Question

A square sheet of aluminum is uniformly 3.8mm thick. It was cut to the shape shown along the function y = xb (b = 0.59), and retaining the part below the curve. The density of aluminum is ? = 2.70x103 kg/m3. (a) Calculate Iy, the moment-of-inertia about the y-axis, of the piece of aluminum sheet shown in the figure above. (b) Calculate Ix, the moment-of-inertia about the x-axis, of the piece of aluminum sheet shown in the figure above. I'd love a detailed explanation to this problem. Thanks is advance for your help!
A square sheet of aluminum is uniformly 3.8mm thick. It was cut to the shape shown along the function y = x^b (b = 0.59), and retaining the part below the curve. The density of aluminum is ? = 2.70x10³ kg/m³.

(a) Calculate I_y, the moment-of-inertia about the y-axis, of the piece of aluminum sheet shown in the figure above.

(b) Calculate I_x, the moment-of-inertia about the x-axis, of the piece of aluminum sheet shown in the figure above.

I'd love a detailed explanation to this problem. Thanks is advance for your help!

Answer 1

As I can understand, I believe you are asking for the moment of inertia about the x-axis of the figure.

$I=\int r^{2}dm=\int \rho r^{2}dV$

We will consider the following:

$dV=y\times w\times dx$    where: w is the width

In this case: $r=y$

Then: $I=\rho w\int_{0}^{1}y^{3}dx=\rho w\int_{0}^{1}x^{3b}dx$

Solving the integral: $I=\frac{p\times w}{3b+1}=\frac{2.7\times 10^{3}\times 3.8\times 10^{-3}}{(3\times 0.59)+1}$

Finally: $I=3.7$