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{excerpt}An object's moment of inertia is a measure of the effort required to change that object's rotational velocity about a specified axis of rotation. {excerpt}

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h2. Everyday Definition

It is clear that some obejcts are more difficult to set into rotation or to stop from rotating than others.  Take, for example, the cases of 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 [masses|mass].  Thus, mass is one factor that plays into moment of inertia.

Moment of inertia is _not_ the same as mass, however, as can be seen in a straightforward experiment.  Find a desk chair that swivels fairly easily and grab a pair of dumbbells or other objects with significant mass.  Sit on the chair holding the dumbbells at your chest and swivel back and forth a few times.  Get a sense of the effort your feet exert to start and stop your motion.  Next, hold the dumbbells out to your sides at your full arms' length.  Repeat the experiment and note the effort required in the new configuration.  Note that your mass (plus the chair and dumbbells) has not changed in this exercise, only the position of the mass has changed.

h2. Mathematical Definition

{anchor:der}
h4. An Important Assumption

For an introductory course, it is sufficient to consider the definition of the moment of inertia of a [rigid body] executing pure rotation (no tranlation relative to the axis) about an [axis of rotation] that maintains a fixed distance from that [rigid body's|rigid body] center of mass.  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].  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.

h4. 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 [angular acceleration] and the [torque]:

{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 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 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}

h2. Uses of the Moment of Inertia

h4. Role in Rotational Analog of Newton's 2nd Law

This expression lets us equate the sum of torques to:

{latex}\begin{large} \[ \sum \tau = I\alpha \]\end{large}{latex}

This is the rotational analog of Newton's 2nd Law, with the [torque] taking the place of the [force], the [angular acceleration] taking the place of the (linear) [acceleration] and the [moment of inertia] taking the place of the [mass].

h4. Role in Angular Momentum

Under the assumption we discussed at the beginning of the [derivation above|#der], the moment of inertia is a constant.  Thus, using the definition of [angular acceleration], 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 [conserved].  By analogy with the linear case, we refer to _L_ as the *angular momentum* of the rigid body about the specified axis.

h4. Role in Rotational Kinetic Energy

We can similarly define a quantity analogous to the translational kinetic energy.  We start with a relationship from [angular kinematics]:

{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-Energy Theorem], and noting that each side has the units of Joules, a likely definition of rotational kinetic energy is:

{latex} \begin{large}\[ K_{\rm rot} = \frac{1}{2} I\omega^{2} \] \end{large}{latex}

The consistency of this definition with the [principle of conservation of energy] can be seen in example problems like:

{contentbylabel:energy_conservation,rotational_energy|operator=AND|maxResults=20|showSpace=false|excerpt=true}

h2. 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.

{table}{tr}{th:align=center|bgcolor=#F2F2F2}Description{th}{th:align=center|bgcolor=#F2F2F2}Linear Formula{th}{th:align=center|bgcolor=#F2F2F2}Angular Formula{th}{tr}
{tr}{td}Newton's 2nd Law / Angular Version{td}{td:align=center}{latex}\begin{large}\[\sum \vec{F} = m\vec{a}\]\end{large}{latex}{td}{td:align=center}{latex}\begin{large}\[\sum \tau = I\alpha\]\end{large}{latex}{td}{tr}
{tr}{td}Momentum / Angular Momentum{td}{td:align=center}{latex}\begin{large}\[\vec{p}= m\vec{v}\]\end{large}{latex}{td}{td:align=center}{latex}\begin{large}\[L = I\omega\]\end{large}{latex}{td}{tr}
{tr}{td}Kinetic Energy / Rotational Kinetic Energy{td}{td:align=center}{latex}\begin{large}\[K = \frac{1}{2}mv^{2}\]\end{large}{latex}{td}{td:align=center}{latex}\begin{large}\[K_{\rm rot} = \frac{1}{2}I\omega^{2}\]\end{large}{latex}{td}{tr}
{table}

h2. Calculating Moment of Inertia



h2. Summary of Commonly Used Moments of Inertia



h2. The Parallel Axis Theorem