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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

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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)

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

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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.

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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

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

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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. h

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L W x y dy w/2 w/2 b

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

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

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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 ). Final distance of Jumbo from x c :

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9.2. Momentum Total momentum: M constant

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Conservation of Momentum Conservation of Momentum: Total momentum of a system is a constant if there is no net external force.

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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? K.E. is not conserved. E mech = K.E. + P.E. grav is not conserved. E tot = E mech + U chem is conserved.

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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.

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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: E mech not conserved

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Example 9.5. 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:

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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? = force exerted by water on window = Rate of momentum transfer to window Momentum transfer to a plane stream:

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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.

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Application: Rockets Thrust:

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9.3. Kinetic Energy of a System

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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 )

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

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

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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

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

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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?

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Example 9.8. 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.

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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)

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9.6. Elastic Collisions Momentum conservation: Energy conservation: 2-D case: number of unknowns = 2 2 = 4 ( final state: 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 ( final state: v 1fx, v 1fy, v 1fz, v 2fx, v 2fy, v 2fz ) number of equations = 3 +1 = 4 2 more conditions needed. Implicit assumption: particles have no interaction when they are in the initial or final states. ( E i = K i )

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

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(a) m 1 << m 2 : (b) m 1 = m 2 : (c) m 1 >> m 2 :

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Example 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.

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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.

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

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Example 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:

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Center of Mass Frame

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