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
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