Physics 181. Particle Systems


 Whitney Bradley
 3 years ago
 Views:
Transcription
1 Physcs 181 Partcle Systems Overvew In these notes we dscuss the varables approprate to the descrpton of systems of partcles, ther defntons, ther relatons, and ther conservatons laws. We consder a system of dscrete partcles for smplcty, although a contnuous dstrbuton of matter has most of the same propertes; the dfference s manly that wth partcles we have sums over them, whle n the contnuous case we have ntegrals over the volume occuped by the materal. Some of the results we wll obtan have already been used n systems of two partcles. We wll not assume the system s closed, allowng for external forces. The nternal forces wll be assumed to obey the 3 rd law n the form F(1 on 2) =!F(2 on 1). Center of Mass and Momentum of the System We denote the ndvdual partcles by an ndex = 1,2,3,..., N.Ther postons at any tme wll be denoted by r and ther masses by m. The total mass s M =! m. The center of mass (CM) of the system s defned by specfyng ts locaton: r CM = 1 M There may or may not be any of the partcles at that pont. The velocty of the CM s the tme dervatve v CM = dr CM /dt, whch gves! m r Mv CM =! m!r. The rght sde s the total momentum of all the partcles, so we have a general result: p tot = Mv CM. Wth respect to momentum, the system behaves lke a sngle partcle wth the total mass M, movng wth the velocty of the CM. Another tme dervatve gves!p tot =! m!!r. 1
2 But for each partcle the 2 nd law says m!!r = F tot, so the sum on the rght s the sum of all the forces on the partcles. Ths ncludes forces arsng from outsde the system (external forces) as well as nteractons between the partcles (nternal forces). For a gven partcle we separate these types of forces: F tot = F ext + F nt, where each term on the rght represents the total of that type of force actng on ths partcle. But the nternal forces are mutual nteractons wth the other partcles, so F nt = " F( j on ). j! The total force on the whole system then becomes F tot ext nt =! F +! F = F tot + F( j on ) ext!!. The double sum conssts entrely of symmetrc pars, such as F(1 on 2) + F(2 on 1), all of whch add to zero by the 3 rd law. Therefore we have the smplfyng result F tot = F tot ext. Wth respect to the total force, only the external forces need to be counted. That s not generally true for the work done by the forces, as we wll see later. Returnng to the momentum, we have thus shown that!p tot = F tot ext. Of course we also have!p tot = Ma CM. That s, the center of mass moves lke a sngle partcle of mass M under the nfluence of the total external force. An mmedate consequence of ths s the law of conservaton of lnear momentum: If the total external force s zero, the total momentum s constant. Ths holds component by component, as was noted earler. The CM Reference Frame So far all the quanttes have been defned wth respect to some nertal reference frame, whch we wll call the lab frame. It s useful, as we wll see, to defne a reference frame wth ts orgn at the CM of the system; ths may not be an nertal frame. In the CM frame, denote the postons of the partcles by r!. Then we have r = r CM + r!. The veloctes obey!r = v CM +!r!. Clearly " m r! = 0, so we also have " m!r! = 0,.e., the total momentum s zero n the CM frame. That the total momentum s always zero n ths frame means that the total force n that frame must be zero, but one must nclude nertal forces f the reference frame s acceleratng. 2 j"
3 Angular Momentum of the System For a sngle partcle the angular momentum about a reference pont s L = r! p, where r s the vector from the reference pont to the partcle. For a gven reference pont (whch we wll take to be the orgn of the lab system) the total angular momentum s L tot = " r! m!r. (1) Usng r = r CM + r! and!r = v CM +!r!, we get four terms n the sum:! m r CM " v CM = Mr CM " v CM = r CM " p tot. Ths term s the same as f we had a sngle partcle wth the total momentum of the system, located at the CM. We call ths the angular momentum of the CM moton, L(of CM). " m r! # v CM. Ths s zero because " m r! = 0. r CM! # m!r ". Ths s zero because the sum (the total momentum n the CM frame) s zero. # r! " m!r!. Ths s the total angular momentum as measured n the CM frame, usng the CM as reference pont. We call ths the angular momentum about the CM, L(about CM). The net result s L tot = r CM! p tot + # r "! m!r ", or L tot = L(of CM) + L(about CM). Ths breakup of an mportant dynamcal varable nto ts value when only the CM moton s consdered, plus ts value as measured relatve to the CM, s a general feature of classcal mechancs. Now we consder how the angular momentum s changed by the forces actng on the partcles. We start from Eq (1) above and take a tme dervatve:!l tot = " m [!r!!r + r!!!r ]. The frst term n [ ] s zero of course. Now we nvoke the 2 nd law: m!!r = F tot, and as before separate external and nternal forces:!l tot ext = " r! F + " r! " F( j on ). j# 3
