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 density ρ=1+2 x2+3 y2/10, and is in the shape of R, the region bounded by x=y2 and y=x−2. See Example 6.5.7.
The relevant calculations are in Table 6.6.7(a).
m=∫−12∫y2y+2ρ ⅆx ⅆy = 61831400
Ix=∫−12∫y2y+2ρ⋅y2 ⅆx ⅆy = 84511400
Iy=∫−12∫y2y+2ρ⋅x2 ⅆx ⅆy = 16983616
Rx=Iym=16983/6166183/1400 = 52519⁢1584451 ≐ 2.50
Ry=Ixm=8451/14006183/1400 = 1229⁢71677 ≐ 1.17
Table 6.6.7(a) Moments of inertial and radii of gyration
Maple Solution - Interactive
Obtain the intersections of the bounding curves
Write a sequence of two equations and press the Enter key.
Context Panel: Solve≻Solve
A solution from first principles is detailed in Table 6.6.7(b).
Define the density function ρx,y
Context Panel: Assign Name
ρ=1+2 x2+3 y2/10→assign
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
∫−12∫y2y+2ρ ⅆx ⅆy = 61831400→assign to a namem
Obtain Ix, the moment of inertia about the x-axis
Context Panel: Assign to a Name≻Ix
∫−12∫y2y+2ρ⋅y2 ⅆx ⅆy = 84511400→assign to a nameIx
Obtain Iy, the moment of inertia about the y-axis
Context Panel: Assign to a Name≻Iy
∫−12∫y2y+2ρ⋅x2 ⅆx ⅆy = 16983616→assign to a nameIy
Context Panel: Approximate≻5 (digits)
Iy/m = 52519⁢1584451→at 5 digits2.4986
Ix/m = 1229⁢71677→at 5 digits1.1691
Table 6.6.7(b) Moments of inertia and radii of gyration from first principles
Maple Solution - Coded
A solution from first principles is provided in Table 6.6.7(c).
Define the density function ρx,y.
ρ≔1+2 x2+ 3 y2/10:
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 6.6.7(c) Moments of inertia and radii of gyration from first principles
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