Ch0122 - Chapter 22 Center of Mass Moment of Inertia 22...

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Chapter 22 Center of Mass, Moment of Inertia 142 22 Center of Mass, Moment of Inertia A mistake that crops up in the calculation of moments of inertia, involves the Parallel Axis Theorem. The mistake is to interchange the moment of inertia of the axis through the center of mass, with the one parallel to that, when applying the Parallel Axis Theorem. Recognizing that the subscript “CM” in the parallel axis theorem stands for “center of mass” will help one avoid this mistake. Also, a check on the answer, to make sure that the value of the moment of inertia with respect to the axis through the center of mass is smaller than the other moment of inertia, will catch the mistake. Center of Mass Consider two particles, having one and the same mass m , each of which is at a different position on the x axis of a Cartesian coordinate system. Common sense tells you that the average position of the material making up the two particles is midway between the two particles. Common sense is right. We give the name “center of mass” to the average position of the material making up a distribution, and the center of mass of a pair of same-mass particles is indeed midway between the two particles. How about if one of the particles is more massive than the other? One would expect the center of mass to be closer to the more massive particle, and again, one would be right. To determine the position of the center of mass of the distribution of matter in such a case, we compute a weighted sum of the positions of the particles in the distribution, where the weighting factor for a given particle is that fraction, of the total mass, that the particle’s own mass is. Thus, for two particles on the x axis, one of mass m 1 , at x 1 , and the other of mass m 2 , at x 2 , y x #1 #2 m m y x m 1 m 2 ( x 1 , 0 ) ( x 2 , 0 )
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Chapter 22 Center of Mass, Moment of Inertia 143 the position x of the center of mass is given by 2 2 1 2 1 2 1 1 x m m m x m m m x + + + = (22-1) Note that each weighting factor is a proper fraction and that the sum of the weighting factors is always 1. Also note that if, for instance, m 1 is greater than m 2 , then the position x 1 of particle 1 will count more in the sum, thus ensuring that the center of mass is found to be closer to the more massive particle (as we know it must be). Further note that if m 1 = m 2 , each weighting factor is 2 1 , as is evident when we substitute m for both m 1 and m 2 in equation 22-1: 2 1 x m m m x m m m x + + + = 2 1 2 1 2 1 x x x + = 2 2 1 x x x + = The center of mass is found to be midway between the two particles, right where common sense tells us it has to be. The Center of Mass of a Thin Rod Quite often, when the finding of the position of the center of mass of a distribution of particles is called for, the distribution of particles is the set of particles making up a rigid body. The easiest rigid body for which to calculate the center of mass is the thin rod because it extends in only one dimension. (Here, we discuss an ideal thin rod. A physical thin rod must have some non- zero diameter.
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