thumb: object deflects to the right of v ' MOVIE:
33 33 Earth as a Rotating Frame The fact that we live on the surface of the Earth which rotates Means that we are subject to these fictitious forces (Note: For an object on the surface of the Earth, the origin of the body frame is not fixed so that R changes and we need to put back 0 but we will assume Earth to have a fixed rate of rotation, i.e., ) ω ω a R R a a a 2 ωv' ω ωr' f R r
34 34 Earth as a Rotating Frame To calculate, consider R ω a R It moves in the fixed frame but it is stationary in the body frame. R Considering the rate of change of R in the fixed frame, we have dr dr dt dt fixed body ωr Taking the time derivative as measured in the fixed frame again, we have, dr dr R a ω ωr R dt dt fixed body a ω ωr R
35 35 Earth as a Rotating Frame Plugging this result into our equation for, we have, a a 2 ωv' ω ω Rr' f r And the effective force in the rotating frame becomes, Letting (gravity) be the only actual physical force, and assuming that we re interested in objects near the surface of the Earth so that a f F F2 m ωv' mω ω Rr' ma eff F mg 0 ω v Feff m g0 mω R 2 m ω ' r R r' R This is the effective gravity g (actual gravity + centrifugal force)
36 36 Earth as a Rotating Frame Now, let look closer at this effective gravity as a function of latitude on Earth, North Pole ω R - For 90 (North Pole), ωr 0 Then, we have g g 0 - For 0 (Equator), equator 2 gg0 R R ˆ g 0 grˆ South Pole Thus, due to centrifugal effects, gravity g is about ms the poles equator
37 37 Earth as a Rotating Frame In general, also notice that a mass on a string doesn t hang straight down: North Pole ω g 0 ω ωr R equator Actual direction of g (way exaggerated) gg ω ωr 0 South Pole HW: find deviation angle as a function of latitude
38 38 Coriolis Effect on Projectiles: Case 1: At North Pole North Pole R ω 2 m ωv' North Pole ω (out of page) v ' ωv' equator Top View Projectile will defect to the right (as in South Pole the merry-go-round situation).
39 39 Coriolis Effect on Projectiles: 2 m ωv' Case 2: At Equator sending projectile to the East North Pole ω R equator v ' (into of page) ωv' South Pole Coriolis force is up (from the surface) but it is tiny compared to regular gravity.
40 40 Coriolis Effect on Projectiles: 2 m ωv' Case 3: At intermediate northern latitudes sending projectile to the East North Pole ω R v ' (into of page) ωv' equator In addition to the up direction, Coriolis force also has a component deflecting the orbit to the right wrt its original direction of South Pole motion (toward South in this case).
41 41 Coriolis Effect on Projectiles: 2 m ωv' Case 4: Southern latitudes sending projectile to the East North Pole ω The situation is reverse. Coriolis force has a component deflecting the orbit to the left wrt its original direction of motion (toward North in this case). South Pole R equator ωv' v ' (into of page)
42 42 Coriolis Effect on Projectiles: 2 m ωv' Coriolis effect on weather North Pole ω Low equator Northern hemisphere direction of wind will be defected to the right always resulting into forming a counter-clockwise circulating low pressure cell. South Pole Northern hurricane Southern hurricane
Additional Problem (HW 10)
1 Housekeeping - Three more lectures left including today: Nov. 20 st, Nov. 27 th, Dec. 4 th - Final Eam on Dec. 11 th at 4:30p (Eploratory Planetary 206) 2 Additional Problem (HW 10) z h y O Choose origin
More informationNon-inertial systems - Pseudo Forces
Non-inertial systems - Pseudo Forces ?? ame Accelerating frame Newton's second law F = ma holds true only in inertial coordinate systems. However, there are many noninertial (that is, accelerating) frames
