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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
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 m d dt F ex r r We apply the integral on the right side of the equality to obtain the Impulse – Momentum equation: ( ) ) ( v m dt F ex r r = 3. Impulse. The expression on the left side of the equality is called the Impulse. One may notice a similarity to the definition of work. There is however a major difference. The Impulse is a vector time path function. To evaluate the Impulse, one must know how the external vector forces vary as a function of time along the time path from initial to final time. Typically, the letter “J” is used as a shorthand for this integral. dt F J ex ) ( r r Similar to work, the impulse may be viewed as the area under a Fnet vs.

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Collections of Particles - Collections of Particles C...

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