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Introduction to the ModelDescription and AssumptionsThis model applies to a rigid body which is executing pure rotation confined to the xy plane about the origin. Learning ObjectivesStudents will be assumed to understand this model who can:
Relevant DefinitionsRelationship between Angular and Tangential Quantities Unknown macro: {latex} \begin Unknown macro: {large} [ \vec Unknown macro: {v}
_ Unknown macro: {rm tan}
= \vec Unknown macro: {omega}
\times \vec Unknown macro: {r}
= \omega r \;\hat Unknown macro: {theta}
] Unknown macro: {a}
_ = \vec Unknown macro: {alpha}
\times \vec Unknown macro: {r}
= \alpha r \;\hat Unknown macro: {theta}
]\end Centripetal Acceleration Unknown macro: {latex} \begin Unknown macro: {large} [ \vec Unknown macro: {a}
_ Unknown macro: {c}
= -\frac{v_ Unknown macro: {rm tan}
^{2}} Unknown macro: {r}
\hat = -\omega^ Unknown macro: {2}
r\;\hat Unknown macro: {r}
]\end Magnitude of Total Acceleration Unknown macro: {latex} \begin Unknown macro: {large} [ a = \sqrt{a_ Unknown macro: {tan}
^ Unknown macro: {2}
+a_ Unknown macro: {c}
{2}} = r\sqrt{\alpha +\omega^{4}} ]\end 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. S.I.M. Structure of the ModelCompatible SystemsThis model applied to a single rigid body or to a single point particle constrained to move in a circular path. Relevant InteractionsThe system will be subject to a position-dependent centripetal acceleration, and may also be subject to an angular (or equivalently, tangential) acceleration. Laws of ChangeMathematical RepresentationDifferential Form Unknown macro: {latex} \begin Unknown macro: {large} [ \frac Unknown macro: {domega}
Unknown macro: {dt}
= \alpha ] Unknown macro: {dtheta} = \omega] Integral Form Unknown macro: {latex} \begin Unknown macro: {large} [ \omega_ Unknown macro: {f}
= \omega_ Unknown macro: {i}
+\int_{t_{i}}^{t_{f}} \alpha \;dt] = \theta_ Unknown macro: {i}
+\int_{t_{i}}^{t_{f}} \omega\;dt]\end 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:
Unknown macro: {latex} \begin Unknown macro: {large} [ \omega_ Unknown macro: {f}
= \omega_ Unknown macro: {i}
+ \alpha(t_ -t_ Unknown macro: {i}
)] Unknown macro: {f}
= \theta_ + \frac Unknown macro: {1}
Unknown macro: {2}
(\omega_ Unknown macro: {i}
+\omega_ Unknown macro: {f}
)(t_ -t_ )] Unknown macro: {f}
= \theta_ Unknown macro: {i}
+ \omega_ (t_ -t_ Unknown macro: {i}
) +\frac Unknown macro: {2}
\alpha(t_ Unknown macro: {f}
-t_ Unknown macro: {i}
)^ ] Unknown macro: {f}
^ Unknown macro: {2}
=\omega_ Unknown macro: {i}
^ + 2\alpha(\theta_ -\theta_ Unknown macro: {i}
)]\end Diagrammatic Representations
Relevant ExamplesAll Examples Using the Model
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