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Centre of Mass

Centre of Mass. Definition Total momentum of a system of particles Motion of the centre of mass. Serway and Jewett 9.6 - 9.7. Review: Newton’s Second Law. For a particle: (Net external force) = m a. For a particle or a system of particles:

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Centre of Mass

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  1. Centre of Mass • Definition • Total momentum of a system of particles • Motion of the centre of mass Serway and Jewett 9.6 - 9.7 Physics 1D03 - Lecture 16

  2. Review: Newton’s 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 = Dp Physics 1D03 - Lecture 16

  3. Apply Newton’s 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. Physics 1D03 - Lecture 16

  4. Centre of Mass Recall: Definition: m1 CM x or, m2 rCM m3 For continuous objects, (Recall the position vectorr has components x, y, z.) Physics 1D03 - Lecture 16

  5. Dynamics of a system of particles CM definition: Differentiate with respect to time: The total momentum of any collection of particles is equal to the total mass times the velocity of the CM point—amazing but true! So Newton’s second law gives (differentiate above): 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. Physics 1D03 - Lecture 16

  6. v0 CM x Example 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: Physics 1D03 - Lecture 16

  7. 1/3v0 CM 1/3v0 v0 Physics 1D03 - Lecture 16

  8. Examples: • A springboard diver does a triple reverse dive with one and a half twists. Her CM follows a smooth parabola (external force is gravity). • 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. • Pendulum cart (demonstration). Physics 1D03 - Lecture 16

  9. MOON 384,000 km Quiz F 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 In which case will the space station get to the moon faster? Physics 1D03 - Lecture 16

  10. MOON 384,000 km Quiz F 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 Physics 1D03 - Lecture 16

  11. Still more amazing CM theorems: • 1) Kinetic Energy = ½ Mv 2CM + (K.E. relative to CM) • (e.g., rigid body: K = ½ Mv 2CM + ½ ICM w 2 ) • 2) (Torque about CM) = ICMa , even for an accelerated body “alpha” (angular acceleration) Physics 1D03 - Lecture 16

  12. Summary • ptotal = M vCM • (net external force) = M aCM For practice: Chapter 9Problems 39, 41, 58(6th ed)Problems 41, 43, 57, 69 Physics 1D03 - Lecture 16

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