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Chapter 9 (Cont.) Linear Momentum and Collisions Copyright © 2010 Pearson Education, Inc. Units of Chapter 9 • Linear Momentum • Momentum and Newton’s Second Law • Impulse • Conservation of Linear Momentum • Inelastic Collisions • Elastic Collisions • Center of Mass Copyright © 2010 Pearson Education, Inc. Review: 9-1 Linear Momentum Momentum is a vector; its direction is the same as the direction of the velocity. Copyright © 2010 Pearson Education, Inc. Review: 9-2 Momentum and Newton’s Second Law Newton’s second law, Form I: Form II: Copyright © 2010 Pearson Education, Inc. Review: 9-3 Impulse Impulse is a vector, in the same direction as the average force. Copyright © 2010 Pearson Education, Inc. Review: 9-3 Impulse We can rewrite (from Newton’s 2nd law) as So we see that The impulse is equal to the change in momentum. Copyright © 2010 Pearson Education, Inc. Review: 9-4 Conservation of Linear Momentum Internal Versus External Forces: Internal forces act between objects within the system. As with all forces, they occur in action-reaction pairs. As all pairs act between objects in the system, the internal forces always sum to zero: Therefore, the net force acting on a system is the sum of the external forces acting on it. net external force=0Æ conservation of momentum Copyright © 2010 Pearson Education, Inc. Review: 9-5 Inelastic Collisions Time of collision is short enough that external forces may be ignored Inelastic collision: momentum is conserved but kinetic energy is not Completely inelastic collision: objects stick together afterwards Copyright © 2010 Pearson Education, Inc. 9-6 Elastic Collisions In elastic collisions, both kinetic energy and momentum are conserved. One-dimensional elastic collision: Copyright © 2010 Pearson Education, Inc. 9-6 Elastic Collisions We have two equations (conservation of momentum and conservation of kinetic energy) and two unknowns (the final speeds). Solving for the final speeds: Q: what if m1>>m2? Or m2>>m1? Copyright © 2010 Pearson Education, Inc. Question 9.10a Elastic Collisions I Consider two elastic collisions: 1) a golf ball with speed v hits a stationary bowling ball head-on. 2) a bowling ball with speed v hits a stationary golf ball head-on. In which case does the golf ball have the greater speed after the collision? v a) situation 1 b) situation 2 c) both the same at rest at rest 1 v 2 Question 9.10a Elastic Collisions I Consider two elastic collisions: 1) a golf ball with speed v hits a stationary bowling ball head-on. 2) a bowling ball with speed v hits a stationary golf ball head-on. In which case does the golf ball have the greater speed after the collision? Remember that the magnitude of the relative velocity has to be equal before and after the collision! a) situation 1 b) situation 2 c) both the same v 1 In case 1 the bowling ball will almost remain at rest, and the golf ball will bounce back with speed close to v. In case 2 the bowling ball will keep going with speed close to v, hence the golf ball will rebound with speed close to 2v. v 2v 2 9-6 Elastic Collisions Two-dimensional collisions can only be solved if some of the final information is known, such as the final velocity of one object: Copyright © 2010 Pearson Education, Inc. Copyright © 2010 Pearson Education, Inc. (Ki=Kf) (Py=0) Copyright © 2010 Pearson Education, Inc. 9-7 Center of Mass Q: Where do you draw the gravitational force for an object in a free-body diagram? The center of mass of a system is the point where the system can be balanced in a uniform gravitational field. Copyright © 2010 Pearson Education, Inc. 9-7 Center of Mass For two objects: The center of mass is closer to the more massive object. Copyright © 2010 Pearson Education, Inc. Question 9.20 Center of Mass The disk shown below in (1) clearly has its center of mass at the center. a) higher b) lower Suppose the disk is cut in half and the pieces arranged as shown in (2). Where is the center of mass of (2) as compared to (1) ? c) at the same place d) there is no definable CM in this case (1) X CM (2) Question 9.20 Center of Mass The disk shown below in (1) clearly has its center of mass at the center. a) higher b) lower Suppose the disk is cut in half and the pieces arranged as shown in (2). c) at the same place Where is the center of mass of (2) as compared to (1) ? The CM of each half is closer to the top of the semicircle than the bottom. The CM of the whole system is located at the midpoint of the two semicircle CMs, which is higher than the yellow line. d) there is no definable CM in this case (1) X CM (2) CM 9-7 Center of Mass The center of mass need not be within the object: Copyright © 2010 Pearson Education, Inc. Center of mass for a system of many objects: Copyright © 2010 Pearson Education, Inc. Copyright © 2010 Pearson Education, Inc. (a) (b) Copyright © 2010 Pearson Education, Inc. 9-7 Center of Mass Motion of the center of mass: Copyright © 2010 Pearson Education, Inc. 9-7 Center of Mass The total mass multiplied by the acceleration of the center of mass is equal to the net external force: The center of mass accelerates just as though it were a point particle of mass M acted on by Copyright © 2010 Pearson Education, Inc. Question 9.19 Motion of CM Two equal-mass particles (A and B) are located at some distance from each other. Particle A is held stationary while B is moved away at speed v. What happens to the center of mass of the two-particle system? a) it does not move b) it moves away from A with speed v c) it moves toward A with speed v d) it moves away from A with speed ½v e) it moves toward A with speed ½v Question 9.19 Motion of CM Two equal-mass particles (A and B) are located at some distance from each other. Particle A is held stationary while B is moved away at speed v. What happens to the center of mass of the two-particle system? a) it does not move b) it moves away from A with speed v c) it moves toward A with speed v d) it moves away from A with speed ½v e) it moves toward A with speed ½v Let’s say that A is at the origin (x = 0) and B is at some position x. Then the center of mass is at x/2 because A and B have the same mass. If v = Δx/Δt tells us how fast the position of B is changing, 1 2 then the position of the center of mass must be changing like Δ(x/2)/Δt, which is simply v. Copyright © 2010 Pearson Education, Inc. (a) (b) (conservation of momentum) Copyright © 2010 Pearson Education, Inc. Summary of Chapter 9 •Elastic collision: kinetic energy is conserved • Center of mass: Copyright © 2010 Pearson Education, Inc. Summary of Chapter 9 • Center of mass: • Motion of center of mass: Copyright © 2010 Pearson Education, Inc.