Copyright ©2008 by Paul Niquette. All rights reserved.

et
us start with the bicycle in the puzzle,
assigning
some value as the height of the center
of gravity of bike
and rider in the vertical position. For
simplicity, it is preferable
to take 'moment arms' about the point of contact of the
surface with the
wheels, which removes both H
and LFRICTION
from consideration. Let
represent the angle by which the bicycle is leaning at
any given moment.
The weight F
develops a moment
arm W sin around the contact point of the wheels
with the surface.
This is the 'torque' that tends to tip the bicycle over
into a calamitous
fall.
HFrom elementary physics, the
centrifugal force FCENTRIFUGAL
= ( ^{2 }/
),
where g = 32.2 ft/sec/sec, R= speed
in ft/sec, and V
= the local radius (in feet) of the curve being
followed by the bicycle.
Observe thatRCENTRIFUGAL
develops its moment arm from F
cos ,
and we write...
H...noting, after some algebra, the independence of with
respect to both R
and W.
HTurning now (no pun intended) to the
commercial aircraft
overhead at angle-of-bank ,
observe that, unlike in the bicycle calculations,
there is no
cos = L,
and the horizontal component must satisfy W
sin = LCENTRIFUGAL.
Perhaps amazingly, the expression for the radius of
curvature for the commercial
airliner...
F...is not only independent of
but is also mathematically Wthe same as that for the
bicycle.
Exclamatory punctuation optional.
he puzzle calls for both the bike and the plane each to travel in a complete circle. Two equations are most relevant... Time intervals to complete the circular pathways are given by... ...and following substitutions, we learn that... ...which means that...
...suggesting a slightly different case that may be of interest to some solvers. If the bicycle were traveling at 12 mph and leans, say, 6 degrees, while a plane overhead traveling 10 times faster banks 45 degrees (a common practice for airliners), the two will arrive back at the starting point at about the same time.
on't try this at home. Depicted here is a rather extreme case of a left-hand turn being executed on a two-wheel vehicle. Unlike Einstein's bike being ridden casually in a courtyard -- oh, and unlike a jet-plane maneuvering in the open sky -- the motorcycle is constrained to complete every turn within the confines of the racetrack. The objective, of course, is speed, higher the better. The only variable is the lean-angle , shown here to be about 45 degrees. Assuming the requirement for a turn of
90 degrees ( ...and can be used to confirm that at 75 mph, the turn can be completed in 5 seconds. |

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