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}{composition-setup} {table:rules=cols|cellpadding=8|cellspacing=0|border=1|frame=void} {tr:valign=top} {td} {excerpt:hidden=true}*[System|system]:* Any number of [rigid bodies|rigid body] or [point particles|point particle] whose angular momentum is constrained to lie along a certain axis. --- *[Interactions|interaction]:* Any that respect the [one-dimensional angular momentum|angular momentum about a single axis].{excerpt} h4. Introduction to the Model h5. Description and Assumptions 1-D Angular Momentum and Torque is a subclass of the general [Angular Momentum and External Torque] model in which a system of rigid bodies is constrained to move only in a plane (usually taken to be the _xy_ plane) with each body's angular momentum therefore directed along an axis perpendicular to the plane (along the z-axis). Under these conditions, the angular momentum is a one-dimensional vector, and the directional subscript (z) is generally omitted. h5. Learning Objectives Students are assumed to understand this model who can: * Describe the conditions that must be satisfied for the valid selection of an [axis of rotation] in a physics problem. * Cacluate the [moment of inertia] of a [system] composed purely of basic objects like rods and spheres. * Calculate the [angular momentum|angular momentum about a single axis] of a [rigid body] rotating about a fixed axle. * Calculate the [angular momentum|angular momentum about a single axis] of a rotating and translating [rigid body] about any [axis|axis of rotation] parallel to the body's [angular velocity] about its [center of mass]. * Determine the net [external|external force] [torque|torque (single-axis)] on a [system]. * Describe the conditions for [angular momentum|angular momentum about a single axis] to be conserved. * Describe how internal changes to the configuration of a [system] will affect its [angular velocity]. * Analyze collisions involving rotational and translational motion of the participants. h5. Relevant Definitions Angular momentum about axis _a_: {latex}

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Excerpt

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System: Any number of rigid bodies or point particles whose angular momentum is constrained to lie along a certain axis. — Interactions: Any that respect the one-dimensional angular momentum.

Introduction to the Model

Description and Assumptions

1-D Angular Momentum and Torque is a subclass of the general Angular Momentum and External Torque model in which a system of rigid bodies is constrained to move only in a plane (usually taken to be the xy plane) with each body's angular momentum therefore directed along an axis perpendicular to the plane (along the z-axis). Under these conditions, the angular momentum is a one-dimensional vector, and the directional subscript (z) is generally omitted.

Learning Objectives

Students are assumed to understand this model who can:

Describe the conditions that must be satisfied for the valid selection of an axis of rotation in a physics problem.
Cacluate the moment of inertia of a system composed purely of basic objects like rods and spheres.
Calculate the angular momentum of a rigid body rotating about a fixed axle.
Calculate the angular momentum of a rotating and translating rigid body about any axis parallel to the body's angular velocity about its center of mass.
Determine the net external torque on a system.
Describe the conditions for angular momentum to be conserved.
Describe how internal changes to the configuration of a system will affect its angular velocity.
Analyze collisions involving rotational and translational motion of the participants.

Relevant Definitions

Angular momentum about axis a:

Latex

\begin{large}\[ L_{a} = I_{cm}\omega + m\vec{r}_{{\rm cm},a}\times \vec{v}_{{\rm cm}} \]\end{large}

{latex} h4.

S.I.M.

Structure

of

the

Model

h5.

Compatible

Systems

The

[

system

]

can

be

composed

of

any

number

of

[

rigid

bodies

|rigid body]

and

[

point

particles

|point particle]

.

The

system

must

either

be

constrained

to

move

in

such

a

way

that

the

[

angular

momentum

|angular momentum about a single axis]

will

be

one-dimensional,

or

else

the

symmetries

of

the

situation

(

[

system

]

plus

[

interactions

|interaction]

)

must

guarantee

that

the

[angular momentum|

angular

momentum

about a single axis]

will

remain

one

dimensional.

h5.

Relevant

Interactions

External

interactions

must

be

explicitly

given

as

torques,

or

as

forces

with

their

point

of

application

or

[

moment

arm

]

about

a

chosen

[

axis

of

rotation

]

specified

along

with

their

magnitude

and

direction.

(Internal

interactions

do

not

change

the

angular

momentum

of

the

system.)

h4.

Laws

of

Change

h5.

