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
|
(1) |
But is simply , so
|
(2) |
Solving (1) and (2) for gives
|
(3) |
Expressing the length of the Arc in terms of the central angle,
But is given by
|
(5) |
so plugging in gives
|
(6) |
|
(7) |
For ,
|
(8) |
Therefore,
Keeping only terms to order ,
|
(10) |
|
(11) |
so
|
(12) |
and
|
(13) |
If we take
and 1 foot, then
feet.
© 1996-9 Eric W. Weisstein
1999-05-25