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{excerpt}Angular momentum (1-D) is the circulation of [linear momentum|momentum] about some specified axis, being proportional to the component of momentum or each mass along a circle about the axis and the radius of the circle.  Angular momentum is changed by external [torques|torque (single-axis)], and therefore is constant when these sum to zero. The angular momentum of a rigid body is proportional to its [moment of inertia] times its angular velocity.{excerpt}
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h2. Motivation for Concept

Imagine a satellite spinning in space. While there is a gravitational force on the satellite which causes its linear momentum to change, the satellite will continue to spin at the same speed about its axis of rotation.  To change its rate of spin would require the application of an external force that did not act on the center of mass, but rather acted at some point off the axis of rotation.  Moreover, the force would not change the spin rate if it acted directly toward or away from the axis - only if it had some component along or against the rotational velocity of the point where it was applied. 

For a rigid body, angular momentum is analogous to the linear momentum, but is a more complicated topic than conservation of linear momentum because internal motions of the system can change the rate of rotation of a system whereas internal motions cannot change its total linear momentum.  For example, if the satellite spread our solar panels, its rate of rotation would change, must as a twirling skater who pulls in his arms will rotate faster.  Thus angular velocity can change even when angular momentum about some point is conserved due to the fact that there is no applied torque about that axis.


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h2. Conditions for One-Dimensional Angular Momentum

With the exception of certain specific systems like the [gyroscope], angular momentum problems in introductory physics will be confined to one dimension.  To ensure that a problem can be treated using 1-D angular momentum, the following conditions must hold.

h4. Movement in a Plane

Rotational motion in introductory physics will be concerned only with [systems|system] where the [position] of the [center of mass] and the linear [momentum] of each constituent is confined to a plane (by our convention, the _xy_ plane).

h4. Angular velocity and angular momentum Perpendicular to the Plane

Each rigid body in this system that is rotating must only rotate such that its angular velocity and angular acceleration are directed perpendicular to the _xy_ plane containing the centers of mass, that is in the \+z or \-z direction.

h4. Rotational Symmetry

{anchor:rotational symmetries}
Finally, each [rigid body] in the system must have reflection symmetry about the _xy_ plane if it does not lie within the plane.

{note}Each class of objects in the table of common [moments of inertia|moment of inertia#table] _and their_ _[parallel axis theorem]_ _tranformations_ possess the second symmetry (some also possess the first).
{note}

h4. Valid Systems

Systems that can be treated as having one-dimensional angular momentum include:
* Ball rolling from rest down an inclined plane.
* A yo-yo.
* Non-spinning baseball moving horizontally strikes bat pivoted horizontally at the handle with ball center of mass at the same height as bat center of mass.

h4. Invalid Systems

Systems that require more than 1-D angular momentum:
* Styrofoam cup (wider on the top than the bottom) rolling down an inclined plane (the cup will begin to turn as it rolls).
* Baseball moving horizontally with spin along a horizontal axis strikes bat pivoted horizontally at the handle with the ball center of mass at same height as bat center of mass (ball and bat angular momenta perpendicular).
* Baseball moving horizontally with no spin strikes bat pivoted horizontally at the handle with ball center of mass above or below the bat center of mass (ball center of mass and bat center of mass move in different planes).

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h2. Conditions on the Chosen Axis of Rotation

For a consistent definition of angular momentum, the [axis of rotation] must satisfy one of two possibilities:
# The axis is fixed (does not move) in an inertial reference frame.
# The axis intersects the _xy_ plane at the point of the *system's* center of mass.  (Note that the system's center of mass may be moving and even accelerating.)

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h2. Definition of Angular Momentum


h4. Angular Momentum of a Point Particle

Beginning with [Newton's 2nd Law|Newton's Second Law] for a point particle:
{latex}\begin{large} \[ \sum \vec{F} = \frac{d(m\vec{v})}{dt} \] \end{large}{latex}
we take the [cross product] of the particle's position measured in the _xy_ plane from the point of intersection with the chosen [axis of rotation]:
{latex}\begin{large}\[ \sum \vec{r}\times\vec{F} = \vec{r}\times \frac{d(m\vec{v})}{dt}\]\end{large}{latex}
Assuming _m_ is constant, we can pass it out of the derivative.  We cannot assume that it is reasonable to pass _r_ _into_ the derivative, however, since it certainly has a time dependence (unless the velocity is zero).  We can, however, *prove* that it is reasonable to pass _r_ into the derivative.  To see this, we evaluate:
{latex}\begin{large} \[ \frac{d(\vec{r}\times\vec{v})}{dt} = \frac{d\vec{r}}{dt}\times \vec{v} + \vec{r}\times\frac{d\vec{v}}{dt} = \vec{v}\times\vec{v} + \vec{r}\times\frac{d\vec{v}}{dt}\]\end{large}{latex}
Noting that the cross product of any vector with itself is zero, we then see that if the particle's mass is constant:
{latex}\begin{large} \[ \sum \vec{r}\times\vec{F} = \frac{d(m\vec{r}\times\vec{v})}{dt} \] \end{large}{latex}
The left hand side is simply the net [torque (single-axis)] on the particle about the chosen axis.  Thus, in the absence of net torque, the quantity:
{latex}\begin{large}\[ L_{z} = m\vec{r}\times\vec{v}\]\end{large}{latex}
is constant in time.  We therefore choose to define this quantity as the angular momentum of the point particle about the chosen axis.  Note that _L_ is a vector quantity, but it is one-dimensional (it must point perpendicular to the _xy_ plane because of the cross product and our assumptions).  We have therefore indicated the vector nature by a component subscript instead of a vector arrow.

