|MadSci Network: Physics|
Train wheels are mounted onto wheel units, with two wheels firmly attached to an axle. Bearings on the axle allow the wheel unit to rotate, at one angular rate, regardless of whether the train is travelling in a straight line or around a curved track.
Note that this is quite different to the wheels of a car or truck, which can rotate at different rates when turning corners. For non-powered road wheels, this is fine - a seperate bearing in each wheel solves the problem. Powered road wheels also have independent bearings, but also require the use of a "differential gear", to enable power to be given to the wheels regardless of the rate at which they need to rotate, in order to maintain traction and avoid slip. There's an explanation of how differential gears works here: differential gear.
Without independent bearings or a differential gear on the train, at least one wheel of a rotating wheel unit must slip - at least to some extent - when a corner is being negotiated. I would suggest that it's likely this slip mainly occurs on the inner wheel of a moving train, since the centripetal force of cornering would place more of the mass of the train onto the outer wheels.
In general this slip is not a problem for two reasons. Firstly, curves are wide in radius; and secondly the coefficient of sliding friction between two steel surfaces is quite low.
Taking the radius issue first. As I understand it, the tightest curvature for British Rail lines (where I live) relies on a radius of about 40m. With a railway gauge - the distance between the tracks - of 1.44 metres, a 45-degree curve, taking up 31 metres of track, would mean that the outer rail is just over 1.1 metres longer than the inner rail. This equates to under 4% of the rail track for this curve, so the slip required is fairly low. Faster track uses larger radii curves, and therefore minimises the slip - and subsequently the wear - on the wheels.
Secondly the sliding friction. We've all seen trains in films spinning their wheels on starting up, or during braking. It's not good with regards to wear-and-tear on either the wheels or the track, but it's easily done. If you've looked at railway track you can see that it's polished by the action of having rolling stock using it. Sliding a wheel along a track must therefore be relatively easy - certainly much easier than skidding a rubber tyre over a rough bitumen road - and this is the reason that gradients are low on a railway line: it's hard to maintain traction (i.e. avoid slipping) on anything like a steep slope. I'd not thought of the sliding-wheel issue for cornering trains before, but it's clearly a common process.
And finally! The different distances travelled by the wheels on the left or the right side of a train over the course of a journey. I suspect that if the train ends up at its destination pointing in the same direction it was at the start, and therefore the number of "lefts" equals the number of "rights", then the distance travelled by the left wheels compared to the right wheels will be only a tiny fraction of the distance of the journey - possibly the difference could be just a few metres over the course of many kilometres.
I hope the above answers helps you with your question
[note added by MadSci Admin: There is an excellent paper here which discusses some tests of a different design for railcar trucks, one of the features being that the two axles steer on a curve, but, like conventional wheelsets, each of the two wheels on an axle are firmly connected to the axle.]
Try the links in the MadSci Library for more information on Physics.