Collections of Particles

Collections of Particles - Collections of Particles C...

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Collections of Particles C George Kapp 2002 Table of Contents I. Newton’s Second Law. Page 1. Introduction. ............................................................ 2 2. Impulse – Momentum Derived. . ............................... 2 3. Impulse. ................................................................... 3 4. Change in Momentum. ............................................. 3 5. General Application. . ............................................... 4 6. Example in Application. . .......................................... 6 II. Center of Mass. 1. Introduction. ............................................................ 7 2. Location. .................................................................. 7 3. Beyond Location. ..................................................... 8 4. Properties. ............................................................... 9 III. The Flow F=ma. 1. Development. ......................................................... 11 2. Example Application. ............................................. 13 G. Kapp, 2/20/04 1
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Newton’s Second Law. 1. Introduction. Newton’s' second law is customarily presented to beginning students of physics as a m F ex r v = where the left side of the equation represents the sum of all the external force vectors and on the right side, m represents the system's mass (resistance to being accelerated), and "a" is the resulting vector acceleration of the system. Although the above relationship is true, it is not what Newton said. Newton expressed his second law in a much more general statement. The second law of Newton is: dt v m d F ex ) ( r r = In words, the sum of the external force vectors equals the time rate of change of momentum; the mv product representing momentum. One can deduce F = ma from Newton’s second law by considering the mass to be constant, and as such, the mass can be removed from the differential and the resulting derivative, dv/dt, interpreted as acceleration. ( ) a m dt v d m dt v m d F ex r r r r = = = One may argue that the two statements above are for all purposes equivalent however the very fact that the mass of the system does not need to be constant lends generality and power to Newton’s law. There are at least two interesting variations of Newton’s law which lend a systematic approach to the solution of many problems in engineering. They are both derived from the 2 nd law above. The first variation is called Impulse – Momentum and the second is called The Flow F=ma . We shall derive each variation. 2. Impulse – Momentum. Starting with the second law above, G. Kapp, 2/20/04 2
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dt v m d F ex ) ( r r = , we multiply both sides of the equation by dt. ) ( v m d dt F ex r r = Next, we select a time interval so that we may integrate both sides of the equation. ( ) = ) ( v
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This note was uploaded on 01/17/2011 for the course PHY PHY 211 taught by Professor Kapp during the Spring '10 term at Washtenaw.

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Collections of Particles - Collections of Particles C...

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