Moment of Inertia for Square Plate on X Axis

[fightnchikn]

helllo, I am having trouble finding the moment of inertia for a square uniform plate with length h on the sides...how do you go about it when the interia is about the x axis...the z axis is perpendicular to the square in this case... i understand how to do it when it is with respect to the z axis...the double intergral and all but when its on the xaxis i get confused...how do you set that up.

 

[OlderDan]

helllo, I am having trouble finding the moment of inertia for a square uniform plate with length h on the sides...how do you go about it when the interia is about the x axis...the z axis is perpendicular to the square in this case... i understand how to do it when it is with respect to the z axis...the double intergral and all but when its on the xaxis i get confused...how do you set that up.

It should be easier to do it for the x-axis than for the z axis. I assume the origin is in the center of the plate, with the x-axis being a line that divides the plate into two equal rectangles. Consider a mass element dm = σdxdy and integrate x^2dm over the area of the plate. The y integral is very simple since there is no function of y in the integrand. You just get h, which is constant for all x. All that is left is the integral of x^2.

 

[fightnchikn]

Now when I integrate what do I integrate from 0 to h?

 

[OlderDan]

Now when I integrate what do I integrate from 0 to h?

If the axis is as I assumed, it is from -h/2 to +h/2. By symmetry, you could do twice the integral from 0 to h/2 and get the same result.