4 The frst term s the total torque on all the partcles due to external forces only, whch we denote by N tot ext. In the double sum we have pars agan, such as r 1! F(2 on 1) + r 2! F(1 on 2). Wrtng F(2 on 1) =!F(1 on 2) from the 3 rd law, ths becomes (r 2! r 1 ) " F(1 on 2). The vector r 2! r 1 s drected from partcle 1 toward partcle 2. If the nteracton between the partcles s along that lne (.e., an attracton or repulson) then the cross product s zero, and the whole double sum gves zero. Mcroscopc analyses usng quantum mechancs show that ths assumpton about the forces s vald. The result s a smple law: Ths s often called the rotatonal 2 nd law.!l tot = N tot ext. An mmedate consequence s the law of conservaton of angular momentum: If the total external torque s zero, the total angular momentum s conserved. Of course the torque and the angular momentum must have the same reference pont. Also, lke conservaton of lnear momentum, ths holds component by component. The total torque can be dvded nto two parts, just lke the angular momentum: ext " r! F = "(r CM + r #)! F ext = r CM! F tot ext + " r #! F ext. The frst term on the rght s the torque of the total external force appled at the CM. The other term s the total external torque about the CM. The former changes the angular momentum of the CM moton; the latter changes the angular momentum about the CM:!L(of CM) = r CM! F tot ext, L(about! CM) = N tot ext (about CM). The second of these holds even though the CM frame may be nonnertal. The reason s that any nertal forces act effectvely at the CM, so they produce no torque about that pont. Knetc Energy of the System Of course the total knetc energy s the sum of the ndvdual knetc energes, so Usng the fact that v = v CM + v!, we fnd T tot = 1 2 m 2! v. T tot = 1 m 2# [v 2 CM + v! 2 + 2v " v! ]. CM Because " m v! = 0 (the total momentum s zero n the CM frame) the last term above gves zero. So we have 4
5 T tot = 1 2 Mv CM " m v! 2. The frst term s called the knetc energy of CM moton. The second term s the total knetc energy measured n the CM frame. So we have T tot = T(of CM) + T(about CM). Ths breakup nto two parts s lke that of angular momentum. Total Mechancal Energy of the System Frst, consder the total work done by all forces, n nfntesmal form: dw tot = " F tot! dr = " m a! dr. Now durng tme dt the th partcle moves through dsplacement dr = v dt, and the acceleraton s a = dv /dt, so we have a! dr = (dv /dt)! v dt = v! dv. But also d(v 2 ) = 2v! dv, so we fnd m a! dr = 1 2 m d(v 2 ) = dt. So each term n the sum above s the (nfntesmal) change n that partcles knetc energy. The sum thus represents the change n the total knetc energy, and we have the workenergy theorem for the system: dw tot = dt tot. Now dvde the forces nto conservatve and nonconservatve. We fnd dw cons + dw non!cons = dt tot. But the conservatve forces are related to potental energy by dw cons =!du tot, where U tot represents the total potental energy for all conservatve forces (nternal and external). Rearrangng the above equaton, we have dw non!cons = dt tot + du tot = de tot. Here E tot = T tot + U tot s the total mechancal energy of the system. We see that ths quantty s changed only by work done by nonconservatve forces. If only conservatve do work, we have the law of conservaton of mechancal energy: If only conservatve forces do work, the total mechancal energy s conserved. Note however that any nternal forces that do work must be conservatve too. A note about gravty near earth s surface, where we use the approxmaton that g s unform. Ths feld produces a force m g on each partcle, so the total s Mg. But what about torques? Usng the above dvson of the total external torque nto that actng at the CM plus that about the CM, we note that the latter s " m r! # g. But " m r! = 0, so ths part of the torque vanshes. Ths gves us a smple rule: 5
6 For purposes of torques, gravty acts at the CM. In a nonunform gravtatonal feld ths argument fals. One result s tdal torques on planets and moons. As for the potental energy, any constant force s conservatve, wth potental energy gven by U(r) =!F "r + const., so we have (omttng addtve constants) U grav =!m g " r. The total potental energy s thus U grav =!# m g "r. But! m r = Mr CM, so U grav =!Mg " r CM. In a unform gravtatonal feld, the formula for the potental energy of a system s the same as for a sngle partcle wth the total mass located at the CM. (If one chooses the usual coordnate system wth the yaxs postve upward from the surface of the earth, then g! r CM = "gy CM and U grav = Mgy CM. Ths s the formula one uses n ntroductory courses.) 6
PHYS 705: Classical Mechanics. Newtonian Mechanics
1 PHYS 705: Classcal Mechancs Newtonan Mechancs Quck Revew of Newtonan Mechancs Basc Descrpton: An dealzed pont partcle or a system of pont partcles n an nertal reference frame [Rgd bodes (ch. 5 later)]
More informationPhysics 53. Rotational Motion 3. Sir, I have found you an argument, but I am not obliged to find you an understanding.
