Presentation on theme: "9. Systems of Particles Center of Mass Momentum"— Presentation transcript:
19. Systems of Particles Center of Mass Momentum Kinetic Energy of a SystemCollisionsTotally Inelastic CollisionsElastic Collisions
2Ans. His center of mass (CM) As the skier flies through the air,most parts of his body follow complex trajectories.But one special point follows a parabola.What’s that point, and why is it special?Ans. His center of mass (CM)Rigid body: Relative particle positions fixed.
39.1. Center of Mass N particles: = total mass= Center of mass= mass-weighted average positionwith3rd law Cartesian coordinates:Extension: “particle” i may stand for an extended object with cm at ri .
4Example 9.1. Weightlifting Find the CM of the barbell consisting of 50-kg & 80-kg weights at opposite ends of a 1.5 m long bar of negligible weight.CM is closer to the heavier mass.
5Example 9.2. Space Station 2: 2m x L 1: m 3:m y A space station consists of 3 modules arranged in an equilateral triangle, connected by struts of length L & negligible mass.2 modules have mass m, the other 2m.Find the CM.Coord origin at m2 = 2m & y points downward.2: 2mx30LCMobtainable by symmetry1: m3:my
6Continuous Distributions of Matter Discrete collection:Continuous distribution:Let be the density of the matter.
7Example 9.3. Aircraft Wing y W x L A supersonic aircraft wing is an isosceles triangle of length L, width w, and negligible thickness.It has mass M, distributed uniformly.Where’s its CM?Density of wing = .Coord origin at leftmost tip of wing.By symmetry,ydxhWxL
9CMfuselageCMplaneCMwingA high jumper clears the bar,but his CM doesn’t.
10Got it? 9.1. A thick wire is bent into a semicircle. Which of the points is the CM?
11Example Circus TrainJumbo, a 4.8-t elephant, is standing near one end of a 15-t railcar,which is at rest on a frictionless horizontal track.Jumbo walks 19 m toward the other end of the car.How far does the car move?1 t = 1 tonne= 1000 kgFinal distance of Jumbo from xc :Jumbo walks, but the center of mass doesn’t move (Fext = 0 ).
13Conservation of Momentum Conservation of Momentum:Total momentum of a system is a constant if there is no net external force.
14GOT IT! 9.2. K.E. is not conserved. A 500-g fireworks rocket is moving with velocity v = 60 j m/s at the instant it explodes.If you were to add the momentum vectors of all its fragments just after the explosion,what would you get?K.E. is not conserved.Emech = K.E. + P.E. grav is not conserved.Etot = Emech + Uchem is conserved.
15Conceptual Example 9.1. Kayaking Jess (mass 53 kg) & Nick (mass 72 kg) sit in a 26-kg kayak at rest on frictionless water.Jess toss a 17-kg pack, giving it a horizontal speed of 3.1 m/s relative to the water.What’s the kayak’s speed after Nick catches it?Why can you answer without doing any calculations ?Initially, total p = 0.frictionless water p conservedAfter Nick catches it , total p = 0.Kayak speed = 0Simple application of the conservation law.
16Making the ConnectionJess (mass 53 kg) & Nick (mass 72 kg) sit in a 26-kg kayak at rest on frictionless water.Jess toss a 17-kg pack, giving it a horizontal speed of 3.1 m/s relative to the water.What’s the kayak’s speed while the pack is in the air ?InitiallyWhile pack is in air:Note: Emech not conserved
17Example 9.5. Radioactive Decay A lithium-5 ( 5Li ) nucleus is moving at 1.6 Mm/s when it decays intoa proton ( 1H, or p ) & an alpha particle ( 4He, or ). [ Superscripts denote mass in AMU ] is detected moving at 1.4 Mm/s at 33 to the original velocity of 5Li.What are the magnitude & direction of p’s velocity?Before decay:After decay:
18Example 9.6. Fighting a Fire A firefighter directs a stream of water to break the window of a burning building.The hose delivers water at a rate of 45 kg/s, hitting the window horizontally at 32 m/s.After hitting the window, the water drops horizontally.What horizontal force does the water exert on the window?Momentum transfer to a plane stream:= Rate of momentum transfer to window= force exerted by water on window
19GOT IT? 9.3.Two skaters toss a basketball back & forth on frictionless ice.Which of the following does not change:momentum of individual skater.momentum of basketball.momentum of the system consisting of one skater & the basketball.momentum of the system consisting of both skaters & the basketball.
