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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 out solar panels, its rate of rotation would change, just 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.

Utility of Single-Axis Case

The specification of a single axis of rotation allows a complete description of the angular momentum only in certain cases. To ensure that a problem can be completely described using angular momentum about a single axis, the following conditions must hold.

Movement in a Plane: Rotational motion in introductory physics will be concerned only with systems 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).

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.

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Examples of Valid Systems

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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.

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Examples of Invalid Systems

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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).
• Gyroscopes (the one invalid system that is generally covered in introductory mechanics).

Conditions on the Chosen Axis of Rotation

For a consistent definition of angular momentum, the axis of rotation must satisfy one of two possibilities:

1. The axis is fixed (does not move) in an inertial reference frame.
2. 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.)

Definition of Angular Momentum

Angular Momentum of a Point Particle

Beginning with Newton's 2nd Law for a point particle:

}The rotational equivalent of [linear momentum|momentum].  Angular momentum is often approximately conserved in collisions, and is usually conserved when external [torques|torque] sum to zero.  The action of purely internal forces can sometimes create a change in the system [moment of inertia] while conserving the angular momentum, which leads to interesting effects.{excerpt}

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h2. Motivation for Concept

Imagine a satellite spinning in space.  In the absence of external influences, the satellite will continue to spin.  To stop the spinning would require the application of force.  The faster the satellite is spinning, the more work the force will have to perform to halt the satellite.  These concepts are completely analogous to the linear ideas of force and work.  As such, there must be some quantity analogous to the linear momentum which is conserved in the absence of external influences.  Angular momentum is that quantity.  

Conservation of angular momentum is a more complicated topic than conservation of linear momentum, however.  It is very rare for an everyday system to experience a significant change in mass.  Thus, in physics problems involving conservation of linear momentum, the center of mass velocity of the system is usually constant.  It is not difficult, however, for a system to alter its moment of inertia.  In a case where angular momentum is conserved, such an alteration must produce an inverse alteration in the rotational speed.  This is easily seen in sporting events like figure skating and diving.  When an athlete is rotating essentially free of external forces (such as a figure skater in a spin on slippery ice or a diver rotating in mid-air) they can still affect a dramatic change in their rotational rate simply by tucking in their arms and legs or, conversely, by extending their arms and legs.    

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 where the center of mass of each constituent is confined to move in a plane (we will refer to this as the _xy_ plane).  

h4. Rotation Perpendicular to the Plane

Each rigid body in the system that is rotating must only rotate such that their angular velocity and angular acceleration are directed perpendicular to the _xy_ plane containing the centers of mass.  (They must point in the +z or -z direction.)

h4. Rotational Symmetry
{anchor:rotational symmetry}

Finally, each [rigid body] in the system must obey *one* of the following rotational symmetry constraints:

# The object has a rotational symmetry about the axis of rotation.
# The object has a reflection symmetry about the _xy_ 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).


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

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take

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the

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cross

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product

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of

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the

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particle's

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position

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measured

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in

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the

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xy

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plane

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from

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the

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point

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of

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intersection

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with

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the

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chosen

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axis

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of

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rotation

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:

}\begin{large}\[ \sum \vec{r}\times\vec{F} = \vec{r}\times \frac{d(m\vec{v})}{dt}\]\end{large}{latex}

Assuming _m_ is 

Assuming m is constant,

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we

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can

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pass

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it

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out

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of

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the

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derivative.

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We

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cannot

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assume

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that

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it

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is

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reasonable

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to

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pass

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r

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into

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the

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derivative,

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however,

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since

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it

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certainly

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has

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a

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time

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dependence

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(unless

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the

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velocity

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is

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zero).

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We

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can,

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however,

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prove

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that

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it

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is

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reasonable

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to

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pass

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r

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into

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the

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derivative.

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To

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see

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this,

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we

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evaluate:

}\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

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that

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the

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cross

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product

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of

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any

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vector

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with

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itself

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is

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zero,

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we

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then

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see

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that

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if

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the

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particle's

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mass

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is

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constant:

}\begin{large} \[ \sum \vec{r}\times\vec{F} = \frac{d(m\vec{r}\times\vec{v})}{dt} \] \end{large}

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:

\begin{large}\[ L_{z} = m\vec{r}\times\vec{v}\]\end{large}

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.

Angular Momentum of a Rigid Body in Pure Rotation

By dividing a rigid body into N tiny mass elements Δm_i (i ranging from 1 to N), we can express its angular momentum about a specific axis as:

\begin{large}\[ L_{z} = \sum_{i=1}^{N} \vec{r}_{i}\times\vec{v}_{i} (\Delta m_{i}) \]\end{large}

The assumption of pure rotation, however, means that the velocity is completely tangential (perpendicular to r) and equal in magnitude to the magnitude of rω where ω is the angular velocity associated with the rotation. We can therefore write:

\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}

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:

\begin{large}\[ \sum r_{i}^{2} \Delta m_{i} \rightarrow \int r^{2} dm \]\end{large}

which is simply the definition of the body's moment of inertia about the chosen axis.

Thus, for a rigid body:

\begin{large}\[ L_{z} = I_{a} \omega\]\end{large}

where I_a is the body's moment of inertia about the chosen axis.

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 Δm_i, we note that these mass elements must execute pure rotation about the rigid body's center of mass.

Because of this property of rigid bodies, it makes sense to restate the sum:

\begin{large}\[ L_{z} = \sum_{i=1}^{N} \vec{r}_{i}\times\vec{v}_{i} (\Delta m_{i}) \]\end{large}

by defining body coordinates r' such that:

\begin{large}\[ \vec{r} = \vec{r}\:' + \vec{r}_{cm}\]\end{large}

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:

\begin{large}\[ \vec{v} = \vec{v}\:' + \vec{v}_{cm}\]\end{large}

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:

\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}

Inserting this expression into the sum for the body's angular momentum gives:

\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}

In the primed coordinates, however, the center of mass velocity and the center of mass position are by definition equal to zero. Thus:

\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}

Therefore, we have:

\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}

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.