Physcs 53 Rotatonal Moton 3 Sr, I have found you an argument, but I am not oblged to fnd you an understandng. Samuel Johnson Angular momentum Wth respect to rotatonal moton of a body, moment of nerta plays
More informationStudy Guide For Exam Two
Study Gude For Exam Two Physcs 2210 Albretsen Updated: 08/02/2018 All Other Prevous Study Gudes Modules 0106 Module 07 Work Work done by a constant force F over a dstance s : Work done by varyng force
More informationSo far: simple (planar) geometries
Physcs 06 ecture 5 Torque and Angular Momentum as Vectors SJ 7thEd.: Chap. to 3 Rotatonal quanttes as vectors Cross product Torque epressed as a vector Angular momentum defned Angular momentum as a vector
More informationWeek 11: Chapter 11. The Vector Product. The Vector Product Defined. The Vector Product and Torque. More About the Vector Product
The Vector Product Week 11: Chapter 11 Angular Momentum There are nstances where the product of two vectors s another vector Earler we saw where the product of two vectors was a scalar Ths was called the
More informationFirst Law: A body at rest remains at rest, a body in motion continues to move at constant velocity, unless acted upon by an external force.
Secton 1. Dynamcs (Newton s Laws of Moton) Two approaches: 1) Gven all the forces actng on a body, predct the subsequent (changes n) moton. 2) Gven the (changes n) moton of a body, nfer what forces act
More informationPhysics 207: Lecture 20. Today s Agenda Homework for Monday
Physcs 207: Lecture 20 Today s Agenda Homework for Monday Recap: Systems of Partcles Center of mass Velocty and acceleraton of the center of mass Dynamcs of the center of mass Lnear Momentum Example problems
More informationLinear Momentum. Center of Mass.
Lecture 6 Chapter 9 Physcs I 03.3.04 Lnear omentum. Center of ass. Course webste: http://faculty.uml.edu/ndry_danylov/teachng/physcsi Lecture Capture: http://echo360.uml.edu/danylov03/physcssprng.html
More informationChapter 11 Angular Momentum
Chapter 11 Angular Momentum Analyss Model: Nonsolated System (Angular Momentum) Angular Momentum of a Rotatng Rgd Object Analyss Model: Isolated System (Angular Momentum) Angular Momentum of a Partcle
More informationConservation of Angular Momentum = "Spin"
Page 1 of 6 Conservaton of Angular Momentum = "Spn" We can assgn a drecton to the angular velocty: drecton of = drecton of axs + rght hand rule (wth rght hand, curl fngers n drecton of rotaton, thumb ponts
More informationA particle in a state of uniform motion remain in that state of motion unless acted upon by external force.
The fundamental prncples of classcal mechancs were lad down by Galleo and Newton n the 16th and 17th centures. In 1686, Newton wrote the Prncpa where he gave us three laws of moton, one law of gravty,
More informationClassical Mechanics ( Particles and Biparticles )
Classcal Mechancs ( Partcles and Bpartcles ) Alejandro A. Torassa Creatve Commons Attrbuton 3.0 Lcense (0) Buenos Ares, Argentna atorassa@gmal.com Abstract Ths paper consders the exstence of bpartcles
More informationPart C Dynamics and Statics of Rigid Body. Chapter 5 Rotation of a Rigid Body About a Fixed Axis
Part C Dynamcs and Statcs of Rgd Body Chapter 5 Rotaton of a Rgd Body About a Fxed Axs 5.. Rotatonal Varables 5.. Rotaton wth Constant Angular Acceleraton 5.3. Knetc Energy of Rotaton, Rotatonal Inerta
More informationSpring Force and Power
Lecture 13 Chapter 9 Sprng Force and Power Yeah, energy s better than orces. What s net? Course webste: http://aculty.uml.edu/andry_danylov/teachng/physcsi IN THIS CHAPTER, you wll learn how to solve problems
More informationPhysics 111: Mechanics Lecture 11
Physcs 111: Mechancs Lecture 11 Bn Chen NJIT Physcs Department Textbook Chapter 10: Dynamcs of Rotatonal Moton q 10.1 Torque q 10. Torque and Angular Acceleraton for a Rgd Body q 10.3 RgdBody Rotaton
More informationNEWTON S LAWS. These laws only apply when viewed from an inertial coordinate system (unaccelerated system).
