...
Moment
...
of
...
Inertia
...
| Excerpt |
|---|
...
A |
...
measure |
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of |
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the |
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tendency |
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of |
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an |
...
object |
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to |
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maintain |
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its |
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rotational |
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velocity |
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about |
...
a |
...
specified |
...
axis |
...
of |
...
rotation. |
...
The |
...
moment |
...
of |
...
inertia |
...
depends |
...
linearly |
...
on |
...
the |
...
mass |
...
and |
...
quadratically |
...
on |
...
the |
...
distance |
...
of |
...
that |
...
mass |
...
from |
...
the |
...
axis |
...
of |
...
rotation. |
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It |
...
plays |
...
the |
...
same |
...
role |
...
for |
...
rotational |
...
motion |
...
as |
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mass |
...
plays |
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for |
...
translational |
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motion, |
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being |
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both |
...
the |
...
ratio |
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of |
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angular |
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momentum |
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to |
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angular |
...
velocity |
...
and |
...
the |
...
ratio |
...
of |
...
torque |
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to |
...
resultant |
...
angular |
...
acceleration, |
...
whereas |
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mass |
...
is |
...
the |
...
ratio |
...
of |
...
(linear) |
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momentum |
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to |
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velocity |
...
and |
...
the |
...
ratio |
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of |
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force |
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to |
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resultant |
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linear |
...
acceleration. |
Motivation for Concept
It is clear that some objects are more difficult to set into rotation or to stop from rotating than others. Consider four very different objects that are often rotated: a CD, a bicycle wheel, a merry-go-round
...
in
...
a
...
park,
...
and
...
a
...
carousel
...
at
...
an
...
amusement
...
park.
...
Rotating
...
a
...
CD
...
about
...
its
...
natural
...
axis
...
is
...
trivial
...
(simply
...
brush
...
it
...
with
...
your
...
finger),
...
and
...
stopping
...
its
...
rotation
...
is
...
similarly
...
trivial.
...
Rotating
...
a
...
bicycle
...
wheel
...
is
...
fairly
...
easy
...
(a
...
push
...
with
...
your
...
hand)
...
and
...
stopping
...
its
...
rotation
...
is
...
similarly
...
straightforward.
...
Rotating
...
a
...
park
...
merry-go-round
...
requires
...
some
...
effort
...
(a
...
full
...
push
...
with
...
your
...
legs)
...
and
...
stopping
...
it
...
takes
...
some
...
thought
...
if
...
you
...
wish
...
to
...
avoid
...
injury.
...
Starting
...
an
...
amusement
...
park
...
carousel
...
requires
...
a
...
large
...
motor
...
and
...
stopping
...
it
...
requires
...
sturdy
...
brakes.
...
These
...
objects
...
have
...
distinctly
...
different
...
moments
...
of
...
inertia.
...
Of
...
course,
...
they
...
also
...
have
...
very
...
different
...
...
.
...
Thus,
...
mass
...
is
...
one
...
factor
...
that
...
plays
...
into
...
moment
...
of
...
inertia.
...
Moment
...
of
...
inertia
...
is
...
not
...
the
...
same
...
as
...
mass,
...
however,
...
because
...
it
...
depends
...
quadratically
...
on
...
size
...
as
...
well.
...
This
...
can
...
be
...
seen
...
in
...
a
...
straightforward
...
experiment.
...
Find
...
two
...
boards
...
that
...
have
...
the
...
same
...
weight
...
but
...
different
...
lengths
...
-
...
for
...
example
...
a
...
1"
...
X
...
2"
...
board
...
that
...
is
...
12'
...
long
...
and
...
a
...
2"
...
by
...
4"
...
board
...
that
...
is
...
3'
...
long.
...
Grab
...
each
...
by
...
the
...
center
...
in
...
each
...
hand
...
and
...
rotate
...
them.
...
It
...
will
...
require
...
dramatically
...
more
...
effort
...
to
...
rotate
...
the
...
longer
...
board
...
-
...
16
...
times
...
as
...
much,
...
in
...
fact.
...
Note
...
that
...
the
...
mass
...
has
...
not
...
changed
...
in
...
this
...
exercise,
...
only
...
the
...
distance
...
between
...
the
...
mass
...
and
...
the
...
axis
...
of
...
rotation.
...
