Moment of Inertia

Explore new capabilities covering physical parameters of a rigid body, including rotational inertia.

In[1]:=

wrench = ExampleData[{"Geometry3D", "Wrench"}, "Region"]

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Pick a point for the wrench to rotate about.

In[2]:=

point = {-8, -0.168, 0};

In[3]:=

Show[wrench, Graphics3D[{PointSize[Large], Point[point]}], ViewPoint -> {0, -\[Infinity], 0}]

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The inertia matrix centered at this point.

In[4]:=

\[ScriptCapitalI] = MomentOfInertia[wrench, point]

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Find the principal axes, the axes a body tends to rotate about.

In[5]:=

principalaxes = Eigenvectors[\[ScriptCapitalI]]

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Find the inertia ellipsoid whose radii indicate the relative resistance to accelerate about a principal axis.

In[6]:=

inertiaellipsoid = Ellipsoid[point, 1000 Inverse[\[ScriptCapitalI]]]

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Show the wrench with its principal axes and inertia ellipsoid. The wrench arm stretches the ellipsoid along its axis.

In[7]:=

axes = Graphics3D[{Opacity[1], Arrowheads[{-0.04, 0.04}], {Specularity[Gray, 10], Gray, inertiaellipsoid}, MapThread[{RGBColor[#2], Arrow[Tube[{point - 4 #, point + 4 #}, .06]]} &, {principalaxes, IdentityMatrix[3]}]}];

In[8]:=

Show[wrench, axes, BaseStyle -> Opacity[0.3]]

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