EWTO S LAWS Consder two partcles. 1 1. If 1 0 then 0 wth p 1 m1v. 1 1 2. 1.. 3. 11 These laws only apply when vewed from an nertal coordnate system (unaccelerated system). consder a collecton of partcles
More informationCHAPTER 10 ROTATIONAL MOTION
CHAPTER 0 ROTATONAL MOTON 0. ANGULAR VELOCTY Consder argd body rotates about a fxed axs through pont O n xy plane as shown. Any partcle at pont P n ths rgd body rotates n a crcle of radus r about O. The
More information11. Dynamics in Rotating Frames of Reference
Unversty of Rhode Island DgtalCommons@URI Classcal Dynamcs Physcs Course Materals 2015 11. Dynamcs n Rotatng Frames of Reference Gerhard Müller Unversty of Rhode Island, gmuller@ur.edu Creatve Commons
More informationPhysics 5153 Classical Mechanics. Principle of Virtual Work1
P. Guterrez 1 Introducton Physcs 5153 Classcal Mechancs Prncple of Vrtual Work The frst varatonal prncple we encounter n mechancs s the prncple of vrtual work. It establshes the equlbrum condton of a mechancal
More informationPhysics 141. Lecture 14. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 14, Page 1
Physcs 141. Lecture 14. Frank L. H. Wolfs Department of Physcs and Astronomy, Unversty of Rochester, Lecture 14, Page 1 Physcs 141. Lecture 14. Course Informaton: Lab report # 3. Exam # 2. MultPartcle
More informationPY2101 Classical Mechanics Dr. Síle Nic Chormaic, Room 215 D Kane Bldg
PY2101 Classcal Mechancs Dr. Síle Nc Chormac, Room 215 D Kane Bldg s.ncchormac@ucc.e Lectures stll some ssues to resolve. Slots shared between PY2101 and PY2104. Hope to have t fnalsed by tomorrow. Mondays
More informationRigid body simulation
Rgd bod smulaton Rgd bod smulaton Once we consder an object wth spacal etent, partcle sstem smulaton s no longer suffcent Problems Problems Unconstraned sstem rotatonal moton torques and angular momentum
More informationWeek 9 Chapter 10 Section 15
Week 9 Chapter 10 Secton 15 Rotaton Rgd Object A rgd object s one that s nondeformable The relatve locatons of all partcles makng up the object reman constant All real objects are deformable to some extent,
More informationLecture 20: Noether s Theorem
Lecture 20: Noether s Theorem In our revew of Newtonan Mechancs, we were remnded that some quanttes (energy, lnear momentum, and angular momentum) are conserved That s, they are constant f no external
More informationThe classical spinrotation coupling
LOUAI H. ELZEIN 2018 All Rghts Reserved The classcal spnrotaton couplng Loua Hassan Elzen Basher Khartoum, Sudan. Postal code:11123 louaelzen@gmal.com Abstract Ths paper s prepared to show that a rgd
More informationChapter 8. Potential Energy and Conservation of Energy
Chapter 8 Potental Energy and Conservaton of Energy In ths chapter we wll ntroduce the followng concepts: Potental Energy Conservatve and nonconservatve forces Mechancal Energy Conservaton of Mechancal
More informationSpring 2002 Lecture #13
4450 Sprng 00 ecture # Dr. Jaehoon Yu. Rotatonal Energy. Computaton of oments of nerta. Parallelas Theorem 4. Torque & Angular Acceleraton 5. Work, Power, & Energy of Rotatonal otons Remember the mdterm
More informationChapter 3 and Chapter 4
Chapter 3 and Chapter 4 Chapter 3 Energy 3. Introducton:Work Work W s energy transerred to or rom an object by means o a orce actng on the object. Energy transerred to the object s postve work, and energy
More informationConservation of Energy
Lecture 3 Chapter 8 Physcs I 0.3.03 Conservaton o Energy Course webste: http://aculty.uml.edu/andry_danylov/teachng/physcsi Lecture Capture: http://echo360.uml.edu/danylov03/physcsall.html 95.4, Fall 03,
More informationWeek3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity
Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1s tme nterval. The velocty of the partcle
More informationELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM
ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look
More informationPhysics 607 Exam 1. ( ) = 1, Γ( z +1) = zγ( z) x n e x2 dx = 1. e x2
Physcs 607 Exam 1 Please be wellorganzed, and show all sgnfcant steps clearly n all problems. You are graded on your wor, so please do not just wrte down answers wth no explanaton! Do all your wor on
More informationPhysics 207 Lecture 13. Lecture 13
Physcs 07 Lecture 3 Goals: Lecture 3 Chapter 0 Understand the relatonshp between moton and energy Defne Potental Energy n a Hooke s Law sprng Develop and explot conservaton of energy prncple n problem
More information10/23/2003 PHY Lecture 14R 1
Announcements. Remember  Tuesday, Oct. 8 th, 9:30 AM Second exam (coverng Chapters 94 of HRW) Brng the followng: a) equaton sheet b) Calculator c) Pencl d) Clear head e) Note: If you have kept up wth
More informationCelestial Mechanics. Basic Orbits. Why circles? Tycho Brahe. PHY celestialmechanics  J. Hedberg
PHY 454  celestalmechancs  J. Hedberg  207 Celestal Mechancs. Basc Orbts. Why crcles? 2. Tycho Brahe 3. Kepler 4. 3 laws of orbtng bodes 2. Newtonan Mechancs 3. Newton's Laws. Law of Gravtaton 2. The
More informationχ x B E (c) Figure 2.1.1: (a) a material particle in a body, (b) a place in space, (c) a configuration of the body
Secton.. Moton.. The Materal Body and Moton hyscal materals n the real world are modeled usng an abstract mathematcal entty called a body. Ths body conssts of an nfnte number of materal partcles. Shown
More information12. The HamiltonJacobi Equation Michael Fowler
1. The HamltonJacob Equaton Mchael Fowler Back to Confguraton Space We ve establshed that the acton, regarded as a functon of ts coordnate endponts and tme, satsfes ( ) ( ) S q, t / t+ H qpt,, = 0, and
More informationPhysics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian1
P. Guterrez Physcs 5153 Classcal Mechancs D Alembert s Prncple and The Lagrangan 1 Introducton The prncple of vrtual work provdes a method of solvng problems of statc equlbrum wthout havng to consder the
More information1 What is a conservation law?
