## 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.