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 informationLecture-XV. Noninertial systems
Lecture-XV Noninertial systems Apparent Force in Rotating Coordinates The force in the ating system is where The first term is called the Coriolis force, a velocity dependent force and the second term,
More informationPhysics 351 Monday, February 26, 2018
Physics 351 Monday, February 26, 2018 You just read the first half ( 10.1 10.7) of Chapter 10, which we ll probably start to discuss this Friday. The midterm exam (March 26) will cover (only!) chapters
More informationPHYS 705: Classical Mechanics. Rigid Body Motion Introduction + Math Review
1 PHYS 705: Classical Mechanics Rigid Body Motion Introduction + Math Review 2 How to describe a rigid body? Rigid Body - a system of point particles fixed in space i r ij j subject to a holonomic constraint:
More informationLecture D15 - Gravitational Attraction. The Earth as a Non-Inertial Reference Frame
J. Peraire 16.07 Dynamics Fall 2004 Version 1.3 Lecture D15 - Gravitational Attraction. The Earth as a Non-Inertial Reference Frame Gravitational attraction The Law of Universal Attraction was already
More informationPhysics 351 Wednesday, February 21, 2018
Physics 351 Wednesday, February 21, 2018 HW5 due Friday. For HW help, Bill is in DRL 3N6 Wed 4 7pm. Grace is in DRL 2C2 Thu 5:30 8:30pm. It is often convenient to treat the inertial (pseudo)force F inertial
More informationRotational motion of a rigid body spinning around a rotational axis ˆn;
Physics 106a, Caltech 15 November, 2018 Lecture 14: Rotations The motion of solid bodies So far, we have been studying the motion of point particles, which are essentially just translational. Bodies with
More informationGeneral Physics I. Lecture 9: Vector Cross Product. Prof. WAN, Xin ( 万歆 )
General Physics I Lecture 9: Vector Cross Product Prof. WAN, Xin ( 万歆 ) xinwan@zju.edu.cn http://zimp.zju.edu.cn/~xinwan/ Outline Examples of the rotation of a rigid object about a fixed axis Force/torque
More informationGeneral Physics I. Lecture 9: Vector Cross Product. Prof. WAN, Xin ( 万歆 )
General Physics I Lecture 9: Vector Cross Product Prof. WAN, Xin ( 万歆 ) xinwan@zju.edu.cn http://zimp.zju.edu.cn/~xinwan/ Outline Examples of the rotation of a rigid object about a fixed axis Force/torque
More informationChapter 8 Rotational Motion
Chapter 8 Rotational Motion Chapter 8 Rotational Motion In this chapter you will: Learn how to describe and measure rotational motion. Learn how torque changes rotational velocity. Explore factors that
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 informationPhysics 351 Wednesday, March 1, 2017
Physics 351 Wednesday, March 1, 2017 HW7 due this Friday. Over spring break, you ll read 10.1 10.7 of Ch 10 (rigid body rotation). A copy of this Ch 10 is on Canvas so that you don t need to take your
More informationFictitious Forces and Non-inertial Frames: The Coriolis Force *
OpenStax-CNX module: m42142 1 Fictitious Forces and Non-inertial Frames: The Coriolis Force * OpenStax This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License
More informationd v 2 v = d v d t i n where "in" and "rot" denote the inertial (absolute) and rotating frames. Equation of motion F =
Governing equations of fluid dynamics under the influence of Earth rotation (Navier-Stokes Equations in rotating frame) Recap: From kinematic consideration, d v i n d t i n = d v rot d t r o t 2 v rot
More informationPHYS 705: Classical Mechanics. Euler s Equations
1 PHYS 705: Classical Mechanics Euler s Equations 2 Euler s Equations (set up) We have seen how to describe the kinematic properties of a rigid body. Now, we would like to get equations of motion for it.
