Transcription of Newton’s laws of Motion
1 Newton s laws of MotionLecture of Newtonian MechanicsThree fundamental quantities: (i) Mass, (ii) Motion & (iii) ForceExcerpts from Newton s Principia (Book 1 )MassThequantityofmatteristhemeasureofth esamearisingfromit quantity of Motion is the measure of the same arising from the velocity and quantity of matter (Definiton# 1)Thevisinsita:aninnateforcesofmatter,is apowerofresisting,bywhicheverybody,asmuc hasinitlies,continuesinitspresentstate,w hetheritbeofrest,orofmovinguniformlyforw ardinarightline. Force (Definition # 2)Animpressedforceisanactionexertedupona body,inordertochangeitsstate,eitherofres t, definitions gave rise to the famous three laws : known as Newton s laws of 1 Everybodycontinuesinit 2 The change of Motion is proportional to the applied force and takes place in the direction of the straight line along which that force 3 Toeveryactionthereisalwaysanequalandcont raryreaction; solving Newton s laws we shall find r(t).
2 R(t) = 0: implies that the body is in rest for all general, r(t)=(x(t), y(t), z(t))or (r(t), , )Examplerepresentsuniformmotioninthex-di rectionwithasthevelocity,inastateofresti nthey-directionandishavingauniformveloci tyandafreefallinthegravitationalfield.() t ()t 0( ) (;xr tv t x 0;20/ 2)zv t zgt xvzvMechanics of particlesClassical MechanicsNon-relativisticRelativisticQua ntum MechanicsNon-Relativistic(SchroedingerEq uation)Relativistic(Dirac Equation)(Newton s laws )(Special Theoryof Relativity)Newton s first law of a definition of (zero) force an inertial frame: If the relative velocity between the two reference frames is constant, then the relative acceleration between the two reference frames is zero, and the reference frames are considered to be inertial reference frames. The inertial frame is then simply a frame of reference in which the first law first law does not hold in an arbitrary frame.
3 For example, it fails in the frame of a rotating :Whenabodymoveswithconstantvelocityinast raightline, ,thentheremustbeanacceleration, SS Rr'r',dRrr vtvdt Galilean transformationIs Earth an inertial frame?Newton s Second law of Motion :Ifanyforcegeneratesachangeinmotio n,adoubleforcewillgeneratedoublechangein themotion,atripleforcewillcorrespondtotr iplechangeinthemotion, that a force is applied to a body for a time interval t . The impressed force or impulse produces a change in the momentum of the body, The instantaneous action of the total force acting on a body at a time t is defined by taking the mathematical limit as the time interval t becomes smaller and smaller, F1m11maF2m22ma1221mama Inertial massChange of Motion is described by the change in momentum of body. For a point mass particle, the momentum is defined asp mv totalddvFmvmdtdt Inertial mass Gravitational massNewton s third law of , ,2F2,1 FIftheaccelerationofabodyistheresultofan outsideforce, ,butaslongastheforcesareequalandopposite ,Newton' real Forces arise due to interaction!
4 1212121221212121212212 Gravitational force: Coulomb force: mmFGrrrrqqFkrFFr Newton s 3rdlaw emphasizes Conservation of Momentum Validity of Newton s laws Validity of the first two laws -The first law is always valid (add a pseudo force). -The second law F = holds but Fand p have different expressions in the relativistic limit. The 3rdlaw is not valid in the relativistic limit. Why????Consider two positive chargesMomentum conservation is not validApplication of Newton s laws : PrescriptionStep 1: Divide a composite system into constituent systems each of which can be treated as a point 2: Draw free body force diagrams for each point 3: Introduce a coordinate system, the inertial frame, and write the equations of 4: Motion of a body may be constrained to move along certain path or plane. Express each constraint by an equation called constraint 6: Identify the number of unknown quantities.
5 There must be enough number of equations ( Equations of Motion + constraint equations) to solve for all the unknown quantities. Example 1 Identify the constraintsEOM in x and y-directions2 Velocity and acceleration in cylindrical polar coordinates : cossinrij zkzk cossin sincos ijijzk Example ,BisheldstationaryandArotatesatconstantr adiusrowithsteadyangularvelocity ,whatisitsaccelerationimmediatelyafterwa rd?Two movable bodies and their free body diagrams:Cylindrical polar coordinate after B is released = 0and Equations of Motion :Constraint equations:Unknowns:,, ,ra a z T Four unknowns and four 0 Example ,pulleyismassless, body diagrams:MFT2 MmF1mT2m2mMF2mgTEquations of Motion :11xm aT 2 MmxF T FMa 22222 2m MxyFm am g Tm a 3rdlaw of Motion :22 Mmm MFF Constraint equations:22112 0xxxxyx xaax xy Laaa Specification of coordinates:Unknowns:12122, ,,,,,xxxyT F F a a a aThere are seven equations and seven : 1212221212xxF mmm m gaaM mmmm m 22121212xm FM m gaM mmmm m 2122221212ym M mm g m FaM mmmm m Example 4 Find the velocity of the mass at subsequent times.
