Chapter08 - Chapter 8: Linear momentum, collisions, and CM...

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Physics 1301: Chapter 8, Pg 1 Chapter 8: Linear momentum, collisions, and CM ¾ Systems of particles: internal and external forces ¾ Momentum and its conservation ¾ Impulse ¾ Inelastic and elastic collisions ¾ Center of mass (CM) ¾ Collisions in CM reference frame
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Physics 1301: Chapter 8, Pg 2 Momentum (Linear) Momentum: ¾ Single particle: ± As kinetic energy, used to quantify motion ± Vector quantity (direction of velocity)! ± SI dimension: ± SI units: ¾ System of particles: Relation to kinetic energy: ¾ Single particle: ¾ System of particles: v p m = = i i i i i m v p p [] m/s kg [ ] ML/T m mv mv K 2 ) ( 2 2 2 m m p K 2 2 2 2 p i i i i i m p K K 2 2
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Physics 1301: Chapter 8, Pg 3 Momentum and Newton’s laws: single particle Newton’s 2 nd law: ± This is more general formulation (mass may not be constant) Newton’s 1 st law: ¾ If there no net force is acting on an object, the momentum remains constant (conserved) ± Note 1: This is momentum conservation law for single particle, but… ± Note 2: This is just a bit more general formulation of the firstlaw for a single particle (earlier we assumed constant mass and arrived at “conservation” of velocity) ¾ We want to study more complicated systems, consisting of several bodies which can interact (“collide”) with each other! = = = dt m d dt md m net ) ( v v a F dt d net p F = = = 0 0 dt d net p F const = p
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Physics 1301: Chapter 8, Pg 4 Example 1: two-body system Sum equations of motion: ¾ Nature of internal forces is not important! They will always cancel out! 0 law 3 d 21 12 2 1 21 2 12 1 = + = + = = 43 42 1 1 F F p p p F p F p dt d dt d dt d dt d dt d const 2 1 = + = p p p
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Physics 1301: Chapter 8, Pg 5 Example 2: three-body system Sum equations of motion: ¾ Q: What about N bodies? Hint: all internal forces are action-reaction pairs – they will always cancel out 32 31 23 21 13 12 3 2 1 32 31 3 23 21 2 13 12 1 ; ; F F F F F F p p p F F p F F p F F p + + + + + = + + + = + = + = dt d dt d dt d dt d dt d dt d const 3 2 1 = + + = p p p p
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Physics 1301: Chapter 8, Pg 6 Ninteracting particles: internal and external forces Internal force: ¾ Force on a particle from (particles) inside the system ± It is the sum of all forces from all ( N -1) other particles External force: ¾ Force on a particle from outside the system Second law for one particle in the system: ± We explicitly write internal and external forces ± Note: With respect to this particle all forces are external i F = N i j ij i F F ' i i j ij i dt d F F p + =
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Physics 1301: Chapter 8, Pg 7 Ninteracting particles: Conservation of momentum ¾ sum over all bodies, ¾ … and apply 3d law for each pair – 2 nd law for a system ¾ If there are no external forces, momentum of the system of N interacting particles is conserved: ± No external force means zero net force when system is considered as a whole Total momentum of an isolated system is constant regardless of any internal forces which might be present (all internal forces are action-reaction pairs ) net net i i ii j ij i i i i dt d dt d dt d dt d F p F F F p p p = = = = = ∑∑ ; 0 ;
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Chapter08 - Chapter 8: Linear momentum, collisions, and CM...

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