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You are watching: Moment of inertia of a cube

just how perform ns calculate ns minute of inertia of a unicreate hard cube about a axis passong via its facility that mass?

ns additionally want to understand if the moment of inertia the a body ins independenns of itns shape. Also, newly ns reADVERTISEMENT somewhere the ns minute the inertia the a uniform hard cube is minimum around a axins passong through itns COM Because the mass ins even more focused at its center. Doens the statement do any kind of sense?

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initially you should define ns orientation of ns cube family member come the axis friend want come measure. Frequently a 3×3 rotatitop top matrix $E$ doens the job transforming neighborhood coordinates along the principal axes come ns people coordinates. The usage ns transdevelopment $E I_body E^\intercal$


a solitary rotation $\theta$ around ns people $z$ axis is

$$E = \beginpmatrix \cos\theta & -\sin\theta & 0 \\ \sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \endpmatrix $$

If ns masns minute the inertia matrix around ns major axes $(x,y,z)$ is

$$I_body = \beginvmatrix I_xx & & \\ & I_yy & \\ & & I_zz \endvmatrix $$

climate ns mass minute that inertia around the civilization coordinates is

$$ I_world = E\,I_body\, E^\intercal $$

wbelow $E^\intercal$ ins ns inverse rotati~ above found by the transpose operator. Ns outcome is $$ I_world = \beginvmatrix I_yy+(I_xx-I_yy)\cos^2\theta & (I_xx-I_yy)\sin\theta\cos\theta & 0\\ (I_xx-I_yy)\sin\theta\cos\theta & I_xx+(I_yy-I_xx)\cos^2\theta & 0\\0 & 0& I_zz \endvmatrix $$

This represent ns masns moment the inertia about the three world coordinates. Come get ns MMOi about a certain axis $\hate$ you calculate $$I_ee = \hate^\intercatogether I_world \hate $$

for this reason to get the MMOns about ns civilization $X$ axins through $\hate=(1,0,0)$ you discover the $$I_XX = \beginpmatrix1 & 0 & 0 \endpmatrix I_world \beginpmatrix 1 \\ 0 \\ 0 \endpmatrix = I_yy+(I_xx-I_yy)\cos^2\theta $$

Alernativelygirlfriend deserve to find ns regional works with the ns world $X$ axins together $\hatx = E^\intercatogether \hate$ and also then compute $$\boxed I_XX = \hatx^\intercal I_body \hatx $$