We would now like to compare
the moment of inertia
for a rigid body.
Let's take an arbitrary rigid
body about the center of mass.
So let's say the rigid body
is rotating about this axis.
And what we'd like
to compare that is
to the moment of inertia, say,
about a parallel axis that's
also going through
the rigid body.
Now, let's recall how we
define moment of inertia.
We first choose a
mass element dm.
What I'd like to show
is if this object
is rotating about this axis,
then what is that object doing.
Well, that object is
undergoing a little bit
of circular motion.
And this distance here is
what we call the perpendicular
distance about that axis.
And let's indicate this
is for our element dm.
So this perpendicular
distance is
what shows up in our definition
for the center of mass moment
of inertia about that axis--
it's the interval of dm r cm
perp.
Now again, quantity squared.
What is this distance?
This is the perpendicular
distance from our dm
and to the axis of rotation.
Imagine it's doing
a circle and that's
the radius of that circle.
So if we were to calculate
the moment of inertia
about another axis,
then about this
axis the perpendicular
distance here
that I'll write as rs perp.
And you can see these
perpendicular distances are not
the same.
And the moment of
inertia about that
other axis is equal to the
integral of dm rs perp quantity
squared.
Now, how do we relate those
perpendicular distances?
Well, there's a couple
of ways to do it.
And notice that the
distance between these axes
is given by d.
And I'm going to call
this the distance r cm.
Now, let's just call this-- the
x direction-- I'll call that x.
So how do I relate
these distances?
Well, d is a fixed distance.
And you can see from
my diagram that rs perp
is equal to d plus r cm x.
And if I square this, I
get d-squared plus 2d rcm x
plus rcm x-squared.
And that r cm x-squared
is precisely what we're
calling perpendicular distance.
So when I put those into
my moment of inertia Is,
I get dm times d-squared plus 2d
times rcm x plus, parentheses,
rcm perp squared.
Now, I'll separate
this into three terms.
The first term is
dm times d-squared.
This is an integral
over the body.
The second term is
2d-- and I'm going
to hold off on the
interval, because the 2d is
the same for every
piece-- dm rcm x.
And the third piece is
integral over the body of dm
r-- since rcm x is
the r perp, I'll
write it as r perp squared.
And you can see that
this term is precisely
the moment of inertia
about the center of mass.
Now, what I'd like
to focus on is
this terribly, in
particular, dm rcm
x that appears in our
integral expression.
Recall, that we
define center of mass.
We had the condition that
the sum of mj rcmj was 0.
Now for an integral
relationship,
this is dm rcmj cm equal to 0.
So when you sum up the position
of every object with respect
to the vector from the center
of mass to your dm element, 0.
What does this say in
terms of components?
In terms of components,
each component
separately vanishes so we have
the condition that cm x is 0.
So that term is 0, which is
precisely this term-- that's 0.
And so we can
conclude that Is-- now
in this term, where d is the
same piece for every object--
so we're just pulling
out the total mass.
So it's m total d-squared.
And let's remind
ourselves that d
is the distance between
the two parallel axes
plus the moment of inertia
about the center of mass.
And this result is called
the parallel axis theorem.
