A train’s wheels may look simple, but there is a slight difference to a regular wheel that maths helps explain. The ‘flat’ bit of the wheel that rests on the railhead is actually a truncated cone. The reason for this is when going around a corner, the outer edge of anything will need to travel a further distance, as proved when you see circles with a radius of 5 and 10 have a circumference of 31.4 and 62.8 — double the distance to cover in that case. Because of this, any vehicle that will be turning needs to accommodate this and allow that wheel to cover that extra distance at the same time. Cars achieve this by a set of gears known as a differential, which spins the outer wheel faster than the inner wheel, thus allowing it to turn. In trains however, truncated cone wheels are used so that when the train reaches a corner, the part of the wheel that is in contact with the rail moves to either side, so that the inner side of the train has the outer part of the cone moving it, and vice versa, thus for each revolution the outer wheel is covering a greater distance than the inner wheel. The steeper the cone, the tighter the curve that can be handled. But too steep a cone and hunting oscillation occurs, which is where on the straight, the train is extra bumpy for reasons too complex to list in this space.
