Sharp Turning & Changing Direction: Physics Explained

[modeman]

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

 

[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.

 

[bomba923]

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

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}<br /> \vec T = \frac{{\vec r\,&#039;}}<br /> {{\left| {\vec r\,&#039;} \right|}} = \frac{{\vec v}}<br /> {{\left| {\vec v} \right|}} \Rightarrow \vec v = \left| {\vec v} \right|\vec T \hfill \\<br /> \kappa = \frac{{\left| {\vec T\,&#039;} \right|}}<br /> {{\left| {\vec r\,&#039;} \right|}} = \frac{{\left| {\vec T\,&#039;} \right|}}<br /> {{\left| {\vec v} \right|}} \Rightarrow \left| {\vec T\,&#039;} \right| = \kappa \left| {\vec v} \right| \hfill \\<br /> \vec N = \frac{{T\,&#039;}}<br /> {{\left| {T\,&#039;} \right|}} \Rightarrow T\,&#039; = \vec N\left| {T\,&#039;} \right| = \kappa \left| {\vec v} \right|\vec N \hfill \\ <br /> \end{gathered} \right\}

And so, making the necessary substitution,
\begin{gathered}<br /> \vec a = \vec v \, &#039; = \left| {\vec v} \right| &#039; \vec T + \left| {\vec v} \right|\vec T&#039; \Rightarrow \hfill \\<br /> \vec a = \left| {\vec v} \right| &#039;\vec T + \kappa \left| {\vec v} \right|^2 \vec N \hfill \\ \end{gathered}

Hope this helps

 

[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