Engineering Calculus Notes 122

Engineering Calculus Notes 122 - momenta will not change...

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110 CHAPTER 1. COORDINATES AND VECTORS Remark 1.7.5. The (spatial) velocity −→ v of a point P under a steady rotation (about the axis ) with angular velocity −→ ω is −→ v = −→ ω × −−→ P 0 P (1.31) where P 0 is an arbitrary point on , the axis of rotation. Associated to the analysis of rotation of rigid bodies are the rotational analogues of momentum and force, called moments . Recall that the momentum of a (constant) mass m moving with velocity −→ v is m −→ v ; its angular momentum or moment of momentum about a point P 0 is deFned to be −−→ P 0 P × m −→ v . More generally, the moment about a point P 0 of any vector quantity −→ V applied at a point P is deFned to be −−→ P 0 P × −→ V . ±or a rigid body, the “same” force applied at di²erent positions on the body has di²erent e²ects on its motion; in this context it is the moment of the force that is relevant. Newton’s ±irst Law of Motion [ 39 , Law 1(p. 416)] is usually formulated as conservation of momentum : if the net force acting on a system of bodies is zero, then the (vector) sum of their
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Unformatted text preview: momenta will not change with time: put dierently, their center of mass will move with constant velocity. A second conservation law is conservation of angular momentum , which says that in addition the (vector) sum of the angular momenta about the center of mass will be constant. This net angular momentum speciFes an axis (through the center of mass) and a rotation about that axis. or a rigid body, the motion can be decomposed into these two parts: the displacement motion of its center of mass, and its rotation about this axis through the (moving) center of mass. Exercises for 1.7 Practice problems: 1. ind an equation for the plane P described in each case: (a) P goes through (1 , 2 , 3) and contains lines parallel to each of the vectors v = (3 , 1 , 2) and w = (1 , , 2) ....
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