class13 - 1 Systems of particles REFERENCE: Arya, Chapter...

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1 Systems of particles REFERENCE: Arya, Chapter 8. The ideas of Newtonian mechanics and the conservation theorems can be straightfor- wardly extended to systems of N particles. 1.1 Center of Mass Consider a system of N particles 1 , 2 , 3 ...N , with masses m 1 ,m 2 ,...,m N located a distance r 1 , r 2 ,..., r 3 from the origin O . The center of mass is at a point R ( x,y,z ) from the origin, de±ned by N ± k =1 m k R = N ± k =1 m k r k (1) Where we can de±ne the total mass M = N k =1 m k . The velocity and acceleration of the center of mass is V = ˙ R = 1 M N ± k =1 m k ˙ r k (2) A = ¨ R = 1 M N ± k =1 m k ¨ r k (3) For example, the center of mass of a pair of particles is R = m 1 r 1 + m 2 r 2 m 1 + m 2 (4) The center of mass lies on the line connecting the two particles, and divides the line in the ratio m 2 : m 1 . The center of mass of three identical particles, R = m r 1 + m r 2 + m r 3 m + m + m = r 1 + r 2 + r 3 3 (5) For a continuous distribution of mass, the summation is replaced by an integral: R = 1 M ² r dm (6) example:
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class13 - 1 Systems of particles REFERENCE: Arya, Chapter...

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