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Short Version : 9. Systems of Particles. 9.1. Center of Mass. N particles:. . = total mass. = Center of mass = mass-weighted average position. with. 3 rd law . Cartesian coordinates:. Extension: “particle” i may stand for an extended object with cm at r i.
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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.
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 m2 = 2m & y points downward. 2: 2m x 30 H 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 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 dm h w x L
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 ).
Alternative Solution relative to car relative to ground 19m + xc 19m xc
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.
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 After Nick catches it :
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:
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.
9.5. Totally Inelastic Collisions Totally inelastic collision: colliding objects stick together maximum energy loss consistent with momentum conservation.
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
mom. cons. rel. v reversed (a) m1 << m2 : (b) m1=m2 : (c) m1 >> m2 : Mathematica
Elastic Collision in 2-D Impact parameter b : additional info necessary to fix the collision outcome. Mathematica
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:
