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Moments - Two Dimensions


Figure 6.6.8

A mass m at the point (x0, y0) in the (x, y) plane will have moments Mx about the x-axis and My about the y-axis (Figure 6.6.8). They are defined by

Mx = my0,       My = mx0.

Consider a vertical length of wire of mass m and constant density which lies on the line x = x0 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,

ΔMX ≈ y Δm = yρ Δy (compared to Δy),

ΔMy ≈ x0 Δm = xoρΔy (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 x-coordinate. 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

ΔMy ≈ x Δm ≈ xρ(x) f(x) Δx              (compared to Δx),

ΔMx ≈ 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 two-dimensional object is defined as the point (x, y) with coordinates

x = My/m,       y = Mx/m.

A single mass m at the point (x, y) will have the same moments as the two-dimensional body, Mx = my, My, = mx. The object will balance on a pin placed at the center of mass.

If a two-dimensional 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


Example 4

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.

Consider the y-axis. Suppose a plane object occupies the region under the curve y = f(y) from -a to a and its density at a point (x, y) is p(x) (Figure 6.6.12). The object is symmetric about the y-axis, so for all x between 0 and a,

f(-x) = f(x), ρ(-x) = ρ(x).




x = My/m = 0.


Figure 6.6.12 Symmetry about the y-axis

Example 5

Last Update: 2010-11-25