How it works
How a train goes around a bend
D. Indumathi, The Institute of Mathematical Sciences, Chennai

In the last issue of JM, we looked at how the differential in a car helps to provide different speeds to the wheels on the two sides, so that a car can turn. This is because, when it turns, the inside wheel moves slower than the outside wheel (see figure). The differential works by making available these different speeds on the two sides. A reader of JM has now written in, asking if a train turns the same way.
It is a very interesting question. Trains do not have differentials. Trains have wheels that are connected together by a fixed axle. This means that the wheels on both sides of the train always turn at the same speed. This can present problems when turning, because one wheel has to cover more distance than the other. How then can a train turn?
The wheels are specifically designed so that when the train goes around a corner it stays on the tracks. The part of the wheel that has to travel a greater distance has a greater diameter, and the end result is a train that stays on the tracks. Let us examine closely how this happens.
Train wheels aren't perfect cylinders. They're beveled, that is, the flat surface of the rim is sloped, to make them wider on the inside. This means that when the train moves left or right on the track, the diameter of the wheels can change. But because the wheels are connected by an axle, they still spin at the same rate. Because of this, the wheels will travel different distances per revolution.
A simple way to understand this is to assume that train wheels are conical in shape; actually a cone that has its top chopped off. A cone has different diameters at different heights from the base. That allows the wheels to have a varying diameter at different points of contact. See the figure. By the way, did you notice that the track rails are also not flat? The rails are embedded a little inclined inwards. This maintains the angle between the wheel and supporting track to be always 90 degrees.
Coming back to the wheels, how does having a conical shape help? When the tracks are going straight, the wheels are symmetrically placed on the rails. So both of them rest on the track rail at a point where their diameters are the same. So as the train moves ahead, both wheels turn with the same speed (which they always do); in addition, they travel the same distance in equal times since their diameters are the same. See the figure.
Now suppose the track turns right. The train’s left wheels now have to travel more than the right wheels because at the turn the track on the left is longer. But the train has no differential, so both wheels are still turning at the same speed. Now, both the wheels move left, so that the portion of the left wheel touching the rails has a larger diameter (and so moves further in one rotation) while the wheel on the right moves less. The figure explains it very nicely. Now both the wheels move different distances in equal times so that the wheel which has to travel a larger distance smoothly moves around the curve and "catches" up with the inner wheel.
What happens if the curve is so sharp that the wheels are not able to adjust their diameters sufficiently? The observant reader may have noticed that train wheels have flanges: a kind of protruding part that doesn't allow the wheels to go off the rails. Flanges are a safety mechanism to keep the train on its track only if the main mechanism fails. This could happen if the train was going too fast around a bend. In fact, railway companies specify a "minimum railway curve radius" which states how sharp a bend can be if the train has a certain maximum speed.
For a very clear video explanation of how the conical shape of train wheels helps it to turn, see the web-site, http://www.etudes.ru/en/etudes/train-wheelset/ from where these pictures are taken. As pointed out there, it's a very nice application of geometry, isn't it?