1. 9. Systems of Particles 1. Center of Mass 2. Momentum 3. Kinetic Energy of a System 4. Collisions 5. Totally Inelastic Collisions 6. Elastic Collisions

2. 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? Rigid body: Relative particle positions fixed. Ans. His center of mass (CM)

3. 9.1. Center of Mass  3 rd law  Cartesian coordinates: = total mass r cm = Center of mass = mass-weighted average position of the collection of particles.  Extension: “particle” i may stand for an extended object with cm at r i.

4. Example 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.

5. Example 9.2. Space Station 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 m 2 = 2m & y points downward. obtainable by symmetry 2: 2m 1: m3:m L x y CM 30 

6. Continuous Distributions of Matter Continuous distribution : Discrete collection : Let  be the density of the matter.

7. Example 9.3. Aircraft Wing 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 = . By symmetry, L W x y dx Coord origin at leftmost tip of wing.

8. L W x y dy w/2  w/2

9. A high jumper clears the bar, but his CM doesn’t. CM fuselage CM wing CM plane

10. Got it? 9.1. A thick wire is bent into a semicircle. Which of the points is the CM?

11. Example 9.4. Circus Train Jumbo, 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 kg Jumbo walks, but the center of mass doesn’t move (F ext = 0 ). 

12. 9.2. Momentum Total momentum: M constant 

13. Conservation of Momentum  Conservation of Momentum: Total momentum of a system is a constant if there is no net external force. GOT IT! 9.2. 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?

14. Example 9.5. 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 while the pack is in the air & after Nick catches it? Initially While pack is in air: After Nick catches it:

15. Example 9.6. Radioactive Decay A lithium-5 ( 5 Li ) nucleus is moving at 1.6 Mm/s when it decays into a proton ( 1 H, or p ) & an alpha particle ( 4 He, or  ). [ Superscripts denote mass in AMU ]  is detected moving at 1.4 Mm/s at 33  to the original velocity of 5 Li. What are the magnitude & direction of p’s velocity? Before decay: After decay:

16. Example 9.7. 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? Water loses horizontal momentum completely after hitting window: = force exerted by window on water =  ( force exerted by water on window ) ( water moves in +x direction )

17. GOT IT? 9.3. Two skaters toss a basketball back & forth on frictionless ice. Which of the following does not change: (a) momentum of individual skater. (b) momentum of basketball. (c) momentum of the system consisting of one skater & the basketball. (d) momentum of the system consisting of both skaters & the basketball.

18. 9.3. Kinetic Energy of a System

19. 9.4. Collisions Examples of collision: Balls on pool table. tennis rackets against balls. bat against baseball. asteroid against planet. particles in accelerators. galaxies spacecraft against planet ( gravity slingshot ) Characteristics of collision: Duration: brief. Effect: intense (all other external forces negligible )

20. Momentum in Collisions External forces negligible  Total momentum conserved For an individual particle  t = collision time impulse More accurately,

21. Energy in Collisions Elastic collision: K conserved. Inelastic collision: K not conserved. Bouncing ball: inelastic collision between ball & ground.

22. GOT IT? 9.4. Which of the following qualifies as a collision? Of the collisions, which are nearly elastic & which inelastic? (a) a basketball rebounds off the backboard. (b) two magnets approach, their north poles facing; they repel & reverse direction without touching. (c) a basket ball flies through the air on a parabolic trajectory. (d) a truck crushed a parked car & the two slide off together. (e) a snowball splats against a tree, leaving a lump of snow adhering to the bark. elastic inelastic

23. 9.5. Totally Inelastic Collisions Totally inelastic collision: colliding objects stick together  maximum energy loss consistent with momentum conservation.

24. Example 9.8. Hockey A 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?

25. Example 9.9. Fusion Consider a fusion reaction of 2 deuterium nuclei 2 H + 2 H  4 He. Initially, one of the 2 H is moving at 3.5 Mm/s, the other at 1.8 Mm/s at a 64  angle to the 1 st. Find the velocity of the Helium nucleus.

26. Example 9.10. 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)

27. 9.6. Elastic Collisions Momentum conservation: Energy conservation: 2-D case: number of unknowns = 2  2 = 4 ( v 1fx, v 1fy, v 2fx, v 2fy ) number of equations = 2 +1 = 3  1 more conditions needed. 3-D case: number of unknowns = 3  2 = 6 ( v 1fx, v 1fy, v 1fz, v 2fx, v 2fy, v 2fz ) number of equations = 3 +1 = 4  2 more conditions needed.

28. Elastic Collisions in 1-D 1-D case: number of unknowns = 1  2 = 2 ( v 1f, v 1f ) number of equations = 1 +1 = 2  unique solution.  2-D collision 1-D collision

29.   (a) m 1 << m 2 :  (b) m 1 = m 2 :  (c) m 1 >> m 2 : 

30. Example 9.11. Nuclear Engineering Moderator slows neutrons to induce fission. A common moderator is heavy water ( D 2 O ). Find the fraction of a neutron’s kinetic energy that’s transferred to an initially stationary D in a head-on elastic collision.

31. GOT 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? 1 st one is lighter.

32. Elastic Collision in 2-D Impact parameter b : additional info necessary to fix the collision outcome.

33. Example 9.12. Croquet A 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: 

34. Center of Mass Frame