### Earlier this year, I was lucky enough to go to the new Olympic velodrome in London. I was shocked to see how steep the sides were. Cycling is a sport with a huge amount of variety. There are events based on sprinting, events based on endurance and even events like the Madison where there are people both going fast and going slow. So, how are velodromes designed to maximise the speed of riders in all categories?

There are two main things that lead to the need for banked sides at a velodrome. One of which is a simple property of curved movement, the other is an intrinsic property of bicycles themselves.

So lets start with the first one. We have all experienced a so called "Centrifugal" force. This is the force that, when you are going around a corner in a car or spinning on a children's merry-go-round that you find in a playground, seems to pull you outwards, pushing you into the car door or off the merry-go-round. In fact what you are feeling is not really a force pulling you outwards, it is instead the effect of the force pulling you inwards:

When you want to turn from going in the original direction, you have to induce the force towards the centre of the turn. You feel like you are being dragged away from the corner, because that is the original direction your body was going in and it doesn't want to change. This force does come into play when cyclists have to turn corners in a race and can lead to problems. When you turn on a flat surface, your change in direction relies purely on traction, or friction, that your tires exert on the road. The equation that shows how fast an object can go before it slides off the surface is this:

Which we can simplify to

Where

*v*is velocity,*r*is the radius of the turn and*g*is the acceleration due to gravity. The weird*u*-like character is the Ancient Greek letter Mu, and it represents the coefficient of friction. Despite its complex name, the coefficient of friction is just a number that shows how well two substances grip each other. So lets work out how fast an Olympic cyclist could go around a corner if it was completely flat.
The radius of a turn on an Olympic velodrome is around 20 meters, the acceleration due to gravity is 10 meters per second squared, and the coefficient of friction between rubber and wood is roughly 0.7-0.9. So if we put that into the equation we get:

This roughly equates to 45.5 km/h. This means that a cyclist can only go at about 45.5 km/h around a corner without losing grip and possibly being dragged away from the inside of the track. Racing cyclists get to speeds as high as 80 km/h, which would cause problems on flat tracks. However, banking the corners alleviates the problem. Instead of getting pulled to the outside of the track, instead the cyclist is pulled into it:

The equation used to find the angle of the slope is a reasonably simple one:

Where

*v*is again the maximum velocity of the bike and*r*is the radius of the curve, whilst the 0-like character is the Greek letter Theta, and just represents the angle. So lets work out the maximum velocity that the bike can go. The radius remains 20 meters and the acceleration due to gravity remains around 10 meters per second squared. The angle of an average 250 meter track is around 45 degrees:
Ok, so that isn't yet high enough. However, what we haven't accounted for is the fact that the cyclist leans into the corner, instead of solving for a speed, lets solve for the angle.

This means that for the cyclist not to slip the bike has to be quite close to the ground. The slope of the side helps the bike get this angle.

However, there is another property that requires a tilt in order for the bike to turn at maximum speed. This property is one that applies specifically to bikes and other objects with a similar configuration to bikes. This property is that the bicycle is an inverted pendulum. This means it pivots around a point which is below it's centre of gravity:

So what does this mean? Well the best way to think about it is to imagine that you are riding a bike. When you turn a corner there is one main change to when you are moving straight, you lean in. So why is this? Well it relates back to the pendulum idea. When the bike turns, the centre of gravity has to be over the inside of the turn. Imagine leaning away from the inside of the turn when on a bike, the result is inevitable, you will fall over. The angle that the bike has to lean at is expressed by the formula:

But wait, if we scroll up, that formula is very similar to the one we saw before. In fact, if we rearrange it, we get:

which is exactly what we used to find the angle needed so that the bike won't slide off the track. These two properties are in fact linked, and that means that solving one problem cannot make the other worse. Instead, when you solve one problem, you also solve the other. We already calculated the angle the cyclist has to be at, around 68˚. The slope of the track cannot be this steep, it just isn't practical. Any cyclists in a event like the Madison that need to go slowly will just slide down the sides. Instead the sides are usually around 45˚ and then the cyclist leans in order to make the turn.

There is one more reason a cyclist would prefer for the floor to be sloped, rather than them having to lean steeply into the corner. The leaning into the corner is both a factor that needs to be done before the bike can turn and also a way of turning the bike. The more a cyclist leans, the sharper they can turn. In fact, as a cyclist leans, they are turning the bike a little simply by leaning. The fact that the track slopes, means that it nearly turns the cyclist by itself. This means it is a lot easier for the cyclist to continue moving at speed, instead of having to slow down.

Finally, the reason the straights are also curved is simply so that it would not be as large a change between the angle of the straights and the angle of the turns.

In conclusion, the sides on a velodrome are sloped so that the cyclists won't slide of the track, so that they will not have to lean at extreme angles in order not to fall over and so that they can keep speed whilst turning.

Lastly, I think I should show you a fantastic piece of cycling, to finish off. I was lucky enough to be at the Melbourne velodrome when this happened. The whole race is very good, but watch from 4:00 to see the really amazing stuff. Our boy, Chris Hoy steals the gold beautifully:

We hope you enjoyed this post, it was certainly very interesting to research and write. If you you want to get in touch you can follow and mention us on twitter, @theaftermatter, email us at contactus@theaftermatter.com or search "The Aftermatter"on Facebook.

Ned Summers

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The Physics of Rugby: Forward Passes and Relativity - How can a backwards pass look like they are going forward, and why is it so difficult for referees to judge?

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Your post is very informative and I am here with my general views about physics that is physics is a subject which requires a lot of practice in every part and your blog is very helpful in understanding the basics of Physics Of Cycling.

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