l15anno - Moment of Momentum (Euler s 2nd Law) Consider...

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Moment of Momentum (Euler ` s 2 nd Law) Consider system of N particles as in Euler ` s First Law C : mass center O : fixed point in inertial frame P : arbitrary point (may be moving) P r i m i F i f ij f ji m j F j O x r ji r j i a j a
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Newton ` s 2 nd Law for m i : 1 N ii j i i j m = += Ff a Take moment about P (cross with r i ): 1 N i i j i j m = × + × = × rF r f r a (using distributive rule) Add: r i ! F i = M P { + r i ! f ij j = 1 N " i = 1 N " = 0 1 2 4 3 4 = r i ! m i a i i = 1 N " i = 1 N " 0 (co linearity) r 1 ! f 12 + r 2 ! f 21 (typical pair, recall f ii = 0) r 1 ! f 12 + r 2 ! ( " f 12 ) (equal & opposite - 3rd Law) ( r 1 ! r 2 ) ! f 12 (distributive rule) 11 NN Pi i i m == = × ∑∑ Mr a
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Define: 1 moment of momentum about N Pi i i i mP = = × Hr v () PC i i i m =+ × rv r = r PC ! m i v i i = 1 N ! L 12 4 3 4 + ! m i v i i = 1 N ! H C 1 2 44 3 So: P C × HH r L P r i m i C O x y PC r i r i i i i i mm = × + × ∑∑ v r C v i v i r
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