| Excerpt |
|---|
A relationship between the moment of inertia of a rigid body about an axis passing through the body's center of mass and the moment of inertia about any parallel axis. |
Statement of the Theorem
The parallel axis theorem states that if the moment of inertia of a rigid body about an axis passing through the body's center of mass is Icm then the moment of inertia of the body about any parallel axis can be found by evaluating the sum:
| Latex |
|---|
| Wiki Markup |
The parallel axis theorem states that if the moment of inertia of a rigid body about an axis passing through the body's center of mass is _I_~cm~ then the moment of inertia of the body about any parallel axis can be found by evaluating the sum: {latex}\begin{large}\[ I_{||} = I_{cm} + Md^{2} \] \end{large}{latex} where _d_ is the |
where d is the (perpendicular)
...
distance
...
between
...
the
...
original
...
center
...
of
...
mass
...
axis
...
and
...
the
...
new
...
parallel
...
axis.
...
Motivation
...
for
...
the
...
Theorem
...
We
...
know
...
that
...
the
...
...
momentum about a single axis of a rigid body that is translating and rotating with respect to a (non-accelerating)
...
axis
...
can
...
be
...
written:
| Latex |
|---|
}\begin{large}\[ L = m\vec{r}_{\rm cm,axis}\times\vec{v}_{cm} + I_{cm}\omega_{cm} \]\end{large}{latex} |
Now
...
suppose
...
that
...
the
...
rigid
...
body
...
is
...
executing
...
pure
...
rotation
...
about
...
the
...
axis.
...
In
...
that
...
case,
...
the
...
velocity
...
of
...
the
...
center
...
of
...
mass
...
will
...
be
...
perpendicular
...
to
...
the
...
displacement
...
vector
...
from
...
the
...
axis
...
of
...
rotation
...
to
...
the
...
center
...
of
...
mass.
...
Calling
...
the
...
magnitude
...
of
...
that
...
displacement
...
d
...
,
...
to
...
make
...
contact
...
with
...
the
...
form
...
of
...
the
...
theorem,
...
we
...
then
...
have:
| Latex |
|---|
}\begin{large}\[ L \mbox{ (pure rotation)} = mv_{cm}d + I_{cm}\omega_{cm} \]\end{large}{latex} |
If
...
the
...
body
...
is
...
purely
...
rotating,
...
we
...
can
...
also
...
define
...
an
...
angular
...
speed
...
for
...
rotation
...
about
...
the
...
new
...
parallel
...
axis.
...
The
...
angular
...
speed
...
must
...
satisfy
...
(consider
...
that
...
the
...
center
...
of
...
mass
...
is
...
describing
...
a
...
circle
...
of
...
radius
...
d
...
about
...
the
...
axis):
| Latex |
|---|
}\begin{large} \[ \omega_{\rm axis} = \frac{v_{cm}}{d} \] \end{large}{latex} |
Futher,
...
the
...
rotation
...
rate
...
of
...
the
...
object
...
about
...
its
...
center
...
of
...
mass
...
must
...
equal
...
the
...
rotation
...
rate
...
about
...
the
...
parallel
...
axis,
...
since
...
when
...
the
...
object
...
has
...
completed
...
a
...
revolution
...
about
...
the
...
parallel,
...
its
...
oritentation
...
must
...
be
...
the
...
same
...
if
...
it
...
is
...
executing
...
pure
...
rotation.
...
Thus,
...
we
...
can
...
write:
| Latex |
|---|
}\begin{large}\[ L \mbox{ (pure rotation)} = m\omega_{\rm axis} d^{2} + I_{cm}\omega_{\rm axis} \]\end{large}{latex} |
which
...
implies
...
the
...
parallel
...
axis
...
theorem
...
holds.
...
Derivation
...
of
...
the
...
Theorem
...
From
...
the
...
definition
...
of
...
the
...
moment
...
of
...
inertia:
| Latex |
|---|
}\begin{large}\[ I = \int r^{2} dm \] \end{large}{latex} |
The
...
center
...
of
...
mass
...
is
...
at
...
a
...
position
...
r
...
cm with
...
respect
...
to
...
the
...
desired
...
axis
...
of
...
rotation.
...
We
...
define
...
new
...
coordinates:
| Latex |
|---|
}\begin{large}\[ \vec{r} = \vec{r}\:' + \vec{r}_{cm}\]\end{large}{latex} where _ |
where r'
...
measures
...
the
...
positions
...
relative
...
to
...
the
...
object's
...
center
...
of
...
mass.
...
Substituting
...
into
...
the
...
moment
...
of
...
inertia
...
formula:
| Latex |
|---|
{latex}\begin{large}\[ I = \int\: (r\:'^{2} + 2\vec{r}\:'\cdot\vec{r}_{cm} + r_{cm}^{2})\: dm \]\end{large}{latex} The _r_~cm~ is a constant within the integral over the |
The rcm is a constant within the integral over the body's
...
mass
...
elements.
...
Thus,
...
the
...
middle
...
term
...
can
...
be
...
written:
| Latex |
|---|
}\begin{large}\[ \int\:\vec{r}\:'\cdot \vec{r}_{cm}\:dm = r_{cm}\cdot \int\:\vec{r}\:'\:dm \]\end{large}{latex} |
Since
...
the
...
r
...
'
...
measure
...
deviations
...
from
...
the
...
center
...
of
...
mass
...
position,
...
the
...
integral
...
r
...
'
...
dm
...
must
...
give
...
zero
...
(the
...
position
...
of
...
the
...
center
...
of
...
mass
...
in
...
the
...
r
...
'
...
system).
...
Thus,
...
we
...
are
...
left
...
with:
| Latex |
|---|
}\begin{large}\[ I = \int\: r\:'^{2}\:dm + r_{cm}^{2} \int\:dm = I_{cm} + Mr_{cm}^{2} Which is the parallel axis theorem. }\]\end{large} |
Which is the parallel axis theorem.