MATHEMATICS 7302 (Analytcal Dynamcs) YEAR 2016 2017, TERM 2 HANDOUT #6: MOMENTUM, ANGULAR MOMENTUM, AND ENERGY; CONSERVATION LAWS In ths handout we wll develop the concepts of momentum, angular momentum,
More informationPHYSICS 231 Review problems for midterm 2
PHYSICS 31 Revew problems for mdterm Topc 5: Energy and Work and Power Topc 6: Momentum and Collsons Topc 7: Oscllatons (sprng and pendulum) Topc 8: Rotatonal Moton The nd exam wll be Wednesday October
More informationA Tale of Friction Basic Rollercoaster Physics. Fahrenheit Rollercoaster, Hershey, PA max height = 121 ft max speed = 58 mph
A Tale o Frcton Basc Rollercoaster Physcs Fahrenhet Rollercoaster, Hershey, PA max heght = 11 t max speed = 58 mph PLAY PLAY PLAY PLAY Rotatonal Movement Knematcs Smlar to how lnear velocty s dened, angular
More informationMoments of Inertia. and reminds us of the analogous equation for linear momentum p= mv, which is of the form. The kinetic energy of the body is.
Moments of Inerta Suppose a body s movng on a crcular path wth constant speed Let s consder two quanttes: the body s angular momentum L about the center of the crcle, and ts knetc energy T How are these
More informationCenter of Mass and Linear Momentum
PH 2212A Fall 2014 Center of Mass and Lnear Momentum Lectures 1415 Chapter 9 (Hallday/Resnck/Walker, Fundamentals of Physcs 9 th edton) 1 Chapter 9 Center of Mass and Lnear Momentum In ths chapter we
More informationChapter 11: Angular Momentum
Chapter 11: ngular Momentum Statc Equlbrum In Chap. 4 we studed the equlbrum of pontobjects (mass m) wth the applcaton of Newton s aws F 0 F x y, 0 Therefore, no lnear (translatonal) acceleraton, a0 For
More informationcoordinates. Then, the position vectors are described by
Revewng, what we have dscussed so far: Generalzed coordnates Any number of varables (say, n) suffcent to specfy the confguraton of the system at each nstant to tme (need not be the mnmum number). In general,
More informationCHAPTER 8 Potential Energy and Conservation of Energy
CHAPTER 8 Potental Energy and Conservaton o Energy One orm o energy can be converted nto another orm o energy. Conservatve and nonconservatve orces Physcs 1 Knetc energy: Potental energy: Energy assocated
More informationAngular Momentum and Fixed Axis Rotation. 8.01t Nov 10, 2004
Angular Momentum and Fxed Axs Rotaton 8.01t Nov 10, 2004 Dynamcs: Translatonal and Rotatonal Moton Translatonal Dynamcs Total Force Torque Angular Momentum about Dynamcs of Rotaton F ext Momentum of a
More informationτ rf = Iα I point = mr 2 L35 F 11/14/14 a*er lecture 1
A mass s attached to a long, massless rod. The mass s close to one end of the rod. Is t easer to balance the rod on end wth the mass near the top or near the bottom? Hnt: Small α means sluggsh behavor
More informationPhysics for Scientists and Engineers. Chapter 9 Impulse and Momentum
Physcs or Scentsts and Engneers Chapter 9 Impulse and Momentum Sprng, 008 Ho Jung Pak Lnear Momentum Lnear momentum o an object o mass m movng wth a velocty v s dened to be p mv Momentum and lnear momentum
More informationWeek 8: Chapter 9. Linear Momentum. Newton Law and Momentum. Linear Momentum, cont. Conservation of Linear Momentum. Conservation of Momentum, 2
Lnear omentum Week 8: Chapter 9 Lnear omentum and Collsons The lnear momentum of a partcle, or an object that can be modeled as a partcle, of mass m movng wth a velocty v s defned to be the product of
More informationWork is the change in energy of a system (neglecting heat transfer). To examine what could
Work Work s the change n energy o a system (neglectng heat transer). To eamne what could cause work, let s look at the dmensons o energy: L ML E M L F L so T T dmensonally energy s equal to a orce tmes
More informationArmy Ants Tunneling for Classical Simulations
Electronc Supplementary Materal (ESI) for Chemcal Scence. Ths journal s The Royal Socety of Chemstry 2014 electronc supplementary nformaton (ESI) for Chemcal Scence Army Ants Tunnelng for Classcal Smulatons
More informationRotational Dynamics. Physics 1425 Lecture 19. Michael Fowler, UVa
Rotatonal Dynamcs Physcs 1425 Lecture 19 Mchael Fowler, UVa Rotatonal Dynamcs Newton s Frst Law: a rotatng body wll contnue to rotate at constant angular velocty as long as there s no torque actng on t.
