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 whose shape is that of a semicircle with radius 1, and whose density is equal to the distance from the center of the circle. See Example 6.5.5.
The relevant calculations are in Table 6.6.5(a).
m=∫0π∫01r⋅ρ ⅆr ⅆθ = π/3
Ix=∫0π∫01r⋅ρ⋅r sinθ2 ⅆr ⅆθ = π/10
Iy=∫0π∫01r⋅ρ⋅r cosθ2 ⅆr ⅆθ = π/10
Table 6.6.5(a) Moments of inertia and radii of gyration
Maple Solution - Interactive
A solution from first principles is detailed in Table 6.6.5(b).
Define the density ρ in polar coordinates
Context Panel: Assign Name
Obtain m, the total mass in region R
Iterated double-integral template
Context Panel: Evaluate and Display Inline
Context Panel: Assign to a Name≻m
∫0π∫01r⋅ρ ⅆr ⅆθ = 13⁢π→assign to a namem
Obtain Ix, the moment of inertia about the x-axis
Context Panel: Assign to a Name≻Ix
∫0π∫01r⋅ρ⋅r sinθ2 ⅆr ⅆθ = 110⁢π→assign to a nameIx
Obtain Iy, the moment of inertia about the y-axis
Context Panel: Assign to a Name≻Iy
∫0π∫01r⋅ρ⋅r cosθ2 ⅆr ⅆθ = 110⁢π→assign to a nameIy
Iy/m = 110⁢30
Ix/m = 110⁢30
Table 6.6.5(b) Calculation of moments of inertia and radii of gyration from first principles
Maple Solution - Coded
Use the Int and value commands.
Second Moments (Moments of Inertia)
q≔Intr⋅ρ⋅r sinθ2,r=0..1,θ=0.. π
q≔Intr⋅ρ⋅r cosθ2,r=0..1,θ=0.. π
Radii of gyration
<< Previous Example Section 6.6
Next Example >>
© Maplesoft, a division of Waterloo Maple Inc., 2023. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
Download Help Document