229.4. Collisions Examples of collision: Balls on pool table. tennis rackets against balls.bat against baseball.asteroid against planet.particles in accelerators.galaxiesspacecraft against planet ( gravity slingshot )Characteristics of collision:Duration: brief.Effect: intense(all other external forces negligible )
23Momentum in Collisions External forces negligible Total momentum conservedFor an individual particlet = collision timeimpulseMore accurately,Same sizeAverageCrash test
24Energy in Collisions Elastic collision: K conserved. Inelastic collision: K not conserved.Bouncing ball: inelastic collision between ball & ground.
25GOT IT? 9.4. Which of the following qualifies as a collision? Of the collisions, which are nearly elastic & which inelastic?a basketball rebounds off the backboard.two magnets approach, their north poles facing; they repel & reverse direction without touching.a basket ball flies through the air on a parabolic trajectory.a truck crushed a parked car & the two slide off together.a snowball splats against a tree, leaving a lump of snow adhering to the bark.elasticelasticinelasticinelastic
269.5. Totally Inelastic Collisions Totally inelastic collision: colliding objects stick together maximum energy loss consistent with momentum conservation.
27Example HockeyA Styrofoam chest at rest on frictionless ice is loaded with sand to give it a mass of 6.4 kg.A 160-g puck strikes & gets embedded in the chest, which moves off at 1.2 m/s.What is the puck’s speed?
28Example FusionConsider a fusion reaction of 2 deuterium nuclei 2H + 2H 4He .Initially, one of the 2H is moving at 3.5 Mm/s, the other at 1.8 Mm/s at a 64 angle to the 1st.Find the velocity of the Helium nucleus.
29Example 9.9. Ballistic Pendulum The ballistic pendulum measures the speeds of fast-moving objects.A bullet of mass m strikes a block of mass M and embeds itself in the latter.The block swings upward to a vertical distance of h.Find the bullet’s speed.Caution:(heat is generated when bullet strikes block)
309.6. Elastic CollisionsMomentum conservation:Energy conservation:Implicit assumption: particles have no interaction when they are in the initial or final states. ( Ei = Ki )2-D case:number of unknowns = 2 2 = ( final state: v1fx , v1fy , v2fx , v2fy )number of equations = 2 +1 = 3 1 more conditions needed.3-D case:number of unknowns = 3 2 = 6 ( final state: v1fx , v1fy , v1fz , v2fx , v2fy , v2fz )number of equations = 3 +1 = 4 2 more conditions needed.
31Elastic Collisions in 1-D 1-D collision1-D case:number of unknowns = 1 2 = ( v1f , v2f )number of equations = 1 +1 = 2 unique solution.This is a 2-D collision
33Example 9.10. Nuclear Engineering Moderator slows neutrons to induce fission.A common moderator is heavy water ( D2O ).Find the fraction of a neutron’s kinetic energy that’s transferred to an initially stationary D in a head-on elastic collision.
34GOT IT? 9.5. One ball is at rest on a level floor. Another ball collides elastically with it & they move off in the same direction separately.What can you conclude about the masses of the balls?1st one is lighter.
35Elastic Collision in 2-D Impact parameter b :additional info necessary to fix the collision outcome.
36Example CroquetA croquet ball strikes a stationary one of equal mass.The collision is elastic & the incident ball goes off 30 to its original direction.In what direction does the other ball move?p cons:E cons: