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Dynamics Kinematic Vectors Kinematic quantities refer to the motion only. • Position • Velocity • Acceleration dr v r dt vi xi a v r dxi dt ai vi xi x2 Kinematic vectors of velocity and acceleration are coplanar. a 1 r1 v1 r2 a 2 v2 x1 Force and Mass vA aA vB mA mB Two particles can influence each other. • Masses related to accelerations aB a mA B mB a A p mv pF Momentum is defined be mass and velocity. • Derivative is force • Defines equations of motion Forces on One Particle There is a net force on each particle in an object. This corresponds to the particle’s acceleration. m r 2 2 d r d ( m r ) F m a m 2 dt dt 2 Forces on a System For N particles in a system the forces add. • Particles of constant mass M m 1 FINT ( ) 0 Some forces are internal and some are external to the system. • Internal forces cancel FINT ( ) N FEXT ( ) m r N N d 2 (m r ) Ftotal F dt 2 1 1 N N N d 2 (m r ) Ftotal FEXT ( ) FINT ( ) dt 2 1 1 1 2 M N d (m r ) d2 1 N Fnet M m r M 1 dt 2 dt 2 M 1 Center of Mass The weighted average of the positions of the particles is the center of mass. Fnet N m r d 2 1 M 2 dt M M m r N The system acts like a single particle. • Force at center of mass • Translational change at center of mass rCM 1 M rCM Fnet drCM d d Fnet M MvCM pCM dt dt dt Sliding Inclined Plane F1 • Two frictionless surfaces iˆ mg sin q M mg cos q q Fg (b ) mg m( A a ) Fg (b ) F1 mA cos q ma mg sin q mAsin q 0 F1 mg cosq The block and inclined plane are both free to move. The coordinates should point along the surface. • Normal force is the force of constraint • The motion will be along the surface • Acceleration from plane and block relative to plane Motion of Plane mA cos q ma mg sin q mAsin q F1 mg cosq The inclined plane has two forces from constraints. • Upward from table • Reaction from block F2 M q F1 Fg ( p ) Mg MA Fg (b ) F2 F1 MA F1 sin q 0 Mg F2 F1 cosq iˆ The system of linear equations are solved for the accelerations. A g sin q cos q sin 2q M m cos 2q a g sin q 1 2 sin q M m Newtonian Mechanics The result compares to the simple problem of a fixed plane. There were four unknowns in the problem. • Two accelerations • Two forces of constraint F2 g sin q M F1 F1 q Fg ( p ) Mg Fg (b ) mg The constraint forces can be eliminated by using work. next