CD | Bike Wheel | Merry-Go-Round | Carousel |
|---|---|---|---|
| | | |
Photo courtesy Wikimedia Commons, | Photo courtesy Wikimedia Commons, | Photo by Eric Hart, courtesy Flickr. | Photo courtesy Wikimedia Commons, |
Mathematical Definition
Rigid Body Simplification
For an introductory course, it is sufficient to consider the definition of the moment of inertia of a rigid body executing pure rotation (no translation relative to the axis) about a single axis of rotation. The importance of this statement is that every point in the body will maintain a fixed distance from the axis of rotation. This condition is specified so that the moment of inertia of the body remains constant.
Body as Sum of Point Particles
Under this condition, we can quickly derive the form and the utility of the moment of inertia by considering the rigid body to be a collection of Np point particles. Each of the Np point particles (of mass mi where i runs from 1 to Np) will obey Newton's 2nd Law:
| Latex |
|---|
|| Carousel || | !CD.jpg|height=175! | !bikewheel.jpg|height=175! | !merrygoround.jpg|height=175! | !carousel.jpg|height=175! | | Photo courtesy Wikimedia Commons, \\ by user Ubern00b. | Photo courtesy Wikimedia Commons, \\ by user Herr Kriss. | Photo by Eric Hart, courtesy Flickr. | Photo courtesy Wikimedia Commons, \\ by user KMJ. | h3. Mathematical Definition h4. Rigid Body Simplification For an introductory course, it is sufficient to consider the definition of the moment of inertia of a [rigid body] executing pure rotation (no translation _relative to the axis_) about a single [axis of rotation]. The importance of this statement is that every point in the body will *maintain a fixed distance from the* *[axis of rotation]*. This condition is specified so that the moment of inertia of the body remains constant. h4. Body as Sum of Point Particles Under this condition, we can quickly derive the form and the utility of the moment of inertia by considering the [rigid body] to be a collection of _N{_}{~}p~ [point particles|point particle]. Each of the _N{_}{~}p~ point particles (of mass _m{_}{~}i~ where i runs from 1 to _N{_}{~}p~) will obey [Newton's 2nd Law|Newton's Second Law]: {latex}\begin{large}\[ \sum\_{j=1}\^{N_{\rm f,i}} \vec{F}_{i,j}= m\_{i}\vec{a}_{i}\] \end{large}{latex} |
where
...
N
...
f,
...
i is
...
the
...
number
...
of
...
forces
...
acting
...
on
...
the
...
ith
...
particle.
...
Cross
...
Product
...
with
...
Radius
...
Taking
...
the
...
cross
...
product
...
of
...
each
...
side
...
of
...
this
...
equation
...
with
...
respect
...
to
...
the
...
radial
...
distance
...
from
...
the
...
axis
...
of
...
rotation:
| Latex |
|---|
}\begin{large}\[ \sum\_{j=1}\^{N_{\rm f,i}} \vec{r}_{i}\times \vec{F}_{i,j}= m\_{i}\vec{r}_{i}\times \vec{a}_{i}\]\end{large}{latex} |
We
...
can
...
rewrite
...
this
...
using
...
the
...
definition
...
of
...
the
...
...
...
and
...
the
...
| Latex |
|---|
(single-axis)]: {latex}\begin{large}\[ \sum\_{j=1}\^{N_{\rm f,i}} \tau\_{i,j}= m\_{i}r\_{i}^{2}\alpha \] \end{large}{latex} {note}Note that for |
| Note |
|---|
Note that for a rigid body that is undergoing pure rotation about a certain axis (recall our assumption), all particles will have the same angular acceleration. |
Implementing a sum over the particles that make up the body then gives:
| Latex |
|---|
a [rigid body] that is undergoing pure rotation about a certain axis (recall our assumption), all particles will have the same angular acceleration. {note}Implementing a sum over the particles that make up the body then gives: {latex}\begin{large}\[ \sum\_{i=1}\^{N_{\rm p}} \sum\_{j=1}\^{N_{\rm f,i}} \tau\_{i,j}= \alpha \sum\_{i=1}\^{N_{\rm p}} m\_{i}r\_{i}^{2}\]\end{large}{latex} h4. Moment of Inertia as Sum The left side of this |
Moment of Inertia as Sum
The left side of this equation is simply the sum of all torques acting on the body. On the right side, we define the moment of inertia, I as:
| Latex |
|---|
equation is simply the sum of all torques acting on the body. On the right side, we define the moment of inertia, _I_ as: {latex}\begin{large}\[ I = \sum\_{i=1}\^{N_{\rm p}} m\_{i}r\_{i}^{2}\] \end{large}{latex} h3. Uses of the Moment of Inertia h4. |
Uses of the Moment of Inertia
Role in Rotational Analog of Newton's 2nd Law
The work of the previous section allows us to write:
| Latex |
|---|
Role in Rotational Analog of Newton's 2nd Law The work of the previous section allows us to write: {latex}\begin{large}\[ \sum \tau = I\alpha \]\end{large}{latex} |
This
...
is
...
the
...
rotational
...
analog
...
of
...