h4. Angular Momentum of a Rigid Body in Pure Rotation

By dividing a rigid body into _N_ tiny mass elements Δ{_}m{_}{~}i~ (i ranging from 1 to _N_), we can express its angular momentum about a specific axis as:
{latex}\begin{large}\[ L_{z} = \sum_{i=1}^{N} \vec{r}_{i}\times\vec{v}_{i} (\Delta m_{i}) \]\end{large}{latex}
The assumption of pure rotation, however, means that the velocity is completely tangential (perpendicular to _r_) and equal in magnitude to the magnitude of _r{_}ω where ω is the angular velocity associated with the rotation.  We can therefore write:
{latex}\begin{large}\[ L_{z} = \sum_{i=1}^{N} r_{i}^{2} \omega \Delta m_{i} = \omega \sum_{i=1}^{N} r_{i}^{2} \Delta m_{i}\]\end{large}{latex}
where we have removed ω from the sum since a body in pure rotation has a uniform angular velocity.

In the limit of a continuous body, the sum goes over to an integral:
{latex}\begin{large}\[ \sum r_{i}^{2} \Delta m_{i} \rightarrow \int r^{2} dm \]\end{large}{latex}
which is simply the definition of the body's [moment of inertia] about the chosen axis.

Thus, for a rigid body:
{latex}\begin{large}\[ L_{z} = I_{a} \omega\]\end{large}{latex}
where _I{_}{~}a~ is the body's moment of inertia about the chosen axis.

h4. Angular Momentum of a Rigid Body both Translating and Rotating

A rigid body's center of mass can be translating with respect to the axis chosen.  If we again imagine dividing the body into _N_ mass elements Δ{_}m{_}{~}i~, we note that these mass elements _must_ execute pure rotation about the rigid body's center of mass.
{note}If the mass elements of the rigid body could change their radial separation from the center of mass, then the relative distance between points in the body would change.  Such a change would violate the definition of [rigid body].
{note}
Because of this property of rigid bodies, it makes sense to restate the sum:
{latex}\begin{large}\[ L_{z} = \sum_{i=1}^{N} \vec{r}_{i}\times\vec{v}_{i} (\Delta m_{i}) \]\end{large}{latex}
by defining body coordinates _r_' such that:
{latex}\begin{large}\[ \vec{r} = \vec{r}\:' + \vec{r}_{cm}\]\end{large}{latex}
where _r{_}{~}cm~ is the position of the body's center of mass measured from the axis of rotation.  Taking a time derivative causes us to similarly define:
{latex}\begin{large}\[ \vec{v} = \vec{v}\:' + \vec{v}_{cm}\]\end{large}{latex}
We can now use the constraint of the rigid body (pure rotation about the center of mass) to define an ω{~}cm~ (the angular velocity of the body about the center of mass) which allows us to write:
{latex}\begin{large}\[ \vec{r}\times\vec{v} = \omega_{cm}{r'}^{2} + \vec{r}_{cm}\times\vec{v}\:' + \vec{r}\:'\times\vec{v}_{cm}+\vec{r}_{cm}\times\vec{v}_{cm}\]\end{large}{latex}
Inserting this expression into the sum for the body's angular momentum gives:
{latex}\begin{large}\[ L_{z} = \sum_{i=1}^{N} (\omega_{cm}{r'}_{i}^{2} + \vec{r}_{cm}\times\vec{v}_{i}\:' + \vec{r}_{i}\:'\times\vec{v}_{cm} + \vec{r}_{cm}\times\vec{v}_{cm}) (\Delta m_{i}) \]\end{large}{latex}
In the primed coordinates, however, the center of mass velocity and the center of mass position are _by definition_ equal to zero.  Thus:
{latex}\begin{large} \[ \sum_{i=1}^{N} \vec{v}_{i}\:' \:\Delta m_{i} =\sum_{i=1}^{N} \vec{r}_{i}\:'\:\Delta m_{i} = 0 \]\end{large}{latex}
Therefore, we have:
{latex}\begin{large} \[ L_{z} = \left(\omega_{cm}\sum_{i=1}^{N} {r'}_{i}^{2} \Delta m_{i}\right) + \left(\vec{r}_{cm}\times\vec{v}_{cm}\right) \left(\sum_{i=1}^{N} \Delta m_{i}\right) = I_{cm}\omega_{cm} + M_{tot} \vec{r}_{cm}\times\vec{v}_{cm}\]\end{large}{latex}
where _M{_}{~}tot~ is the total mass of the rigid body.

Comparing this expression to those found in the previous sections, we see that the angular momentum of a rigid body that is both translating and rotating with respect to the chosen axis of rotation can be thought of as the sum of the angular momentum of the center of mass treated as a point particle with the mass of the entire rigid body and the angular momentum contributed by the body's rotation about the center of mass.

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h2. Relevant Examples

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h2. Advanced Section:  Valid Choices for the Axis of Rotation

Under construction.