More informationMIDTERM 1: APPROXIMATE GRADES TOTAL POINTS = 45 AVERAGE = 33 HIGH SCORE = = A = B = C < 20.0 NP
MIDTERM 1: TOTAL POINTS = 45 AVERAGE = 33 HIGH SCORE = 43 APPROXIMATE GRADES 38.0 45.0 = A 30.0 37.5 = B 20.0 29.5 = C < 20.0 NP Forces to consider: 1) Pressure Gradient Force 2) Coriolis Force 3) Centripetal
More informationPHYS 705: Classical Mechanics. Introduction and Derivative of Moment of Inertia Tensor
1 PHYS 705: Classical Mechanics Introduction and Derivative of Moment of Inertia Tensor L and N depends on the Choice of Origin Consider a particle moving in a circle with constant v. If we pick the origin
More informationThe dynamics of high and low pressure systems
The dynamics of high and low pressure systems Newton s second law for a parcel of air in an inertial coordinate system (a coordinate system in which the coordinate axes do not change direction and are
More informationa reference frame that accelerates in a straight line a reference frame that moves along a circular path Straight Line Accelerated Motion
1.12.1 Introduction Go back to lesson 9 and provide bullet #3 In today s lesson we will consider two examples of non-inertial reference frames: a reference frame that accelerates in a straight line a reference
More informationPhysics 351 Wednesday, February 28, 2018
Physics 351 Wednesday, February 28, 2018 HW6 due Friday. For HW help, Bill is in DRL 3N6 Wed 4 7pm. Grace is in DRL 2C2 Thu 5:30 8:30pm. To get the most benefit from the homework, first work through every
More informationLecture 2. Lecture 1. Forces on a rotating planet. We will describe the atmosphere and ocean in terms of their:
Lecture 2 Lecture 1 Forces on a rotating planet We will describe the atmosphere and ocean in terms of their: velocity u = (u,v,w) pressure P density ρ temperature T salinity S up For convenience, we will
More informationLecture-XIV. Noninertial systems
Lecture-XIV Noninertial systems Accelerating frame Newton's second law F= maholds true only in inertial coordinate systems. However, there are many noninertial (that is, accelerating) frames that one needs
More information5.1. Accelerated Coordinate Systems:
5.1. Accelerated Coordinate Systems: Recall: Uniformly moving reference frames (e.g. those considered at 'rest' or moving with constant velocity in a straight line) are called inertial reference frames.
More informationPHYS 211 Lecture 12 - The Earth's rotation 12-1
PHY 211 Lecture 12 - The Earth's rotation 12-1 Lecture 12 - The Earth s rotation Text: Fowles and Cassiday, Chap. 5 Demo: basketball for the earth, cube for Cartesian frame A coordinate system sitting
More informationSMS 303: Integrative Marine Sciences III
SMS 303: Integrative Marine Sciences III Instructor: E. Boss, TA: C. Proctor emmanuel.boss@maine.edu, 581-4378 4 weeks & topics: waves, tides, mixing and Coriolis. From: Persson, 1998, BAMS Real and fictitious
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 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 informationDealing with Rotating Coordinate Systems Physics 321. (Eq.1)
Dealing with Rotating Coordinate Systems Physics 321 The treatment of rotating coordinate frames can be very confusing because there are two different sets of aes, and one set of aes is not constant in
More informationPhysics 209 Fall 2002 Notes 5 Thomas Precession
Physics 209 Fall 2002 Notes 5 Thomas Precession Jackson s discussion of Thomas precession is based on Thomas s original treatment, and on the later paper by Bargmann, Michel, and Telegdi. The alternative
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
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 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 informationA Summary of Some Important Points about the Coriolis Force/Mass. D a U a Dt. 1 ρ
A Summary of Some Important Points about the Coriolis Force/Mass Introduction Newton s Second Law applied to fluids (also called the Navier-Stokes Equation) in an inertial, or absolute that is, unaccelerated,
More information3 Space curvilinear motion, motion in non-inertial frames
3 Space curvilinear motion, motion in non-inertial frames 3.1 In-class problem A rocket of initial mass m i is fired vertically up from earth and accelerates until its fuel is exhausted. The residual mass
More informationClass 11 Physics NCERT Exemplar Solutions Motion in a Straight Line
Class 11 Physics NCERT Exemplar Solutions Motion in a Straight Line Multiple Choice Questions Single Correct Answer Type Q1. Among the four graphs shown in the figure, there is only one graph for which
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 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 informationKinematics vs. Dynamics
NEWTON'S LAWS Kinematics vs. Dynamics Kinematics describe motion paths mathematically No description of the matter that may travel along motion path Dynamics prediction of motion path(s) of matter Requires
More informationMechanics Physics 151
Mechanics Phsics 151 Lecture 8 Rigid Bod Motion (Chapter 4) What We Did Last Time! Discussed scattering problem! Foundation for all experimental phsics! Defined and calculated cross sections! Differential
More informationRigid Bodies. November 5, 2018
Rigid Bodies November 5, 08 Contents Change of basis 3. Einstein summation convention................................... 4. Passive transformation........................................ 6.3 Active transformation........................................
