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Moments  Two Dimensions
Figure 6.6.8 A mass m at the point (x_{0}, y_{0}) in the (x, y) plane will have moments M_{x} about the xaxis and M_{y} about the yaxis (Figure 6.6.8). They are defined by M_{x} = my_{0}, M_{y} = mx_{0}. Consider a vertical length of wire of mass m and constant density which lies on the line x = x_{0} from y = a to y = b. The wire has density The infinitesimal piece of the wire from y to y + Δy shown in Figure 6.6.9 will have mass and moments Δm = ρ Δy, ΔM_{X} ≈ y Δm = yρ Δy (compared to Δy), ΔM_{y} ≈ x_{0} Δm = x_{o}ρΔy (compared to Δy). Figure 6.6.9 The Infinite Sum Theorem gives the moments for the whole wire, We next take up the case of a flat plate which occupies the region R under the curve y = f(x), f(x) ≥ 0, from x = a to x = b (Figure 6.6.10). Assume the density Figure 6.6.10 ρ(x) depends only on the xcoordinate. A vertical slice of infinitesimal width Δx between x and x + Δx is almost a vertical length of wire between 0 and f(x) which has area ΔA and mass Δm ≈ ρ(x) ΔA ≈ ρ(x) f(x) Δx (compared to Δx). Putting the mass Δm into the vertical wire formulas, the moments are ΔM_{y} ≈ x Δm ≈ xρ(x) f(x) Δx (compared to Δx), ΔM_{x} ≈ ½(f(x) + 0) Δm ≈ ½ρ(x)f(x)^{2} Δx (compared to Δx). Then by the Infinite Sum Theorem, the total moments are The center of mass of a twodimensional object is defined as the point (x, y) with coordinates x = M_{y}/m, y = M_{x}/m. A single mass m at the point (x, y) will have the same moments as the twodimensional body, M_{x} = my, M_{y}, = mx. The object will balance on a pin placed at the center of mass. If a twodimensional object has constant density, the center of mass depends only on the region R which it occupies. The centroid of a region R is defined as the center of mass of an object of constant density which occupies R. Thus if R is the region below the continuous curve y = f(x) from x = a to x = b, then the centroid has coordinates where A is the area
The following principle often simplifies a problem in moments. If an object is symmetrical about an axis, then its moment about that axis is zero and its center of mass lies on the axis. PROOF f(x) = f(x), ρ(x) = ρ(x). Then Also, x = M_{y}/m = 0. Figure 6.6.12 Symmetry about the yaxis


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