Given a straight segment of track of length
, add a small segment
so that
the track bows into a circular Arc. Find the maximum displacement
of the bowed
track. The Pythagorean Theorem gives
![\begin{displaymath}
R^2=x^2+({\textstyle{1\over 2}}l)^2,
\end{displaymath}](r_298.gif) |
(1) |
But
is simply
, so
![\begin{displaymath}
R^2=(x+d)^2=x^2+2xd+d^2.
\end{displaymath}](r_300.gif) |
(2) |
Solving (1) and (2) for
gives
![\begin{displaymath}
x={{\textstyle{1\over 4}}l^2-d^2\over 2d}.
\end{displaymath}](r_301.gif) |
(3) |
Expressing the length of the Arc in terms of the central angle,
But
is given by
![\begin{displaymath}
\tan\theta ={{\textstyle{1\over 2}}l\over x} = {{\textstyle{...
...{1\over 4}}l^2-d^2} = {dl\over {\textstyle{1\over 4}}l^2-d^2},
\end{displaymath}](r_306.gif) |
(5) |
so plugging
in gives
![\begin{displaymath}
{\textstyle{1\over 2}}(l+\Delta l)=\left({d^2+{\textstyle{1\...
...)\tan^{-1}\left({dl\over {\textstyle{1\over 4}}l^2-d^2}\right)
\end{displaymath}](r_307.gif) |
(6) |
![\begin{displaymath}
d(l+\Delta l)=(d^2+{\textstyle{1\over 4}}l^2)\tan^{-1}\left({dl\over {\textstyle{1\over 4}}l^2-d^2}\right).
\end{displaymath}](r_308.gif) |
(7) |
For
,
![\begin{displaymath}
{dl\over {\textstyle{1\over 4}}l^2\left({1-{d^2\over 4l^2}}\...
...right)^{-1} \approx {4d\over l}\left({1+{4d\over l^2}}\right).
\end{displaymath}](r_310.gif) |
(8) |
Therefore,
Keeping only terms to order
,
![\begin{displaymath}
dl+\Delta l\approx {4d^3\over l}+dl+{4d^3\over l}-{16\over 3} {d^3\over l}
\end{displaymath}](r_316.gif) |
(10) |
![\begin{displaymath}
\Delta l \approx \left({8-{\textstyle{16\over 3}}}\right){d^3\over l} = {24-16\over 3} {d^3\over l} = {8\over3} {d^3\over l},
\end{displaymath}](r_317.gif) |
(11) |
so
![\begin{displaymath}
d^2={\textstyle{3\over 8}} l\Delta l
\end{displaymath}](r_318.gif) |
(12) |
and
![\begin{displaymath}
d\approx {\textstyle{1\over 2}}\sqrt{{\textstyle{3\over 2}} l\Delta l} = {\textstyle{1\over 4}}\sqrt{6l\Delta l}.
\end{displaymath}](r_319.gif) |
(13) |
If we take
and
1 foot, then
feet.
© 1996-9 Eric W. Weisstein
1999-05-25