More informationA set of N particles forms a rigid body if the distance between any 2 particles is fixed:
Chapter Rigid Body Dynamics.1 Coordinates of a Rigid Body A set of N particles forms a rigid body if the distance between any particles is fixed: r ij r i r j = c ij = constant. (.1) Given these constraints,
More informationChapter 4: Fundamental Forces
Chapter 4: Fundamental Forces Newton s Second Law: F=ma In atmospheric science it is typical to consider the force per unit mass acting on the atmosphere: Force mass = a In order to understand atmospheric
More information1.3 LECTURE 3. Vector Product
12 CHAPTER 1. VECTOR ALGEBRA Example. Let L be a line x x 1 a = y y 1 b = z z 1 c passing through a point P 1 and parallel to the vector N = a i +b j +c k. The equation of the plane passing through the
More informationPhysics 351, Spring 2018, Homework #9. Due at start of class, Friday, March 30, 2018
Physics 351, Spring 218, Homework #9. Due at start of class, Friday, March 3, 218 Please write your name on the LAST PAGE of your homework submission, so that we don t notice whose paper we re grading
More informationCovariant Formulation of Electrodynamics
Chapter 7. Covariant Formulation of Electrodynamics Notes: Most of the material presented in this chapter is taken from Jackson, Chap. 11, and Rybicki and Lightman, Chap. 4. Starting with this chapter,
More informationESS314. Basics of Geophysical Fluid Dynamics by John Booker and Gerard Roe. Conservation Laws
ESS314 Basics of Geophysical Fluid Dynamics by John Booker and Gerard Roe Conservation Laws The big differences between fluids and other forms of matter are that they are continuous and they deform internally
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 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 information28. Motion in a Non-inertial Frame of Reference
8 Motion in a Non-inertial Frame of Reference Michael Fowler The Lagrangian in Accelerating and Rotating Frames This section concerns the motion of a single particle in some potential U( r ) in a non-inertial
More informationCartesian Tensors. e 2. e 1. General vector (formal definition to follow) denoted by components
Cartesian Tensors Reference: Jeffreys Cartesian Tensors 1 Coordinates and Vectors z x 3 e 3 y x 2 e 2 e 1 x x 1 Coordinates x i, i 123,, Unit vectors: e i, i 123,, General vector (formal definition to
More informationContents. Objectives Circular Motion Velocity and Acceleration Examples Accelerating Frames Polar Coordinates Recap. Contents
Physics 121 for Majors Today s Class You will see how motion in a circle is mathematically similar to motion in a straight line. You will learn that there is a centripetal acceleration (and force) and
More informationEULER EQUATIONS. We start by considering how time derivatives are effected by rotation. Consider a vector defined in the two systems by
EULER EQUATIONS We now consider another approach to rigid body probles based on looking at the change needed in Newton s Laws if an accelerated coordinate syste is used. We start by considering how tie
More informationControl Volume. Dynamics and Kinematics. Basic Conservation Laws. Lecture 1: Introduction and Review 1/24/2017
Lecture 1: Introduction and Review Dynamics and Kinematics Kinematics: The term kinematics means motion. Kinematics is the study of motion without regard for the cause. Dynamics: On the other hand, dynamics
More informationLecture 1: Introduction and Review
Lecture 1: Introduction and Review Review of fundamental mathematical tools Fundamental and apparent forces Dynamics and Kinematics Kinematics: The term kinematics means motion. Kinematics is the study
More informationHomework 7: # 4.22, 5.15, 5.21, 5.23, Foucault pendulum
Homework 7: # 4., 5.15, 5.1, 5.3, Foucault pendulum Michael Good Oct 9, 4 4. A projectile is fired horizontally along Earth s surface. Show that to a first approximation the angular deviation from the
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 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 informationChapter 6. Circular Motion and Other Applications of Newton s Laws
Chapter 6 Circular Motion and Other Applications of Newton s Laws Circular Motion Two analysis models using Newton s Laws of Motion have been developed. The models have been applied to linear motion. Newton
More information1/3/2011. This course discusses the physical laws that govern atmosphere/ocean motions.
