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Calculate the inertia tensor of a homogeneous cube of mass M and density D and sides b
Calculate the inertia tensor of a homogeneous cube of mass M and density D and sides b. Let one corner be at the origin and the 3 sides lie along the coordinate axes.
Please show all work including the matrix expansion in detail...I need to see how the tensor equation is used to generate a moment of inertia.
Expert Solution
Please see the attached file.
The moment of inertia of a solid body with density with respect a given axis is defined by the volume integral
,
Where is the perpendicular distance from the axis of rotation.
It can written as
(1)
Where r is the distance to the center and is the kronecker delta.
In the Cartesian axes x, y, z it can be broken into components as
The distance for any point (x, y, z) to the center (0, 0, 0)
Then according to equation (1)
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