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Mathematics-Online course: Vector Calculus - Coordinates

Rotating Frame of Reference

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The left side of the picture given below shows the paths of a rectilineal and a circular motion:

\begin{displaymath} \begin{array}{ll} G:&(x,y) = (1,\,2+t/(2\pi)),\,\\ K:&(x,y) = (1+\cos(t),\,2-\sin(t))\, . \end{array} \end{displaymath}

On the right the two paths are pictured in a frame of reference rotating at an angular velocity $ \omega=1$ . Hence, they are pictured as they would appear to an observer having a ride on a roundabout.

\includegraphics[ width=0.9\linewidth ]{Fig0001}

For the rectilineal motion the transformation of the coordinates yields

$\displaystyle x' = c + (2+t/(2\pi)) s,\quad y' = -s + (2+t/(2\pi)) c $

with

$\displaystyle c=\cos(\omega t),\quad s=\sin(\omega t)\, . $

Since the factor within the brackets increases if the parameter $ t$ does, we obtain a spiral. The $ 2$ pictured windings correspond to the time interval $ 0\le t\le 4\pi$ .

The path's shape of the circular motion in the frame rotating at $ \omega=1$ cannot directly be seen from the transformed coordinates

$\displaystyle x' = c(1+c) + s(2-s),\quad y' = -s(1+c) + c(2-s)\, . $

As our example shows, the direction of the observed motion can change abruptly. Even bends can appear on the observed path. In such a singular point the observed velocity equals zeros with respect to the rotating frame.
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  automatically generated 10/30/2007