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What is the physics of sharp turning; sharp curves and changing direction about those curves?

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What is the physics of sharp turning; sharp curves and changing direction about those curves?

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Tide

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modeman said:What is the physics of sharp turning; sharp curves and changing direction about those curves?

[tex] \kappa = \left| {\frac{{d\vec T}}{{ds}}} \right| = \frac{{\left| {\vec T\,'\left( t \right)} \right|}}{{\left| {\vec r\,'\left( t \right)} \right|}} = \frac{{\left| {\vec r\,'\left( t \right) \times \vec r\, {''}\left( t \right)} \right|}}{{\left| {\vec r\,'\left( t \right)} \right|^3 }} [/tex]

Sharper curves have larger values of [itex] \kappa [/itex].

Curvature has its physical applications; for example, let [tex] \vec r ( t ) [/tex] represent the position of an object at time [itex] t [/itex]. You know that

[tex] \left\{ \begin{gathered}

\vec T = \frac{{\vec r\,'}}

{{\left| {\vec r\,'} \right|}} = \frac{{\vec v}}

{{\left| {\vec v} \right|}} \Rightarrow \vec v = \left| {\vec v} \right|\vec T \hfill \\

\kappa = \frac{{\left| {\vec T\,'} \right|}}

{{\left| {\vec r\,'} \right|}} = \frac{{\left| {\vec T\,'} \right|}}

{{\left| {\vec v} \right|}} \Rightarrow \left| {\vec T\,'} \right| = \kappa \left| {\vec v} \right| \hfill \\

\vec N = \frac{{T\,'}}

{{\left| {T\,'} \right|}} \Rightarrow T\,' = \vec N\left| {T\,'} \right| = \kappa \left| {\vec v} \right|\vec N \hfill \\

\end{gathered} \right\} [/tex]

And so, making the necessary substitution,

[tex] \begin{gathered}

\vec a = \vec v \, ' = \left| {\vec v} \right| ' \vec T + \left| {\vec v} \right|\vec T' \Rightarrow \hfill \\

\vec a = \left| {\vec v} \right| '\vec T + \kappa \left| {\vec v} \right|^2 \vec N \hfill \\ \end{gathered} [/tex]

Hope this helps

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