class15 - 1.3 Angular Momentum L = r p = r mr = r mv The...

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1.3 Angular Momentum The angular momentum of a single particle is deFned as L = r × p = r × m ˙ r = r × m v (46) We can extend this deFnition to a systems of N particles L = N ± k =1 ( r k × p k )= N ± k =1 ( r k × m k ˙ r ) (47) We can take the total angular momentum about any point A instead of the origin, in which case we replace r k by r k - r A . Taking the time derivative d L dt = N ± k =1 r k × m k ˙ r )+ N ± k =1 ( r k × m k ¨ r ) (48) the second term on the right is zero, while the second term is equal to the total force acting on the particle d L dt = N ± k =1 ( r k × [ F e k + F i k ]) = N ± k =1 ( r k × F e k N ± k =1 ( r k × F i k ) (49) We can prove that the second term on the right is zero, using Newton’s third law, giving d L dt = N ± k =1 r k × F e k = N ± k =1 -→ τ k (50) where τ k is the torque on the kth particle, or if τ N ± k =1 τ k (51) The conservation of angular momentum now says: if the net external torques about a given axis vanish, then the total angular momentum of the system about that axis remains constant in time.
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This note was uploaded on 09/20/2010 for the course AMATH 261 taught by Professor Rogermelko during the Spring '10 term at Waterloo.

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class15 - 1.3 Angular Momentum L = r p = r mr = r mv The...

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