More informationMechanics Physics 151
Mechancs Physcs 151 Lecture 3 Lagrange s Equatons (Goldsten Chapter 1) Hamlton s Prncple (Chapter 2) What We Dd Last Tme! Dscussed multpartcle systems! Internal and external forces! Laws of acton and
More informationChapter 11 Torque and Angular Momentum
Chapter Torque and Angular Momentum I. Torque II. Angular momentum  Defnton III. Newton s second law n angular form IV. Angular momentum  System of partcles  Rgd body  Conservaton I. Torque  Vector
More informationPhysics 106 Lecture 6 Conservation of Angular Momentum SJ 7 th Ed.: Chap 11.4
Physcs 6 ecture 6 Conservaton o Angular Momentum SJ 7 th Ed.: Chap.4 Comparson: dentons o sngle partcle torque and angular momentum Angular momentum o a system o partcles o a rgd body rotatng about a xed
More informationSpinrotation coupling of the angularly accelerated rigid body
Spnrotaton couplng of the angularly accelerated rgd body Loua Hassan Elzen Basher Khartoum, Sudan. Postal code:11123 Emal: louaelzen@gmal.com November 1, 2017 All Rghts Reserved. Abstract Ths paper s
More informationGravitational Acceleration: A case of constant acceleration (approx. 2 hr.) (6/7/11)
Gravtatonal Acceleraton: A case of constant acceleraton (approx. hr.) (6/7/11) Introducton The gravtatonal force s one of the fundamental forces of nature. Under the nfluence of ths force all objects havng
More informationCHAPTER 6. LAGRANGE S EQUATIONS (Analytical Mechanics)
CHAPTER 6 LAGRANGE S EQUATIONS (Analytcal Mechancs) 1 Ex. 1: Consder a partcle movng on a fxed horzontal surface. r P Let, be the poston and F be the total force on the partcle. The FBD s: mgk F 1 x O
More informationˆ (0.10 m) E ( N m /C ) 36 ˆj ( j C m)
7.. = = 3 = 4 = 5. The electrc feld s constant everywhere between the plates. Ths s ndcated by the electrc feld vectors, whch are all the same length and n the same drecton. 7.5. Model: The dstances to
More informationPhysics 207: Lecture 27. Announcements
Physcs 07: ecture 7 Announcements akeup labs are ths week Fnal hwk assgned ths week, fnal quz next week Revew sesson on Thursday ay 9, :30 4:00pm, Here Today s Agenda Statcs recap Beam & Strngs» What
More informationMathematical Preparations
1 Introducton Mathematcal Preparatons The theory of relatvty was developed to explan experments whch studed the propagaton of electromagnetc radaton n movng coordnate systems. Wthn expermental error the
More informationPhysics 106a, Caltech 11 October, Lecture 4: Constraints, Virtual Work, etc. Constraints
Physcs 106a, Caltech 11 October, 2018 Lecture 4: Constrants, Vrtual Work, etc. Many, f not all, dynamcal problems we want to solve are constraned: not all of the possble 3 coordnates for M partcles (or
More informationNotes on Analytical Dynamics
Notes on Analytcal Dynamcs Jan Peters & Mchael Mstry October 7, 004 Newtonan Mechancs Basc Asssumptons and Newtons Laws Lonely pontmasses wth postve mass Newtons st: Constant velocty v n an nertal frame
More informationLecture 22: Potential Energy
Lecture : Potental Energy We have already studed the workenergy theorem, whch relates the total work done on an object to the change n knetc energy: Wtot = KE For a conservatve orce, the work done by
More informationMechanics Physics 151
Mechancs Physcs 5 Lecture 7 Specal Relatvty (Chapter 7) What We Dd Last Tme Worked on relatvstc knematcs Essental tool for epermental physcs Basc technques are easy: Defne all 4 vectors Calculate com
More information9 Derivation of Rate Equations from SingleCell Conductance (HodgkinHuxleylike) Equations
Physcs 171/271  Chapter 9R Davd Klenfeld  Fall 2005 9 Dervaton of Rate Equatons from SngleCell Conductance (HodgknHuxleylke) Equatons We consder a network of many neurons, each of whch obeys a set
More informationLagrangian Theory. Severalbody Systems
Lagrangan Theory of Severalbody Systems Ncholas Wheeler, Reed College Physcs Department November 995 Introducton. Let the Ntuple of 3vectors {x (t) : =, 2,..., N} descrbe, relatve to an nertal frame,
More informationChapter 8: Potential Energy and The Conservation of Total Energy
Chapter 8: Potental Energy and The Conservaton o Total Energy Work and knetc energy are energes o moton. K K K mv r v v F dr Potental energy s an energy that depends on locaton. Dmenson F x d U( x) dx
More informationA how to guide to second quantization method.
