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Parallel-Axis Theorem

9 years ago • updated by 9 years ago • 1

A thin uniform rod of mass M and length L on the x axis has one end at the origin. Using the parallel-axis theorem, find the moment of inertia about y' axis, which is parallel to the y axis, and through the center of the rod.

I know the moment of Inertia would simply be the following I = I_cm + Mh^2. The book however doesn't explain how that is possible to do. Is there a reason to be able to say this?

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9 years ago

The derivation of the parallel axis theorem is a google search away, however I may be able to relate the basic idea to you.

Moment of inertia equations are found by integrating over an object.

The moment of inertia about to the z axis is

$I_{\mathrm {cm} }=\int (x^{2}+y^{2})\,dm.$ The moment of inertia relative to an axis which is a perpendicular distance d along the x-axis from the center of mass, is
$I=\int \left[(x+d)^{2}+y^{2}\right]\,dm$ which becomes $I=\int (x^{2}+y^{2})\,dm+d^{2}\int dm+2d\int x\,dm.$ which is $I=I_{\mathrm {cm} }+md^{2}.$ + 0

Hope this helps!

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