Page History

{table:align=right|cellspacing=0|cellpadding=1|border=1|frame=box|width=40%}
{tr}
{td:align=center|bgcolor=#F2F2F2}*[Model Hierarchy]*
{td}
{tr}
{tr}
{td}
{pagetree:root=Model Hierarchy|reverse=true}
{search-box}
{td}
{tr}
{table}


h2. Description and Assumptions

{excerpt}This model applies to a [rigid body] which is executing [pure rotation] confined to the _xy_ plane about the origin.{excerpt}

h2. Problem Cues

Problems in rotational motion often feature an object which is constrained to rotate about some axle or pivot point.  Additionally, the motion of any rigid body which can be treated using the [1-D Angular Momentum and Torque] model can be described as translation of the center of mass plus pure rotation about the center of mass.

----
|| Page Contents ||
| {toc:style=none|indent=10px} |

----
h2. Prerequisite Knowledge

h4. Prior Models

* [Uniform Circular Motion]
* [One-Dimensional Motion with Constant Acceleration|1-D Motion (Constant Acceleration)]

h4. Vocabulary and Procedures

* [centripetal acceleration]
* [tangential acceleration]
* [angular position]
* [angular frequency]
* [angular acceleration]

----
h2. System

h4. Constituents

A single [rigid body].

h4. State Variables

Time (_t_), position (_r_), angular position (θ), tangential velocity (_v_), angular velocity (ω).

----
h2. Interactions

h4. Relevant Types

The system will be subject to a position-dependent centripetal acceleration, and may also be subject to an angular (or equivalently, tangential) acceleration.

h4. Interaction Variables

Angular acceleration (α), tangential acceleration (_a_~tan~) and radial (or centripetal) acceleration (_a_~c~).

----
h2. Model

h4. Relevant Definitions

h5. Relationships between angular and tangential quantities:
\\
{latex}\begin{large}\[ \vec{v}_{\rm tan} = \vec{\omega} \times \vec{r} = \omega r \;\hat{\theta}\]
\[ \vec{a}_{\rm tan} = \vec{\alpha}\times \vec{r} = \alpha r \;\hat{\theta}\]\end{large}{latex}
\\
h5. Centripetal acceleration:
\\
{latex}\begin{large}\[ \vec{a}_{c} = -\frac{v_{\rm tan}^{2}}{r}\hat{r} = -\omega^{2}r\;\hat{r}\]\end{large}{latex}
\\
h5. Magnitude of total acceleration:
\\
{latex}\begin{large}\[ a = \sqrt{a_{tan}^{2}+a_{c}^{2}} = r\sqrt{\alpha^{2}+\omega^{4}} \]\end{large}{latex}
\\
{note}By definition, _every point_ in an object undergoing [pure rotation] will have the same value for all _angular_ quantities (θ, ω, α).  The linear quantities (_r_, _v_, _a_), however, will vary with position in the object.{note}

h4. Laws of Change

\\
h5. PositionDifferential form:
\\
{latex}\begin{large}\[ x(t) = x_{i}\cos(\omega t) + \frac{v_{i}}{\omega}\sin(\omega t)\]\frac{d\omega}{dt} = \alpha \]
\[\frac{d\theta}{dt} = \omega\]
\end{large}{latex}
\\
or, equivalentlyh5. Integral form:
\\
{latex}\begin{large}\[ x(t)\omega_{f} = A\cos(\omega_{i} t + \phi) \]\end{large}{latex}
\\
h5. Velocity:
\\
{latex}\begin{large}\[ v(t) = -\omega x_{i}\sin(\omega t) + v_{i}\cos(\omega t)\int_{t_{i}}^{t_{f}} \alpha \;dt\]
\[ \theta_{f} = \theta_{i} +\int_{t_{i}}^{t_{f}} \omega\;dt\]\end{large}{latex}
\\
or, equivalently:
\\
{note}Note the analogy between these Laws of Change and those of the [One-Dimensional Motion (General)] model.  Thus, for the case of *constant angular acceleration*, the integral form of these Laws are equivalent to:
{latex}\begin{large}\[ v(t)\omega_{f} = -A\omega\sin(\omega t _{i} + \phi)\]\end{large}{latex}
\\
h5. Acceleration:
\\
{latex}\begin{large}\[ a(t) = -\omega^{2} x_{i}\cos(\omega t) - \omega v_{i} \sin(\omega t) = -\omega^{2} x \]\end{large}{latex}
\\
or, equivalently:
\\
{latex}\begin{large}\[ a(t) = -\omega^{2}A\cos(\omega t+\phi) = -\omega^{2} xalpha(t_{f}-t_{i})\]
\[ \theta_{f} = \theta_{i} + \frac{1}{2}(\omega_{i}+\omega_{f})(t_{f}-t_{i})\]
\[ \theta_{f} = \theta_{i} + \omega_{i}(t_{f}-t_{i}) +\frac{1}{2}\alpha(t_{f}-t_{i})^{2}\]
\[ \omega_{f}^{2} =\omega_{i}^{2} + 2\alpha(\theta_{f}-\theta_{i})\]\end{large}{latex} {note}

----
h2. Diagrammatical Representations

* Angular Accelerationposition versus time graph.
* Velocity versus time graph.
* PositionAngular velocity versus time graph.

----
h2. Relevant Examples

None yet.
----
{search-box}
\\
\\
| !copyright and waiver^copyrightnotice.png! | RELATE wiki by David E. Pritchard is licensed under a [Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License|http://creativecommons.org/licenses/by-nc-sa/3.0/us/]. |