Sections 14.2, 15.1, 15.7 PRINCIPLES OF MECHANICS A force F(x,y,z) acts on a particle of mass m and moves it from point P1(x1,y1,4) at time t, to point P2(x2,y2,22) at time t2 along curve C. The velocity of the particle is v, at P1 and v2 at P2. / B ARC LENGTH S TIME E P(x, y, z) u 13" 12 110- TIME t, ARA LENGTH P (x,3,4) 5, 15 1º CURVE C 1- 1. × P2 (x2 48 2 ) TIME +2 ARE LENGTH $ 2 Dy
Sections 14.2, 15.1, 15.7 PRINCIPLE OF IMPULSE AND MOMENTUM m? + Í F dt = m?2 $1 PRINCIPLE OF IMPULSE OF TORQUE AND ANGULAR MOMENTUM á×m?+[?×Fdt=r2×m?2 1 PRINCIPLE OF WORK AND ENERGY 1 2 1 2 -mv + 2 1 J F . d? = - mv 2 2 P1 + P2 : C
Sections 15.2 and 15.3 CONSERVATION OF LINEAR MOMENTUM FOR A SYSTEM OF PARTICLES · The Principle of Impulse and Momentum for a system of particles is ?(m?),+ŽÍF external t2 dt = >(mv)2 t1 · ! (m'v) is the total linear momentum of the system of particles. t2 ¥ external dt is the impulse of all the external forces on the system. t1 ! · Particles experience forces between them (for example, normal forces) and external forces (for example, gravity). Only external forces contribute to the Principle of Impulse and Momentum since when summing over all forces internal forces cancel in pairs due to Newton's Third Law. · When ! F external =0: ! (mv)} = ! (mv)2 · This last equation expresses Conservation of Linear Momentum. The momenta of the individual particles in a system can change due to the forces between them, but the total momentum of the system remains unchanged. This remarkable and powerful tool allows calculation of final quantities from initial data without determining details of the motion.