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Chapter 9: Linear Momentum & Collisions Reading assignment: Chapter 9.5-9.7 Homework : (due Wednesday, Oct. 5, 2005): Problems: 32, 36, 43, 52, 69, 71 • Center of mass • Momentum p mv • Momentum is conserved Center of mass Center of mass for many particles: rCM ______ i M Black board example 9.1 Where is the center of mass of the arrangement of particles below. (m3 = 2 kg and m1 = m2 = 1 kg)? A method for finding the center of mass of any object. - Hang object from two or more points. - Draw extension of suspension line. - Center of mass is at intercept of these lines. Center of mass of a solid body (uniform density) xCM yCM zCM 1 xdV __ 1 ydV __ 1 zdV __ Black board example 9.2 A uniform square plate 6 m on a side has had a square piece 2 m on a side cut out of it. The center of that piece is at x = 2 m, y = 0. The center of the square plate is at x = y = 0. Find the coordinates of the center of mass of the remaining piece. Motion of a System of Particles. Newton’s second law for a System of Particles The ____________ of a system of particles (combined mass M) moves like one equivalent particle of mass M would move under the influence of an external force. Fnet MaCM Fnet , x MaCM , x Fnet , y MaCM , y Fnet , z MaCM , z A rocket is shot up in the air and explodes. Describe the motion of the center of mass before and after the explosion. Linear Momentum The linear momentum of a particle of mass m and velocity v is defined as p __ v The linear momentum is a vector quantity. It’s direction is along v. The components of the momentum of a particle: px m vx py m vy pz m vz dp d (m v ) From Newton’s second law: Fnet __ __ dt dt The time rate of change in linear momentum is equal to the net forces acting on the particle. This is also true for a system of particles: P M vCM Total momentum = Total mass ·velocity of center of mass And: Net external force = ____________ in momentum of the center of mass dP Fnet dt Conservation of _________ momentum Thus: If no _________________ is acting on a particle, it’s momentum is conserved. This is also true for a system of particles: If no external forces interact with a system of particles the total momentum of the system remains constant. P p p1 p2 constant or : Pi Pf p1i p2i p1 f p2 f Black board example 9.3 You (100kg) and your skinny friend (50.0 kg) stand face-to-face on a frictionless, frozen pond. You push off each other. You move backwards with a speed of 5.00 m/s. (a) What is the total momentum of the you-and-your-friend system? (b) What is your momentum after you pushed off? (c) What is your friends speed after you pushed off?