Chapter 6: Applications of Double Integration
Section 6.6: Second Moments
Find the moments of inertial Ix and Iy, the total mass m, and the radii of gyration Rx and Ry of the lamina that
has the shape of R, the region bounded by the graphs of x=0, and x=y−y3, where y∈0,1. (Take ρ=1.)
See Example 6.5.9.
The relevant calculations are in Table 6.6.9(a).
m=∫01∫0y−y3ⅆx ⅆy = 14
Ix=∫01∫0y−y3y2 ⅆx ⅆy = 112
Iy=∫01∫0y−y3x2 ⅆx ⅆy = 1120
Rx=Iy/m=1/1201/4=130 ≐ 0.58
Ry=Ix/m=1/121/4=13 ≐ 0.18
Table 6.6.9(a) Moments of inertia and radii of gyration
Maple Solution - Interactive
A solution from first principles is detailed in Table 6.6.9(b).
Obtain m, the total mass of the lamina
Iterated double-integral template
Context Panel: Evaluate and Display Inline
Context Panel: Assign to a Name≻m
∫01∫0y−y31 ⅆx ⅆy = 14→assign to a namem
Obtain Ix, the moment of inertia about the x-axis
Context Panel: Assign to a Name≻Ix
∫01∫0y−y3y2 ⅆx ⅆy = 112→assign to a nameIx
Obtain Iy, the moment of inertia about the y-axis
Context Panel: Assign to a Name≻Iy
∫01∫0y−y3x2 ⅆx ⅆy = 1120→assign to a nameIy
Context Panel: Approximate≻10 (digits)
Iy/m = 130⁢30→at 10 digits0.1825741858
Ix/m = 13⁢3→at 10 digits0.5773502693
Table 6.6.9(b) Moments of inertia and radii of gyration
Maple Solution - Coded
A solution from first principles is provided in Table 6.6.9(c).
Obtain the total mass of the lamina
Display the unevaluated integral with the Int command, and evaluate the integral with the value command.
Obtain the moments of inertia Ix and Iy
Obtain the radii of gyration
Table 6.6.9(c) Moments of inertia and radii of gyration
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