# Another question

1. Sep 17, 2005

### modeman

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

2. Sep 18, 2005

### Tide

Unless you have a specific unstated issue in mind then the answer is that it's the same physics as for gentle turning, curves and changing direction, i.e. Newton's Laws of Motion.

3. Sep 18, 2005

### bomba923

Curvature:
$$\kappa = \left| {\frac{{d\vec T}}{{ds}}} \right| = \frac{{\left| {\vec T\,'\left( t \right)} \right|}}{{\left| {\vec r\,'\left( t \right)} \right|}} = \frac{{\left| {\vec r\,'\left( t \right) \times \vec r\, {''}\left( t \right)} \right|}}{{\left| {\vec r\,'\left( t \right)} \right|^3 }}$$

Sharper curves have larger values of $\kappa$.

Curvature has its physical applications; for example, let $$\vec r ( t )$$ represent the position of an object at time $t$. You know that
$$\left\{ \begin{gathered} \vec T = \frac{{\vec r\,'}} {{\left| {\vec r\,'} \right|}} = \frac{{\vec v}} {{\left| {\vec v} \right|}} \Rightarrow \vec v = \left| {\vec v} \right|\vec T \hfill \\ \kappa = \frac{{\left| {\vec T\,'} \right|}} {{\left| {\vec r\,'} \right|}} = \frac{{\left| {\vec T\,'} \right|}} {{\left| {\vec v} \right|}} \Rightarrow \left| {\vec T\,'} \right| = \kappa \left| {\vec v} \right| \hfill \\ \vec N = \frac{{T\,'}} {{\left| {T\,'} \right|}} \Rightarrow T\,' = \vec N\left| {T\,'} \right| = \kappa \left| {\vec v} \right|\vec N \hfill \\ \end{gathered} \right\}$$

And so, making the necessary substitution,
$$\begin{gathered} \vec a = \vec v \, ' = \left| {\vec v} \right| ' \vec T + \left| {\vec v} \right|\vec T' \Rightarrow \hfill \\ \vec a = \left| {\vec v} \right| '\vec T + \kappa \left| {\vec v} \right|^2 \vec N \hfill \\ \end{gathered}$$

Hope this helps

4. Sep 18, 2005

### Dr.Brain

This is studied under 'TNB Physics' which constitutes the T=Unit Tangent vector N= Principle vector and B Vector . The curvature given by 'k' is defined by rate of change of unit normal vector per unit length.

BJ

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