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**9. Systems of Particles Center of Mass Momentum**

Kinetic Energy of a System Collisions Totally Inelastic Collisions Elastic Collisions

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**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.

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**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 .

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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 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

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**Continuous Distributions of Matter**

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

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**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

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

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

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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 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 ).

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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. 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.

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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: Emech not conserved

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**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:

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

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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: 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.

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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, Same size Average 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 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

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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 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 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.

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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: 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.

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**Elastic Collisions in 1-D**

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

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

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**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.

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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? 1st 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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