Imagine a thin rod (a straight material wire of negligible thickness) lying on the r-axis from...
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Imagine a thin rod (a straight material wire of negligible thickness) lying on the r-axis from z = a to z = b. If the mass density of the rod (the mass per unit length) is constant, then the total mass M of the rod is simply the density X times the length of the rod: M = X-(b-a). If the density A varies continuously from point to point, say A X(z), then the mass of the rod is the average density of the rod times the length of the rod, which becomes the following integral: = M = (average density) - (length) - (₁-² L² x(x) dx) - A(z) dr dz). (b- (b-a) a - Mix) the = Continue with that same rod. If the rod is homogeneous (constant density), then the center of mass of the rod (we denote this point by TM) is simply the midpoint of the rod: 1 IM = (a + b) (the average of a from a to b). If the rod is not homogeneous, the center of mass is still an average, but now a weighted average, the density-weighted average of z from a to b; namely: IM = IX(T) dr. 1/ Finally, the mass moment about a point ce [a, b] is given by the integral: [₁(2-C). Exercise: show that the mass moment about the center of mass is zero. (r-c)X(r) dr. Imagine a thin rod (a straight material wire of negligible thickness) lying on the r-axis from z = a to z = b. If the mass density of the rod (the mass per unit length) is constant, then the total mass M of the rod is simply the density X times the length of the rod: M = X-(b-a). If the density A varies continuously from point to point, say A X(z), then the mass of the rod is the average density of the rod times the length of the rod, which becomes the following integral: = M = (average density) - (length) - (₁-² L² x(x) dx) - A(z) dr dz). (b- (b-a) a - Mix) the = Continue with that same rod. If the rod is homogeneous (constant density), then the center of mass of the rod (we denote this point by TM) is simply the midpoint of the rod: 1 IM = (a + b) (the average of a from a to b). If the rod is not homogeneous, the center of mass is still an average, but now a weighted average, the density-weighted average of z from a to b; namely: IM = IX(T) dr. 1/ Finally, the mass moment about a point ce [a, b] is given by the integral: [₁(2-C). Exercise: show that the mass moment about the center of mass is zero. (r-c)X(r) dr.
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Related Book For
Fundamentals Of Momentum Heat And Mass Transfer
ISBN: 9781118947463
6th Edition
Authors: James Welty, Gregory L. Rorrer, David G. Foster
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