The surface density of a thin rectangle varies as:
σ(x,y)= 14.0 kg/m2 + 9.00 kg/m4(x2+y2)
The rectangle has a length L = 0.500 m and a width W = 1.500 m. Calculate Iz, the moment of inertia about the z-axis.
To calculate the moment of inertia (Iz) about the z-axis for a thin rectangle with a variable surface density, we use the formula:
Iz = ∬ (ρ(x, y) * (x^2 + y^2)) dA
Where:
- ρ(x, y) is the surface density at the point (x, y)
- (x^2 + y^2) is the distance squared from the z-axis at the point (x, y)
- dA is the differential area element
In our case, the surface density varies as σ(x, y) = 14.0 kg/m^2 + 9.00 kg/m^4(x^2 + y^2).
Given that the rectangle has a length L = 0.500 m and a width W = 1.500 m, we need to integrate the moment of inertia formula over the area of the rectangle.
The integral becomes:
Iz = ∫∫ (σ(x, y) * (x^2 + y^2)) dA
= ∫∫ (14.0 kg/m^2 + 9.00 kg/m^4(x^2 + y^2)) * (x^2 + y^2) dA
To perform this double integral, we need to express the integration limits for x and y:
The rectangle has a length L = 0.500 m, so the x limits will be -L/2 to L/2:
-0.250 ≤ x ≤ 0.250
The rectangle has a width W = 1.500 m, so the y limits will be -W/2 to W/2:
-0.750 ≤ y ≤ 0.750
We can now proceed with the integration.