Moment of Inertia of Rectangles

[al4n]

TL;DR Summary

Why is the axis of rotation used for the moment of inertia of a rectangle about its center of mass, perpendicular to one of its bases and on the same plane instead of say, an axis that goes through the plane/page

I often encounter the formula: I = (1/12)Mbh^2 when dealing with moment of inertia of rectangles and got confused when I was unable to get the same result when figuring it out with integration. It seems that the axis of rotation used is a line perpendicular to one of the bases and on the plane of the page. Wouldn't it be more useful if the axis used is one that points into or out of the page since most problems involve rotations that remain on the same plane as the rectangle?

 

[Ibix]

I think the lesson is not to trust anybody's statement of a moment of inertia until you've checked what axis it is using. Or, more generally, make sure that the question they're answering is the one you're asking.

Also, I believe the second moment of area through axes in the plane is important in the study of beams and the way they bend. So I suspect engineers would disagree with your characterisation of which axes are important.

 

[vanhees71]

Which moment of inertia to use depends of course on the axis of rotation. For a complete description of a rigid rectangle's (or any rigid body's) motion you need the tensor of inertia:

https://en.wikipedia.org/wiki/Moment_of_inertia#Inertia_tensor

 

[Lnewqban]

Please, see:
http://hyperphysics.phy-astr.gsu.edu/hbase/perpx.html#ppx