Math 21a Supplement on Torque and Angular Momentum According to Newton, the position vector, r (t), of a particle changes with time, t, under the influence of a force vector F according to the rule m r ´´ = F . (1) Here, m is the mass of the particle. The momentum of the particle is, by definition, the vector p ≡ m r ´. Thus, momentum is mass times velocity. The angular momentum is defined to be the vector L ≡ m r × r ´ = r × p . (2) Note that L is perpendicular to both the position vector r and the velocity vector r ´. In some sense, angular momentum measures the deviation from motion on a straight line. Indeed if L = 0, then the velocity vector r ´ is proportional to the position vector r , and this implies that the particle travels along a straight line (but maybe back and forth). To see that the condition r ´ being proportional to r means straight line motion, differentiate the unit vector r /|r| under this assumption to see that the latter is constant.
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This note was uploaded on 02/18/2012 for the course MATH 310 taught by Professor Gregoryj.galloway during the Fall '11 term at University of Miami.