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Moment of Inertia
A Nexus mindmap
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Unpacking the meaning of this
integral takes a little practice -
essentially you need to write the
mass as a function of position
and then integrate over the
objects shape.
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(view from above)
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Neglect width of object
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If the axis is in the middel of the beam, then it's the
same calculation just the limits in the integral change
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Example calculation for
moment of inertia - a disk
rotating about two
different axes.
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This is the view sideon
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It's easier to turn a coin by
flipping the faces than
rotating it face up. If you
think about it, more of the
coins mass is concentrated
near the axis of rotation in
the former case.
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This is an alternative way of splitting up the mass
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Since the moment of inertial is given by
an integral, we are free to split up the
mass however convenient. It pays off to
look for symmetries that will simplify
the calculation.
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This would be the calculation by direct integration
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for 1D
object
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for 2D object
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e.g.
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Using scale
(and symmetry)
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Create Cantor dust by repeatedly
removing the middle third of a
line segment ad infinitum...
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Notice that this is larger than for a solid line
segment. We have the same mass but
distributed further out so it makes sense.
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See [Morin,
Introduction
to Classical Mechanics
]
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Create the Sierpinski gasket by
repeatedly removing the middle
nineth of a square and each square
in the surrounding material.
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Again, this is larger than the solid
version as the mass is distributed
further out.
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Fractal objects!
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Moment of Inertia