Chapter 8: Applications of Triple Integration
Section 8.4: Moments of Inertia (Second Moments)
The connection between second moments and moments of inertia for was developed in Section 6.6 for plane regions. The present section deals with moments of inertia and the associated radii of gyration for three-dimensional regions.
Let m be the total mass of a three-dimensional region R having density δx,y,z or δr,θ,z in Cartesian or cylindrical coordinates, respectively. Table 8.4.1 lists expressions for the moments of inertia (second moments) Ix,Iy,Iz. For Cartesian coordinates, dv is one of the six permutations of the differentials dx,dy,dz. For cylindrical coordinates, dv′ is one of the six permutations of the differentials dz,dr,dθ.
Second Moments - Cartesian
Second Moments - Cylindrical
Radii of gyration
Ix=∫∫∫δ r2sin2θ+z2 r dv′
Iy=∫∫∫δ r2cos2θ+z2 r dv′
Iz=∫∫∫δ r2 r dv′
Table 8.4.1 Moments of inertia and radii of gyration
For Ix, the expressions y2+z2 and r2sin2θ+z2 represent, for a point in R, the square of the distance from the x-axis.
For Iy, the expressions x2+z2 and r2cos2θ+z2 represent, for a point in R, the square of the distance from the y-axis.
For Iz, the expressions x2+y2 and r2 represent, for a point in R, the square of the distance from the z-axis.
Because Maple uses I for the imaginary unit −1, it is troublesome to assign to a symbol such as Ix. Hence, whenever such an assignment is needed in the accompanying examples, a symbol such as Ix will be used instead.
The radii of gyration represent distances from the Cartesian coordinate-axes where, if all the mass m in R were concentrated, the rotational properties of the region would be preserved.
In each of the following examples, the region R and the density δ is taken from the corresponding example in Section 8.3.3. For each example, obtain the second moments Ix,Iy,Iz, and the radii of gyration kx,ky,kz.
R is the region in Example 8.1.3
R is the region in Example 8.1.5
R is the region in Example 8.1.8
R is the region in Example 8.1.15
R is the region in Example 8.1.20
R is the region in Example 8.1.22
δr ,θ,z=r z cosθ/6
R is the region in Example 8.1.27
δx,y,z=2 x2+3 y2+4 z2
R is the region in Example 8.1.28
R is the region in Example 8.1.29
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