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### lecture27

Course: PHYSICS 1D03, Spring 2011
School: McMaster
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Word Count: 570

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of Centre Mass Definition Total momentum of a system of particles Motion of the centre of mass Serway and Jewett 9.6 - 9.7 Review: Newtons Second Law For a particle: (Net external force) = ma For a particle or a system of particles: (Net external force) = F = dp/dt (Net external impulse) = I = p Apply Newtons Laws to objects that are not particles: e.g., F F or How will an extended body move (accelerate)...

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of Centre Mass Definition Total momentum of a system of particles Motion of the centre of mass Serway and Jewett 9.6 - 9.7 Review: Newtons Second Law For a particle: (Net external force) = ma For a particle or a system of particles: (Net external force) = F = dp/dt (Net external impulse) = I = p Apply Newtons Laws to objects that are not particles: e.g., F F or How will an extended body move (accelerate) when a force is applied at an arbitrary location? The motion of the centre of mass is simple; in addition, various parts of the object move around the centre of mass. Centre of Mass Recall: Definition: rCM mi ri mi ri = = mi M m1 CM x or, MrCM = mi ri m2 rCM m3 For continuous objects, rCM 1 = r dm M (Recall the position vector r has components x, y, z.) Dynamics of a system of particles CM definition: MrCM = mi ri Differentiate with respect to time: Mv CM = mi v i = p total The total momentum of any collection of particles is equal to the total mass times the velocity of the CM pointamazing but true! So Newtons second law gives (differentiate above): (external forces) = M total a CM This is remarkable. The motion of the centre of mass is the same as if the object were a single particle at the CM, with all external forces applied directly to it. Example v0 CM x A neutron (mass m) travels at speed v0 towards a stationary deuteron (mass 2m). What is the initial velocity of the CM of the system (neutron plus deuteron)? Since: Mv CM = mi v i v CM m v = i M i v0 CM 1 /3v0 /3v A 0 1 Examples: 1) springboard diver does a triple reverse dive with one and a half twists. Her CM follows a smooth parabola (external force is gravity). 2) A paddler in a stationary canoe (floating on the water, no friction) walks from the rear seat to the front seat. The CM of the canoe plus paddler moves relative to the canoe, but not relative to the land (the canoe moves backwards). Here there is no external force. 3) Pendulum cart (demonstration). MOON384,000km Quiz A space station consists of a 1000-kg sphere and a 4000-kg sphere joined by a light cylinder. A rocket is fired briefly to provide a 100-N force for 10 seconds. Compare the velocity (CM) change if the rocket motor is A) at the small sphere B) at the large sphere C) same for either F F In which case will the space station get to the moon faster? MOON384,000km Quiz A space station consists of a 1000-kg sphere and a 4000-kg sphere joined by a light cylinder. A rocket is fired briefly to provide a 100-N force for 10 seconds. In which case will the space station rotate faster? A) at the small sphere B) at the large sphere C) same for either F F Still more amazing CM theorems: 1) Kinetic Energy = Mv 2CM + (K.E. relative to CM) (e.g., rigid body: K = Mv 2CM + ICM 2 ) about CM) = ICM , even for an accelerated body 2) (Torque alpha (angular acceleration) Summary ptotal = M vCM (net external force) = M aCM For practice: Chapter 9 Problems 39, 41, 58 (6th ed) Problems 41, 43, 57, 69
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