9. Systems of Particles Center of Mass Momentum Kinetic Energy of a System Collisions Totally Inelastic Collisions Elastic Collisions

Ans. 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.

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

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.

Example 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: 2m x 30 L CM obtainable by symmetry 1: m 3:m y

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

Example 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, y dx h W x L

y b dy w/2 W x w/2 L

CMfuselage CMplane CMwing A high jumper clears the bar, but his CM doesn’t.

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

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 Final distance of Jumbo from xc :  Jumbo walks, but the center of mass doesn’t move (Fext = 0 ).

9.2. Momentum Total momentum: M constant 

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. 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.

Conceptual 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 conserved After Nick catches it , total p = 0. Kayak speed = 0 Simple application of the conservation law.

Making the Connection 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 ? Initially While pack is in air: Note: Emech not conserved

Example 9.5. Radioactive Decay A lithium-5 ( 5Li ) nucleus is moving at 1.6 Mm/s when it decays into a 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:

Example 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

GOT 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.

Application: Rockets Thrust:

9.3. Kinetic Energy of a System

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 )

Momentum in Collisions External forces negligible  Total momentum conserved For an individual particle t = collision time impulse More accurately, Same size Average Crash test

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

GOT 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. elastic elastic inelastic inelastic

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

Example 9.7. 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?

Example 9.8. Fusion Consider 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.

Example 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)

9.6. Elastic Collisions Momentum 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 = 4 ( 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.

Elastic Collisions in 1-D 1-D collision 1-D case: number of unknowns = 1  2 = 2 ( v1f , v2f ) number of equations = 1 +1 = 2  unique solution. This is a 2-D collision 

  (a) m1 << m2 :  (b) m1 = m2 :  (c) m1 >> m2 : 

Example 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.

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? 1st one is lighter.

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

Example 9.11. 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: 

Center of Mass Frame