1 Big Picture2.003J/1.053JDynamics andControlI, Spring 2007 ProfessorThomasPeacock 2/7/2007 Lecture 1Newtons Laws, Cartesian and Polar Coordinates, Dynamics of a Single Particle Big PictureFirst Half of the Course Momentum Principles (Force, Vectors) NewtonianDynamicsSecondHalf of the Course LagrangianDynamics (Energy,Scalar)Bothgive equations ofmotion.Restrictions:a. Planar (2-D) Dynamics [(3-D) Dynamics 2.032 - Graduate follow oncourse]b. Linear Dynamics [Nonlinear Dynamics I: Chaos 2.050J -Undergraduatefollow oncourse]Learning ObjectivesYoumightcomment,Iveseenthismaterialbeforein18.03,8.01. Youwilluseknowledgefrompreviouscoursesincludingdierentialequationsandmechanics.After this course,youwill be able to:1. Apply knowledge to new problems2. Dene coordinate system3. Obtain equationsof motion4. Solve equationsIsaac Newtons (1642 -1727) Laws of MotionPrincipia,Chapter 1, in CambridgeTrinity College. Originallyin Latin.Cite as: Thomas Peacock and Nicolas Hadjiconstantinou, course materials for 2.003J/1.053J Dynamics andControl I, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology.Downloaded on [DD Month YYYY].Cite as: Thomas Peacock and Nicolas Hadjiconstantinou, course materials for 2.003J/1.053J Dynamics andControl I, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology.Downloaded on [DD Month YYYY].2 Inertial Frame of ReferenceI. 1st Law -A particle remains at rest or continues to move in a straight linewith constantvelocityif there is no resultantforce acting onit.II.2ndLaw- Aparticleacteduponbyaresultantforcemovesinsuchamannerthat the time rate ofchange ofits linear momentum is equalto the force.F =ma for a single particle.where F is the force,m is the mass,anda is the acceleration.III.3rdLaw- Forcesthatresultfrominteractionsofparticlesandsuchforcesbe-tweentwoparticlesareequalinmagnitude,oppositeindirection,andcollinear.a. ReadWilliams Chapter 1 and2 for backgroundi. History ofCalculus: Leibniz and Newtonii. Principia-Mostly geometricdiii. Euler wroteF =ma andF =dt(mv)Calledlawsbecausetheyhavebeentested. Cannotprove,butcandisprove.Inertial Frame of ReferencedpNewtonII:Requiresthe conceptofaninternalframeofreferencebecausedtdependsonreferenceframeinwhichmotionisobserved(thisisnotsoformassofparticle or the force appliedto the particle.)Aninternalreferenceframemustbenon-accelerating. Whatisnon-accelerating?(Newton I)Absolute inertial frame is where Newton I holds andis anidealizedsituation.Surfaceofearthisgoodenough(forshortperiodsoftime: minutes). Earthisrotating. If particle followedlong enough,will curve. (Fictitious force, coriolliseect).Denitions of Velocity and AccelerationThe velocity of a particle B is the time rate ofchange ofits position.r drv= lim =t0 t dtCite as: Thomas Peacock and Nicolas Hadjiconstantinou, course materials for 2.003J/1.053J Dynamics andControl I, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology.Downloaded on [DD Month YYYY].3 Two coordinate systems: Cartesian and Polarrisposition,andtistime. Thedirectionofvisinthedirectionofrast0.The acceleration:dv d2ra= =dt dt2Accelerationis the time rate ofchange ofits velocity.Two coordinate systems: Cartesian and PolarVelocities and accelerations can be expressed using a variety of dierent coor-dinate systems. Here are twoexamples.In Cartesian(rectangular)coordinates(x,y):Figure 1: A Cartesiancoordinatesystem. Figure by MIT OCW. ^means unit vector; means time derivative r=x+y v=r=x+ y a=v = r=x+ yIn polar coordinates(r,):See Figure 2.r=rerCite as: Thomas Peacock and Nicolas Hadjiconstantinou, course materials for 2.003J/1.053J Dynamics andControl I, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology.Downloaded on [DD Month YYYY].4 Two coordinate systems: Cartesian and PolarFigure 2: A polar coordinate system. r is the distance from origin. is theangle made with x-axis. eand erare perpendicular. Figure by MIT OCW.v=dr=drer+rder=rer+redt dt dta=dv= rer+ re+ re+rer2erdta=( rr2) er+ (r+ 2 r) ede=erdtder Proofthatdt=e:Figure3: Dierentiationofunitvectors. Changesinthedirectionofunitvectorercanbe relatedto changes in . Figure by MIT OCW.der er|er| d= lim = lim e= e=edt t0 t t0 t dter=cos+sin d er=sin+cos=(sin+cos)=edtCite as: Thomas Peacock and Nicolas Hadjiconstantinou, course materials for 2.003J/1.053J Dynamics andControl I, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology.Downloaded on [DD Month YYYY].5 Dynamics of a Single Particle (Review)Dynamics of a Single Particle (Review) Figure 4: A pointmass m movesfromr(t0) to r(t). Figure by MIT OCW.Consequences of Newtons Second Law: Linear and Angular Mo-mentum ConservationUsing aninertial frame ofreference,here is the expressionofNewton II:F =ma.Linear Momentum PrincipledF = (mv) =p (p = mv = Linear Momentum)dtIfF =0,p is constant. (ConservationofLinear Momentum)Angular Momentum PrincipleDene Angular Momentumabout B in aninertialframe of reference.hB=rmv is the moment ofthe linear momentum about B.InvestigateProperties(Take the Time Derivative)Cite as: Thomas Peacock and Nicolas Hadjiconstantinou, course materials for 2.003J/1.053J Dynamics andControl I, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology.Downloaded on [DD Month YYYY].6 Dynamics of a Single Particle (Review)Figure 5: Vector relationships between moving point mass m and angularmomentumaroundB. B isageneralpoint(couldbemoving);chooseB appropriately to help solvethe problem. Figure by MIT OCW.-dhB=r mv+rmvdt= ( RRB)mv+rF= (vmv)vBmv+rF (1)=vBmv+rF (2)The rst term in Equation (1) is zero, because the cross product of twoparallelvectorsis zero.DeneResultantTorquearoundB.B=rF isthemomentofthetotalforce aboutB. Combining this denition with Equation(2) yieldsB=hB+vBmv.If B= 0 and vB= 0 or vB mv hB= 0 (Conservation of AngularMomentum)