Phys. 67 (Graduate Quantum Mechancs Sprng 2009 Prof. Pu K. Lam. Verson 3 (4/3/2009 A how to gude to second quantzaton method. > Second quantzaton s a mathematcal notaton desgned to handle dentcal partcle
More informationAngular momentum. Instructor: Dr. Hoi Lam TAM ( 譚海嵐 )
Angular momentum Instructor: Dr. Ho Lam TAM ( 譚海嵐 ) Physcs Enhancement Programme or Gted Students The Hong Kong Academy or Gted Educaton and Department o Physcs, HKBU Department o Physcs Hong Kong Baptst
More informationPhysics 207 Lecture 6
Physcs 207 Lecture 6 Agenda: Physcs 207, Lecture 6, Sept. 25 Chapter 4 Frames of reference Chapter 5 ewton s Law Mass Inerta s (contact and noncontact) Frcton (a external force that opposes moton) Free
More informationand Statistical Mechanics Material Properties
Statstcal Mechancs and Materal Propertes By Kuno TAKAHASHI Tokyo Insttute of Technology, Tokyo 15855, JAPA Phone/Fax +81357343915 takahak@de.ttech.ac.jp http://www.de.ttech.ac.jp/~ktlab/ Only for
More informationModeling of Dynamic Systems
Modelng of Dynamc Systems Ref: Control System Engneerng Norman Nse : Chapters & 3 Chapter objectves : Revew the Laplace transform Learn how to fnd a mathematcal model, called a transfer functon Learn how
More informationImplicit Integration Henyey Method
Implct Integraton Henyey Method In realstc stellar evoluton codes nstead of a drect ntegraton usng for example the RungeKutta method one employs an teratve mplct technque. Ths s because the structure
More informationIn this section is given an overview of the common elasticity models.
Secton 4.1 4.1 Elastc Solds In ths secton s gven an overvew of the common elastcty models. 4.1.1 The Lnear Elastc Sold The classcal Lnear Elastc model, or Hooean model, has the followng lnear relatonshp
More informationChapter 07: Kinetic Energy and Work
Chapter 07: Knetc Energy and Work Conservaton o Energy s one o Nature s undamental laws that s not volated. Energy can take on derent orms n a gven system. Ths chapter we wll dscuss work and knetc energy.
More informationPES 1120 Spring 2014, Spendier Lecture 6/Page 1
PES 110 Sprng 014, Spender Lecture 6/Page 1 Lecture today: Chapter 1) Electrc feld due to charge dstrbutons > charged rod > charged rng We ntroduced the electrc feld, E. I defned t as an nvsble aura
More informationChapter 3. r r. Position, Velocity, and Acceleration Revisited
Chapter 3 Poston, Velocty, and Acceleraton Revsted The poston vector of a partcle s a vector drawn from the orgn to the locaton of the partcle. In two dmensons: r = x ˆ+ yj ˆ (1) The dsplacement vector
More informationChapter 7: Conservation of Energy
Lecture 7: Conservaton o nergy Chapter 7: Conservaton o nergy Introucton I the quantty o a subject oes not change wth tme, t means that the quantty s conserve. The quantty o that subject remans constant
More informationTransfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system
Transfer Functons Convenent representaton of a lnear, dynamc model. A transfer functon (TF) relates one nput and one output: x t X s y t system Y s The followng termnology s used: x y nput output forcng
More information1. Review of Mechanics Newton s Laws
. Revew of Mechancs.. Newton s Laws Moton of partcles. Let the poston of the partcle be gven by r. We can always express ths n Cartesan coordnates: r = xˆx + yŷ + zẑ, () where we wll always use ˆ (crcumflex)
More informationModule 1 : The equation of continuity. Lecture 1: Equation of Continuity
1 Module 1 : The equaton of contnuty Lecture 1: Equaton of Contnuty 2 Advanced Heat and Mass Transfer: Modules 1. THE EQUATION OF CONTINUITY : Lectures 16 () () () (v) (v) Overall Mass Balance Momentum
More informationClassical Field Theory
Classcal Feld Theory Before we embark on quantzng an nteractng theory, we wll take a dverson nto classcal feld theory and classcal perturbaton theory and see how far we can get. The reader s expected to
More informationIterative General Dynamic Model for SerialLink Manipulators
EEL6667: Knematcs, Dynamcs and Control of Robot Manpulators 1. Introducton Iteratve General Dynamc Model for SeralLnk Manpulators In ths set of notes, we are gong to develop a method for computng a general
More informationSymmetric Lie Groups and Conservation Laws in Physics
Symmetrc Le Groups and Conservaton Laws n Physcs Audrey Kvam May 1, 1 Abstract Ths paper eamnes how conservaton laws n physcs can be found from analyzng the symmetrc Le groups of certan physcal systems.