Newton's
...
2nd
...
Law,
...
with
...
the
...
...
taking
...
the
...
place
...
of
...
the
...
...
,
...
the
...
...
...
taking
...
the
...
place
...
of
...
the
...
(linear)
...
...
and
...
the
...
moment
...
of
...
inertia
...
taking
...
the
...
place
...
of
...
the
...
...
.
...
Role
...
in
...
Angular
...
Momentum
...
Under
...
the
...
...
we
...
discussed
...
at
...
the
...
beginning
...
of
...
the
...
derivation
...
above,
...
the
...
moment
...
of
...
inertia
...
is
...
a
...
constant.
...
Thus,
...
using
...
the
...
definition
...
of
...
...
...
,
...
we
...
can
...
write:
| Latex |
|---|
}\begin{large}\[ \sum \tau = \frac{d(I\omega)}{dt}= \frac{dL}{dt}\]\end{large}{latex} |
where,
...
in
...
the
...
absence
...
of
...
a
...
net
...
torque,
...
the
...
quantity:
| Latex |
|---|
}\begin{large}\[ L = I\omega \] \end{large}{latex} |
is
...
...
.
...
By
...
analogy
...
with
...
the
...
linear
...
case,
...
we
...
refer
...
to
...
L
...
as
...
the
...
angular
...
momentum
...
of
...
the
...
rigid
...
body
...
about
...
the
...
specified
...
axis.
...
Role
...
in
...
Rotational
...
Kinetic
...
Energy
...
We
...
can
...
similarly
...
define
...
a
...
quantity
...
analogous
...
to
...
the
...
translational
...
kinetic
...
energy.
...
We
...
start
...
with
...
a
...
relationship
...
from
...
...
| Latex |
|---|
|Rotational Motion]: {latex}\begin{large}\[ \omega\_{f}^{2}= \omega\_{i}^{2}+ 2\alpha(\theta\_{f}\-\theta\_{i}) \] \end{large}{latex} |
We
...
then
...
multiply
...
by
...
the
...
moment
...
of
...
inertia
...
to
...
find:
| Latex |
|---|
}\begin{large}\[ \frac{1}{2}I\omega\_{f}^{2}\- \frac{1}{2}I\omega\_{i}^{2}= I \alpha(\theta\_{f}\- \theta\_{i}) = \Delta\theta \sum \tau \] \end{large}{latex} |
Noting
...
the
...
similarity
...
to
...
the
...
Work-Kinetic
...
Energy
...
Theorem,
...
and
...
noting
...
that
...
each
...
side
...
has
...
the
...
units
...
of
...
Joules,
...
a
...
likely
...
definition
...
of
...
rotational
...
kinetic
...
energy
...
is:
| Latex |
|---|
}\begin{large}\[ K\_{\rm rot}= \frac{1}{2}I\omega\^omega^{2}\] \end{large}{latex} |
The
...
consistency
...
of
...
this
...
definition
...
with
...
the
...
principle
...
of
...
conservation
...
of
...
energy
...
can
...
be
...
seen
...
in
...
example
...
problems
...
like:
...
| Content by Label | ||||
|---|---|---|---|---|
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...
|
Summary of Analogies Between Mass and Moment of Inertia
This table presents a list of formulas in which moment of inertia plays a role in the angular formula analogous to that of mass in the linear formula.
Description | Linear Formula | Angular Formula | |
|---|---|---|---|
Newton's 2nd Law / Angular Version |
|
...
|
...
