DynamicsKinematic VectorsForce and MassForces on One ParticleForces on a SystemCenter of MassSliding Inclined PlaneMotion of PlaneNewtonian MechanicsDynamicsDynamicsKinematic VectorsKinematic VectorsKinematic quantities refer to Kinematic quantities refer to the motion only.the motion only.•PositionPosition•VelocityVelocity•AccelerationAccelerationKinematic vectors of velocity Kinematic vectors of velocity and acceleration are and acceleration are coplanar.coplanar.dtdxxviiidtrdrvrvaiiixvax22r1r1v2v1a2ax1Force and MassForce and MassTwo particles can influence Two particles can influence each other.each other.•Masses related to Masses related to accelerationsaccelerationsMomentum is defined be Momentum is defined be mass and velocity.mass and velocity.•Derivative is forceDerivative is force•Defines equations of motionDefines equations of motionmAAaAvmBBaBvABBAaammvmpFpForces on One ParticleForces on One ParticleThere is a net force on each There is a net force on each particle in an object.particle in an object.This corresponds to the particle’s This corresponds to the particle’s acceleration.acceleration.2222)(dtrmddtrdmamFmrForces on a SystemForces on a SystemFor N particles in a system For N particles in a system the forces add.the forces add.•Particles of constant massParticles of constant massSome forces are internal Some forces are internal and some are external to and some are external to the system.the system.•Internal forces cancelInternal forces cancelNNtotaldtrmdFF1221)(mr)(INTF)(EXTFNNINTNEXTtotaldtrmdFFF1221)(1)()(0)(INTFNNnetrmMdtdMdtrmdMMF1221221)(NmM1Center of MassCenter of MassThe weighted average of The weighted average of the positions of the particles the positions of the particles is the is the center of masscenter of mass..The system acts like a The system acts like a single particle.single particle.•Force at center of massForce at center of mass•Translational change at Translational change at center of masscenter of massMCMrnetF MrmdtdMFNnet122 MrmrNCM1CMCMCMnetpvMdtddtrdMdtdFSliding Inclined PlaneSliding Inclined Plane1FThe block and inclined plane The block and inclined plane are both free to move.are both free to move.•Two frictionless surfacesTwo frictionless surfacesThe coordinates should point The coordinates should point along the surface.along the surface.•Normal force is the force of Normal force is the force of constraintconstraint•The motion will be along the The motion will be along the surfacesurface•Acceleration from plane and Acceleration from plane and block relative to plane block relative to planemgmFbg)(cosmgsinmgiˆM1)()( FFaAmbgsincos mgmamA cos0sin1mgFmA Motion of PlaneMotion of Plane2FThe inclined plane has two The inclined plane has two forces from constraints.forces from constraints.•Upward from tableUpward from table•Reaction from blockReaction from blockThe system of linear The system of linear equations are solved for the equations are solved for the accelerations.accelerations.mgMFpg)(iˆM12)(FFFAMbgsin1FMA cos012FFMg 1Fcossin1mgFmA sincos mgmamA mMgA2sincossinmMga22sincos1sinNewtonian MechanicsNewtonian MechanicsThe result compares to the The result compares to the simple problem of a fixed simple problem of a fixed plane.plane.There were four unknowns in There were four unknowns in the problem.the problem.•Two accelerationsTwo accelerations•Two forces of constraintTwo forces of constraintThe constraint forces can be The constraint forces can be eliminated by using work.eliminated by using