More informationCHAPTER 7 ENERGY BALANCES SYSTEM SYSTEM. * What is energy? * Forms of Energy.  Kinetic energy (KE)  Potential energy (PE) PE = mgz
SYSTM CHAPTR 7 NRGY BALANCS 1 7.17. SYSTM nergy & 1st Law of Thermodynamcs * What s energy? * Forms of nergy  Knetc energy (K) K 1 mv  Potental energy (P) P mgz  Internal energy (U) * Total nergy,
More informationPoisson brackets and canonical transformations
rof O B Wrght Mechancs Notes osson brackets and canoncal transformatons osson Brackets Consder an arbtrary functon f f ( qp t) df f f f q p q p t But q p p where ( qp ) pq q df f f f p q q p t In order
More informationChapter 9. The Dot Product (Scalar Product) The Dot Product use (Scalar Product) The Dot Product (Scalar Product) The Cross Product.
The Dot Product (Scalar Product) Chapter 9 Statcs and Torque The dot product of two vectors can be constructed by takng the component of one vector n the drecton of the other and multplyng t tmes the magntude
More informationLagrangian Field Theory
Lagrangan Feld Theory Adam Lott PHY 391 Aprl 6, 017 1 Introducton Ths paper s a summary of Chapter of Mandl and Shaw s Quantum Feld Theory [1]. The frst thng to do s to fx the notaton. For the most part,
More information1 Derivation of Rate Equations from SingleCell Conductance (HodgkinHuxleylike) Equations
Physcs 171/271 Davd Klenfeld  Fall 2005 (revsed Wnter 2011) 1 Dervaton of Rate Equatons from SngleCell Conductance (HodgknHuxleylke) Equatons We consder a network of many neurons, each of whch obeys
More informationClassical Mechanics Virtual Work & d Alembert s Principle
Classcal Mechancs Vrtual Work & d Alembert s Prncple Dpan Kumar Ghosh UMDAE Centre for Excellence n Basc Scences Kalna, Mumba 400098 August 15, 2016 1 Constrants Moton of a system of partcles s often
More informationRIGID BODY MOTION. Next, we rotate counterclockwise about ξ by angle. Next we rotate counterclockwise about γ by angle to get the final set (x,y z ).
RGD BODY MOTON We now consder the moton of rgd bodes. The frst queston s what coordnates are needed to specf the locaton and orentaton of such an object. Clearl 6 are needed 3 to locate a partcular pont
More informationROTATIONAL MOTION. dv d F m m V v dt dt. i i i cm i
ROTATIONAL MOTION Consder a collecton of partcles, m, located at R relatve to an nertal coordnate system. As before wrte: where R cm locates the center of mass. R Rcm r Wrte Newton s second law for the
More informationEMU Physics Department
Physcs 0 Lecture 8 Potental Energy and Conservaton Assst. Pro. Dr. Al ÖVGÜN EMU Physcs Department www.aovgun.com Denton o Work W q The work, W, done by a constant orce on an object s dened as the product
More informationRecitation: Energy, Phys Energies. 1.2 Three stones. 1. Energy. 1. An acorn falling from an oak tree onto the sidewalk.
Rectaton: Energy, Phys 207. Energy. Energes. An acorn fallng from an oak tree onto the sdewalk. The acorn ntal has gravtatonal potental energy. As t falls, t converts ths energy to knetc. When t hts the
More informationIntegrals and Invariants of EulerLagrange Equations
Lecture 16 Integrals and Invarants of EulerLagrange Equatons ME 256 at the Indan Insttute of Scence, Bengaluru Varatonal Methods and Structural Optmzaton G. K. Ananthasuresh Professor, Mechancal Engneerng,
More informationLecture 4. Macrostates and Microstates (Ch. 2 )
Lecture 4. Macrostates and Mcrostates (Ch. ) The past three lectures: we have learned about thermal energy, how t s stored at the mcroscopc level, and how t can be transferred from one system to another.
More informationThe Feynman path integral
The Feynman path ntegral Aprl 3, 205 Hesenberg and Schrödnger pctures The Schrödnger wave functon places the tme dependence of a physcal system n the state, ψ, t, where the state s a vector n Hlbert space
More informationOpen Systems: Chemical Potential and Partial Molar Quantities Chemical Potential
Open Systems: Chemcal Potental and Partal Molar Quanttes Chemcal Potental For closed systems, we have derved the followng relatonshps: du = TdS pdv dh = TdS + Vdp da = SdT pdv dg = VdP SdT For open systems,
More information