Scientists have been puzzling over what makes a bicycle balance since the days of ye olde penny farthing. Many experts suggested those spinning hoops make the bicycle behave like a gyroscope, but it’s not that simple. A group of engineers from Nottingham University identified 25 separate variables that affect a bicycle’s motion, citing that, ‘A simple explanation does not seem possible because the lean and steer are coupled by a combination of effects, including gyroscopic precession, lateral ground-reaction forces at the front wheel, ground contact point trailing behind the steering axis, gravity and inertial reactions…’
What is known is that as long as a bike is moving at a speed of around 14kmh (9mph), it can remain upright without the presence of a rider. But again, scientists can’t explain why. Against that backdrop, throw in the added dimension of a bend and calculating the angle that you can lean while cornering before you hit the tarmac is clearly a complex affair. In the right conditions it’s possible to see angles of 45°, but how do we get to that point?
‘We know there are three real forces acting on the bike and rider,’ says Rhett Allain, keen cyclist and associate professor of physics at Southeastern Louisiana University in the US. ‘There’s the gravitational force pushing the bike and rider down; there’s the road pushing up, which we call “normal” force, and there’s a frictional force pushing the bike towards the centre of the circular path that it’s moving in.’
The fake force
There’s also centrifugal force. ‘This does have an impact but it’s a fake force,’ says Allain. Many physicists argue that centrifugal force doesn’t exist and is simply a lack of centripetal force – an inward-pulling force that ensures the bike moves in a circle similar to gravity pulling inward on a satellite to keep it in orbit. It’s calculated via the equation F = mv2/r, where F is the centripetal force (Newtons), m is mass of bike and rider (kg), v is velocity (m/s) and r is the radius of the corner in metres.
‘The physics of riding a turn is that you do it by accelerating radially inwards, which is down to centripetal force,’ says David Wilson, emeritus professor of engineering at Massachusetts Institute of Technology. ‘The force has to come from the tyres. The bike has to lean so that the combination of the reaction from the tyre and the radial force is in line with the resulting force of the bike plus rider.’
Also key to how far you can lean is the coefficient of friction, which is the ratio of the force of friction between two bodies and the force applied on them – in this case the tyre and tarmac. Most dry materials have friction values between 0.3 and 0.6, whereas rubber in contact with tarmac can produce a figure of between one and two. When the surfaces are moving relative to each other – as per cycling – this figure decreases slightly.
For the bike to remain upright, the side force (centripetal) must equal the coefficient of friction, and this figure can be surprisingly large. For instance, a 70kg rider on a 10kg bike speeding at 20mph around a curve with a radius of 20m experiences a centripetal force of 316 Newtons. This force has to be generated by the tyres, and if the force didn’t exist, the bike and rider would simply carry on in a straight line.
Using some impressive trigonometric calculations that would fill a whole book, the coefficient of friction is equal to the tangent function of the maximum lean angle. ‘The wheel will slip when the coefficient of friction is exceeded,’ says Marco Arkesteijn, lecturer in sport science at Aberystwyth University. ‘This can be due to friction force increasing [due to tightening the line through a corner for example] or normal force decreasing [due to, say, a depression in the road].’
The coefficient of friction can also change due to a change in surface. That’s why cornering on a white line can be perilous. ‘This is especially true in the wet,’ says Arkesteijn. ‘Paint is less porous so the water doesn’t disperse.’
Rider weight
To complicate matters further is the issue of rider weight. ‘Physics-wise, smaller guys should be able to lean more,’ says Arkesteijn. ‘They’re also usually more agile, which helps.’
Allain is not quite as definite, suggesting that while rider weight matters a ‘little bit’, of greater importance is the rider-plus-bike’s centre of mass. ‘Ultimately, that’s the most important factor,’ he says. Heavier riders tend to be taller riders, especially in the pro peloton, meaning their frame sizes are larger and their centre of mass is higher. You also need to factor in road conditions. If you’re at the limit, a bump in the road can lead to a loss of traction and a fall. UK roads are sometimes grippier than those of our mainland European cousins because they’re more porous to absorb rain and prevent a slippery surface. That’s why our roads are coarser. But they’re often bumpier and in worse condition because of frost damage, hence why cycling and driving in France is an absolute joy when it’s dry.
After all that, what is the maximum lean angle? For mechanical and engineering professor Jim Papadopoulos, that can’t be answered until you throw in one final factor – trail. This is an imaginary line that’s projected down the steerer tube to the ground. If this point is in front of the wheel contact point with the ground, it’s deemed ‘positive’ and is more stable. Behind and the bike is more likely to tip over. Trail reduces the more you lean.
‘Cyclists tend to stay in the positive trail region and don’t exceed 45° of lean,’ he says. ‘It’s usually less, though when the turn is greater than 5m radius, you can reach 45°. That’s because trail becomes less of an issue – then we return to the issue of traction.’
So 45° is possible on a fast, wide, well-surfaced turn, but with so many variables at play, there is, unfortunately, no definitive answer. How far you can lean is a case of trial and (hopefully not too painful) error.