Lecture 1: Introduction and Review Dynamics and Kinematics Kinematics: The term kinematics means motion. Kinematics is the study of motion without regard for the cause. Dynamics: On the other hand, dynamics
More informationPhysics 53. Dynamics 2. For every complex problem there is one solution that is simple, neat and wrong. H.L. Mencken
Physics 53 Dynamics 2 For every complex problem there is one solution that is simple, neat and wrong. H.L. Mencken Force laws for macroscopic objects Newton s program mandates studying nature in order
More informationChapter 4 Kinematics II: Motion in Two and Three Dimensions
Chapter 4 Kinematics II: Motion in Two and Three Dimensions Demonstrations: 1) Ball falls down and another falls out 2) Parabolic and straight line motion from two different frames. The truck with a dropping
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 informationOcean Modeling - EAS Chapter 2. We model the ocean on a rotating planet
Ocean Modeling - EAS 8803 We model the ocean on a rotating planet Rotation effects are considered through the Coriolis and Centrifugal Force The Coriolis Force arises because our reference frame (the Earth)
More informationMATHEMATICAL PHYSICS
MATHEMATICAL PHYSICS Third Year SEMESTER 1 015 016 Classical Mechanics MP350 Prof. S. J. Hands, Prof. D. M. Heffernan, Dr. J.-I. Skullerud and Dr. M. Fremling Time allowed: 1 1 hours Answer two questions
More informationTorque. Introduction. Torque. PHY torque - J. Hedberg
Torque PHY 207 - torque - J. Hedberg - 2017 1. Introduction 2. Torque 1. Lever arm changes 3. Net Torques 4. Moment of Rotational Inertia 1. Moment of Inertia for Arbitrary Shapes 2. Parallel Axis Theorem
More informationNoninertial Reference Frames
Chapter 12 Non Reference Frames 12.1 Accelerated Coordinate Systems A reference frame which is fixed with respect to a rotating rigid is not. The parade example of this is an observer fixed on the surface
More informationChapter 11 Angular Momentum; General Rotation. Copyright 2009 Pearson Education, Inc.
Chapter 11 Angular Momentum; General Rotation ! L = I!! Units of Chapter 11 Angular Momentum Objects Rotating About a Fixed Axis Vector Cross Product; Torque as a Vector Angular Momentum of a Particle
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 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 informationMidterm Feb. 17, 2009 Physics 110B Secret No.=
Midterm Feb. 17, 29 Physics 11B Secret No.= PROBLEM (1) (4 points) The radient operator = x i ê i transforms like a vector. Use ɛ ijk to prove that if B( r) = A( r), then B( r) =. B i = x i x i = x j =
More informationThe coriolis-effect in meterology
The coriolis-effect in meterology Mats Rosengren 29.12.2018 Because of its rotation the Earth was formed in a slightly flattend shape. This shape is such that the gravitational force at any point of the
More informationDynamics II: rotation L. Talley SIO 210 Fall, 2011
Dynamics II: rotation L. Talley SIO 210 Fall, 2011 DATES: Oct. 24: second problem due Oct. 24: short info about your project topic Oct. 31: mid-term Nov. 14: project due Rotation definitions Centrifugal
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
Classical Mechanics Kinematics Velocities Idea 1 Choose the appropriate frame of reference. Useful frames: - Bodies are at rest - Projections of velocities vanish - Symmetric motion Change back to the
More informationHomework 2: Solutions GFD I Winter 2007
Homework : Solutions GFD I Winter 007 1.a. Part One The goal is to find the height that the free surface at the edge of a spinning beaker rises from its resting position. The first step of this process
More informationBalanced Flow Geostrophic, Inertial, Gradient, and Cyclostrophic Flow
Balanced Flow Geostrophic, Inertial, Gradient, and Cyclostrophic Flow The types of atmospheric flows describe here have the following characteristics: 1) Steady state (meaning that the flows do not change
More informationMath review. Math review
Math review 1 Math review 3 1 series approximations 3 Taylor s Theorem 3 Binomial approximation 3 sin(x), for x in radians and x close to zero 4 cos(x), for x in radians and x close to zero 5 2 some geometry
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 informationModels of ocean circulation are all based on the equations of motion.
