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CONCERNING WHEELS.

(Patchwork Puzzles)
There are some curious facts concerning the movements of wheels that are
apt to perplex the novice. For example: when a railway train is
travelling from London to Crewe certain parts of the train at any given
moment are actually moving from Crewe towards London. Can you indicate
those parts? It seems absurd that parts of the same train can at any
time travel in opposite directions, but such is the case.
In the accompanying illustration we have two wheels. The lower one is
supposed to be fixed and the upper one running round it in the direction
of the arrows. Now, how many times does the upper wheel turn on its own
axis in making a complete revolution of the other wheel? Do not be in a
hurry with your answer, or you are almost certain to be wrong.
Experiment with two pennies on the table and the correct answer will
surprise you, when you succeed in seeing it.


Answer:

If you mark a point A on the circumference of a wheel that runs on the
surface of a level road, like an ordinary cart-wheel, the curve
described by that point will be a common cycloid, as in Fig. 1. But if
you mark a point B on the circumference of the flange of a
locomotive-wheel, the curve will be a curtate cycloid, as in Fig. 2,
terminating in nodes. Now, if we consider one of these nodes or loops,
we shall see that "at any given moment" certain points at the bottom of
the loop must be moving in the opposite direction to the train. As there
is an infinite number of such points on the flange's circumference,
there must be an infinite number of these loops being described while
the train is in motion. In fact, at any given moment certain points on
the flanges are always moving in a direction opposite to that in which
the train is going.
[Illustration: 1]
[Illustration: 2]
In the case of the two wheels, the wheel that runs round the stationary
one makes two revolutions round its own centre. As both wheels are of
the same size, it is obvious that if at the start we mark a point on the
circumference of the upper wheel, at the very top, this point will be in
contact with the lower wheel at its lowest part when half the journey
has been made. Therefore this point is again at the top of the moving
wheel, and one revolution has been made. Consequently there are two such
revolutions in the complete journey.










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