Momentum / Angular Momentum |
|
...
|
...
| |||
Kinetic Energy / Rotational Kinetic Energy |
|
...
|
...
|
...
|
...
|
Calculating Moment of Inertia
Integrals in Cylindrical Coordinates
For continuous objects, the summation in our definition of the moment of inertia must be converted to an integral. Because the definition involves the radial distance from a specific axis, the integrals are often best performed in cylindrical coordinates with the z-axis of the coordinate system identified with the axis of rotation. In this case, the sum can be converted to the following integral:
| Latex |
|---|
{latex} | h3. Calculating Moment of Inertia h4. Integrals in Cylindrical Coordinates For _continuous_ objects, the summation in our [definition|#sum] of the moment of inertia must be converted to an integral. Because the definition involves the radial distance from a specific axis, the integrals are often best performed in cylindrical coordinates with the z-axis of the coordinate system identified with the axis of rotation. In this case, the sum can be converted to the following integral: {latex}\begin{large}\[ I = \sum\_{i=1}^{N}r\_{i}^{2}m\_{i}\rightarrow \int\int\int \:r\^r^{2}\: \rho(r,\theta,z)\:r\:dr\:d\theta:dz \]\end{large}{latex} |
where
...
ρ
...
is
...
the
...
density
...
of
...
the
...
object
...
and
...
the
...
quantity:
| Latex |
|---|
}\begin{large}\[ \rho(r,\theta,z)\: r\: dr\: d\theta\: dz = \rho dV = dm \] \end{large}{latex} |
is
...
the
...
differential
...
mass.
...
Thus,
...
the
...
mass
...
of
...
the
...
object
...
can
...
be
...
expressed:
| Latex |
|---|
}\begin{large}\[ M = \int\int\int \:\rho(r,\theta,z)\:r\:dr\:d\theta\:dz \] \end{large}{latex} |
The
...
total
...
mass
...
of
...
the
...
object
...
is
...
usually
...
calculated
...
to
...
allow
...
the
...
moment
...
of
...
inertia
...
to
...
be
...
expressed
...
in
...
a
...
form
...
not
...
involving
...
the
...
density.
...
Integrals
...
in
...
Rectangular
...
Coordinates
...
It
...
is
...
preferable
...
in
...
certain
...
cases
...
to
...
perform
...
the
...
integral
...
in
...
rectangular
...
coordinates
...
instead,
...
where
...
the
...
mass
...
differential
...
is:
| Latex |
|---|
}\begin{large}\[ dm = \rho(x,y,z) dx dy dz \] \end{large}{latex} |
and
...
the
...
axis
...
of
...
rotation
...
is
...
again
...
usually
...
identified
...
with
...
the
...
coordinate
...
z-axis,
...
giving:
| Latex |
|---|
}\begin{large}\[ I = \int \int \int \:(x\^x^{2}\+y\^y^{2})\:\rho(x,y,z)\:dx\:dy\:dz \]\end{large}{latex} h3. Summary of Commonly Used Moments of Inertia Certain objects have simple forms for their moments of inertia. The most commonly referenced such objects are summarized in the table below. Note that the moments reported are only valid about the axis shown (the vertical line in all figures and in some cases shown as an "x" when more than one point of view is provided). In each case, the object has a total mass M and is assumed to be of uniform density. {note}The moment of inertia of composite objects formed of parts that are found in this table can often be found using the [parallel axis theorem].{note} || Description || Illustration || Moment of Inertia || || Thin Hoop or Hollow Cylinder of Radius _R_ | !hollowcylinder.png|height=150! | {latex} |
Summary of Commonly Used Moments of Inertia
Certain objects have simple forms for their moments of inertia. The most commonly referenced such objects are summarized in the table below. Note that the moments reported are only valid about the axis shown (the vertical line in all figures and in some cases shown as an "x" when more than one point of view is provided). In each case, the object has a total mass M and is assumed to be of uniform density.
| Note |
|---|
The moment of inertia of composite objects formed of parts that are found in this table can often be found using the parallel axis theorem. |
Description | Illustration | Moment of Inertia | ||
|---|---|---|---|---|
Thin Hoop or Hollow Cylinder of Radius R | |
|
...
|
...
Disc or Solid Cylinder of Radius R | |
|
|---|
...
|
...
| ||||
Thin Rod or Plane of Length L (rotated about center) | |
|
|---|
...
|
...
Thin Rod or Plane of Length L (rotated about edge) | |
|
|---|
...
|
...
|
|
...
|
...
|
...
|
...
|
|
...
|
...
Thin Hollow Spherical Shell of Radius R | |
|
|---|
...
|
...