Chapter 8: Applications of Triple Integration
Section 8.4: Moments of Inertia (Second Moments)
Example 8.4.6
If R, the region that lies between the paraboloids z=4−x2−y2 and z=3 x2+3 y2, and δr ,θ,z=r z cosθ/6 is the density in R, obtain the moments of inertia and the radii of gyration about the Cartesian coordinate-axes.
(See Example 8.1.22.)
Solution
Maple Solution - Interactive
Initialize
Context Panel: Assign Name
δ=r z cosθ/6→assign
The calculations for the moments of inertia are detailed in Table 8.4.8(a) where the iterated integrals are a modification of the contents of Table 8.1.20(c).
Context Panel: Evaluate and Display Inline
Context Panel: Approximate≻5 (digits)
Ix=∫02 π∫01∫3 r24−r2δ r2sin2θ+z2 r ⅆz ⅆr ⅆθ→assign
Ix = 450152150153→at 5 digits51.928
Iy=∫02 π∫01∫3 r24−r2δ r2cos2θ+z2 r ⅆz ⅆr ⅆθ→assign
Iy = 449968150153→at 5 digits51.908
Iz=∫02 π∫01∫3 r24−r2δ r2 r ⅆz ⅆr ⅆθ→assign
Iz = 1841053→at 5 digits3.0353
Table 8.4.8(a) Calculations for the moments of inertia
The total mass m and the radii of gyration are given in Table 8.4.8(b).
m=∫02 π∫01∫3 r24− r2δ r ⅆz ⅆr ⅆθ→assign
m = 136353
kx=Ix/m→assign
kx = 17293410369817→at 5 digits2.7778
ky=Iy/m→assign
ky = 17293410202078→at 5 digits2.7771
kz=Iz/m→assign
kz = 1511173→at 5 digits0.67155
Table 8.4.8(b) Radii of gyration
Maple Solution - Coded
Define the density.
δ≔r z cosθ/6:
Obtain the moments of inertia
Qx≔Intr δ r2sin2θ+z2,z=3 r2..4− r2,r=0..1,θ=0..2 π
∫02π∫01∫3r2−r2+4r2zcos16θr2sinθ2+z2ⅆzⅆrⅆθ
Ix≔valueQx
450152150153
Qy≔Intr δ r2cos2θ+z2,z=3 r2..4− r2,r=0..1,θ=0..2 π
∫02π∫01∫3r2−r2+4r2zcos16θr2cosθ2+z2ⅆzⅆrⅆθ
Iy≔valueQy
449968150153
Qz≔Intr δ r2,z=3 r2..4− r2,r=0..1,θ=0..2 π
∫02π∫01∫3r2−r2+4r4zcos16θⅆzⅆrⅆθ
Iz≔valueQz
1841053
Obtain the total mass m
M≔Intr δ,z=3 r2..4− r2,r=0..1,θ=0..2 π
∫02π∫01∫3r2−r2+4r2zcos16θⅆzⅆrⅆθ
m≔valueM
136353
Obtain the radii of gyration
kx≔Ix/m
17293410369817
ky≔Iy/m
17293410202078
kz≔Iz/m
1511173
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