Equations of motion Models of ocean circulation are all based on the equations of motion. Only in simple cases the equations of motion can be solved analytically, usually they must be solved numerically.
More informationReview and Notation (Special relativity)
Review and Notation (Special relativity) December 30, 2016 7:35 PM Special Relativity: i) The principle of special relativity: The laws of physics must be the same in any inertial reference frame. In particular,
More informationPhysics 351, Spring 2017, Homework #12. Due at start of class, Friday, April 14, 2017
Physics 351, Spring 2017, Homework #12. Due at start of class, Friday, April 14, 2017 Course info is at positron.hep.upenn.edu/p351 When you finish this homework, remember to visit the feedback page at
More informationBALLISTICS. Uniform motion with respect to an inertial coordinate system.
BALLISTICS Uniform motion with respect to an inertial coordinate system. Picture 1. Animation The blue circles represent pucks sliding over a flat surface. (The slightly larger circle represents a launching
More informationPhysics 351, Spring 2017, Homework #2. Due at start of class, Friday, January 27, 2017
Physics 351, Spring 2017, Homework #2. Due at start of class, Friday, January 27, 2017 Course info is at positron.hep.upenn.edu/p351 When you finish this homework, remember to visit the feedback page at
More informationDifferential Kinematics
Differential Kinematics Relations between motion (velocity) in joint space and motion (linear/angular velocity) in task space (e.g., Cartesian space) Instantaneous velocity mappings can be obtained through
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 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 informationLecture 20 The Effects of the Earth s Rotation
Lecture 20 The Effects of the Earth s Rotation 20.1 Static Effects The Earth is spinning/rotating about an axis with (aside from slight variations) uniform angular velocity ω. Let us examine the effects
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 informationLecture PowerPoints. Chapter 11. Physics for Scientists and Engineers, with Modern Physics, 4 th edition Giancoli
Lecture PowerPoints Chapter 11 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 informationChapter 7. Geophysical Fluid Dynamics of Coastal Region
1 Lecture Notes on Fluid Dynamics (1.63J/2.21J) by Chiang C. Mei, 2006 Chapter 7. Geophysical Fluid Dynamics of Coastal egion [ef]: Joan Brown and six others (Open Univeristy course team on oceanography):
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 informationCurvilinear Coordinates
University of Alabama Department of Physics and Astronomy PH 106-4 / LeClair Fall 2008 Curvilinear Coordinates Note that we use the convention that the cartesian unit vectors are ˆx, ŷ, and ẑ, rather than
More informationPh1a: Solution to the Final Exam Alejandro Jenkins, Fall 2004
Ph1a: Solution to the Final Exam Alejandro Jenkins, Fall 2004 Problem 1 (10 points) - The Delivery A crate of mass M, which contains an expensive piece of scientific equipment, is being delivered to Caltech.
More informationStuff you asked about:
Stuff you asked about: Instrumental illness Can you go over the change in momentum with respect to the change in time being used to calcuate the net force in depth more during the lecture. We never really
More informationMechanics Lecture Notes
Mechanics Lecture Notes Lectures 0 and : Motion in a circle. Introduction The important result in this lecture concerns the force required to keep a particle moving on a circular path: if the radius of
More informationDynamics. Dynamics of mechanical particle and particle systems (many body systems)
Dynamics Dynamics of mechanical particle and particle systems (many body systems) Newton`s first law: If no net force acts on a body, it will move on a straight line at constant velocity or will stay at
More informationLecture 9 - Rotational Dynamics
Lecture 9 - Rotational Dynamics A Puzzle... Angular momentum is a 3D vector, and changing its direction produces a torque τ = dl. An important application in our daily lives is that bicycles don t fall
More informationOCN-ATM-ESS 587. Simple and basic dynamical ideas.. Newton s Laws. Pressure and hydrostatic balance. The Coriolis effect. Geostrophic balance
OCN-ATM-ESS 587 Simple and basic dynamical ideas.. Newton s Laws Pressure and hydrostatic balance The Coriolis effect Geostrophic balance Lagrangian-Eulerian coordinate frames Coupled Ocean- Atmosphere
More information