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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:

{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}\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

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

Mathematical

Representation

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

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

Diagrammatic Representations

• Force diagram.
• Initial-state final-state diagram.

Relevant Examples

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id cons

Examples Involving Constant Angular Momentum

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id rws

Examples Involving Rolling without Slipping

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id

par

Examples

Involving

the

Parallel

Axis

Theorem

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id

all

All

Examples

Using

this

Model

Pictures courtesy of:
Wikimedia Commons user Dobromila
Wikimedia Commons user Vmenkov