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WALTER LEWIN: I now want to discuss with you the parallel axis theorem,
which comes in very handy sometimes when you have to
calculate moments of inertia.
Let us take our potato again, three dimensional object.
And this potato we rotate about an axis through this point P. Random
point P, we rotate it about this axis.
This is the center of mass.
And let's suppose we also rotate it about an axis which is
parallel to this axis.
But this axis goes through the center of mass.
The object has mass M. And the separation between
these two axes is d.
The parallel axis theorem now tells you-- and I give this
to you without proof---
that if I call this axis 1 through the center of mass, and I call this axis 2
through point P, parallel to this axis-- it's very important that they
are parallel--
that the moment of inertia about axis 2 is the moment of inertia about the
center of mass, axis 1-- and these two have to be parallel--
and now I have to add Md squared.
Very powerful theorem, because very often do you find in tables moments of
inertia about centers of mass, because they are often axes of symmetry like
cylinders, and rods, and spheres.
But now you can rotate about an axis which doesn't go through
the center of mass.
And then this parallel axis theorem is very, very powerful.
I could also have rotated this potato about an axis through the center of
mass, which would be perpendicular to the paper.
And then, if I were to rotate it about an axis perpendicular to the paper
through this point, I would get the same results.
I would first have to calculate what the moment of inertia is about this
axis through the center of mass.
And I add now Md squared.
This is now d.
And then I have the moment of inertia about this axis.
Very powerful tool.