Mathematical

Representation

{section}{column} \\ h5. Differential Form \\ {latex}

Section

Column

Differential Form

Latex
\begin{large}\[ \sum_{\rm system}\frac{dL_{a}}{dt} = \sum_{\rm external} \tau_{a} \]\end{large}

{latex} {column}{column} \\ h5. Integral Form \\ {latex}

Column

Integral Form

Latex
\begin{large}\[ \sum_{\rm system}L_{a,f} = \sum_{\rm system}L_{a,i} + \int \:\sum_{\rm external} \tau_{a} \:dt \]\end{large}

{latex}

where

the

last

term

is

called

the

"angular

impulse"

{column}{section} h5. Diagrammatic Representations * [Force diagram|force diagram]. * [Initial-state final-state diagram|initial-state final-state diagram]. h4. Relevant Examples h6. {toggle-cloak:id=cons} Examples Involving Constant Angular Momentum {cloak:id=cons} {contentbylabel:constant_angular_momentum|showSpace=false|showLabels=true|excerpt=true|maxResults=50} \\ {cloak} h6. {toggle-cloak:id=rws} Examples Involving Rolling without Slipping {cloak:id=rws} {contentbylabel:

Diagrammatic Representations

Relevant Examples

Toggle Cloak
id cons
Examples Involving Constant Angular Momentum

Cloak

id	cons

falsetruetrue50

constant_angular_momentum

Toggle Cloak
id rws
Examples Involving Rolling without Slipping

Cloak

id	rws

falsetruetrueAND50

angular_momentum,rolling_without_slipping

|showSpace=

false

|showLabels=

true

|excerpt=

true

|operator=

AND

|maxResults=

50

} {contentbylabel:

constant_angular_momentum,rolling_without_slipping

|showSpace=false|showLabels=true|excerpt=true|operator=AND|maxResults=50} \\ {cloak} h6. {

Toggle Cloak

:

id

=

par

}

Examples

Involving

the

Parallel

Axis

Theorem

{

Cloak

:

id

=par} {contentbylabel:

	par

falsetruetrueAND50

angular_momentum,parallel_axis

|showSpace=

false

|showLabels=

true

|excerpt=

true

|operator=

AND

|maxResults=

50

} {contentbylabel:

constant_angular_momentum,parallel_axis

|showSpace=false|showLabels=true|excerpt=true|operator=AND|maxResults=50} \\ {cloak} h6. {

Toggle Cloak

:

id

=

all

}

All

Examples

Using

this

Model

{

Cloak

:

id

=all} {contentbylabel:

	all

falsetruetrueOR50

angular_momentum,constant_angular_momentum

|showSpace=false|showLabels=true|excerpt=true|operator=OR|maxResults=50} {cloak} \\ \\ {search-box} \\ \\ {td} {td:width=235} !skater.jpg! \\ \\ [!well.jpg|alt="How fast does a bucket fall down a well? Click the image to investigate."!|Down the Well] \\ \\ Pictures courtesy of: [Wikimedia Commons|http://commons.wikimedia.org] user [Dobromila|http://commons.wikimedia.org/wiki/User:Dobromila] \\ [Wikimedia Commons|http://commons.wikimedia.org] user [Vmenkov|http://commons.wikimedia.org/wiki/User:Vmenkov] {td} {tr} {table} \\

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Key

Introduction to the Model

Description and Assumptions

Learning Objectives

Relevant Definitions

S.I.M.

Structure

of

the

Model

Compatible

Systems

The

Relevant

Interactions

External

Laws

of

Change

Mathematical

Representation

Section ColumnDifferential Form Latex\begin{large}\[ \sum_{\rm system}\frac{dL_{a}}{dt} = \sum_{\rm external} \tau_{a} \]\end{large}

Differential Form

Integral Form

Diagrammatic Representations

Relevant Examples

Toggle Cloakidcons Examples Involving Constant Angular Momentum

Toggle Cloakidrws Examples Involving Rolling without Slipping

Toggle Cloak

id

par

Examples

Involving

the

Parallel

Axis

Theorem

Toggle Cloak

id

all

All

Examples

Using

this

Model

Section
Column

Differential Form

Latex
\begin{large}\[ \sum_{\rm system}\frac{dL_{a}}{dt} = \sum_{\rm external} \tau_{a} \]\end{large}

Toggle Cloak
id cons
Examples Involving Constant Angular Momentum

Toggle Cloak
id rws
Examples Involving Rolling without Slipping