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Unformatted text preview: 1 Angular momentum in d=3: some notes. R.Shankar Fall 2006 I thought a few words on angular momentum in d = 3 would be helpful. Before starting convince yourself that the magnitude of A × B is the area of a parallelogram bounded by the two vectors. (Figure(1)). First, like torque or moment of inertia I , L is deﬁned only with respect to some origin. For some origin, a particle located at r and moving with momentum p we deﬁne L = r × p . (1) The vector L is perpendicular to the plane deﬁned by r and p . For example if they both lie in the xy plane, L will point along the zaxis. To decide if it is up or down the zaxis you need to use the right hand rule. For example if r = r i is along x and p = p j is along y , the L is up the zaxis (since i × j = k ) and has magnitude rp . Note that a particle does not have to orbit this origin to have angular momentum around the origin. Look at Figure (2). The dotted line is the linear trajectory of a particle of ﬁxed p . What is L ? First consider the time when it is at A when the line joining it to the origin is perpendicular to its momentum. If this distance is r A then L = r A p k . (To get the direction right, draw the vectors r and p emanating from the same point and turn the screw driver from r to p .) Note that the magnitude of L is twice the are of the triangle of base...
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 Fall '08
 RAMAMURTISHANKAR
 Physics, Angular Momentum, Force, Momentum, dr sin θ

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