Exercise

When I am out riding my bike, I can make wide turns easily but tighter turns get progressively difficult. At some point, I will only be able to just barely make the turn. Any tighter of a turn and I will slip and fall. What kind of number represents how tight a turn is? What is the tightest turn I can make on my bicycle in normal conditions at a typical cruising speed?

Pre-requisites

To know more on this topic, you need to know about speed, acceleration, centripetal motion, friction, and circular geometry. Assume the coefficient of friction is 0.4 and a typical cruising speed of 16 feet per second. Gravity on Earth is 32 feet per second per second. Assume also a slow, safe turning speed of 8 feet per second.

Facts

The tightness of a turn can be expressed as the radius of the circle you make while turning. A big radius is a gentle, safe turn. A small radius is a tight turn.

The smallest circle you can turn while making a gentle turn at 8 feet per second is **5 feet** in radius.

The smallest circle you can turn while making a safe turn at 16 feet per second is **20 feet** in radius.

Thoughts and Opinions

It is normal and healthy to test your limits.

In the Physical world, your limits are set by real Physical conditions. No amount of strength training, mental practice, or insider knowledge will save you from the truth on this matter. Much the same as there are speed limits on roads based on the conditions set by nature, there is also a curve limit. We do these calculations as a virtual exercise, which can only be an approximate substitution for the real thing.

The limit exists because the coefficient of friction between tires and road is constant, and in normal operation your bicycle isn’t pushed in excess of your total combined weight against the road surface. In theory you can beat the limit by engineering a spoiler or similar that squeezes you against the road more, but we ignore these fringe cases for this exercise.

Thus, the frictional force maxes out at a certain value, essentially enforcing a “centripetal acceleration limit” which is speed dependent. You see signs on the road before a curve for this very reason, this limit is not merely a suggestion but is a very real warning. Road engineers have translated the friction force limit directly into a speed limit for your convenience. What happens if you break this limit?

In the case of the gentle turn, you can make a relatively tight turn safely at a radius of 5 feet. That implies you can make the same turn at a radius of 6 feet, 7 feet … or as big as you like. However, you cannot make the same turn with a radius of 4 feet. Your speed will demand a higher centripetal acceleration than the force of friction can supply, and you will skid off and fall over.

In the case of the fast turn, the answer turns out to be 20 feet. Intuitively, if you tried to make a tight turn (say 5 feet radius as before) at a higher speed, you will skid and fall over. In fact, at a comfortable cruising speed of 16 feet per second, the tightest turn you can make safely has a radius of 20 feet. You may make the turn in 21 feet, 22 feet … and so on.

I have learned this lesson the hard way myself. I have tested this limit both on paper with numbers and in the real world with a real bicycle.

Substituting the real world experience for the virtual one will save you scrapes and bruises, but I would argue that the real world experience on the bicycle is a great way to gain intuitive insight on this matter before upgrading to a motor vehicle. We know intuitively the meaning of speed limits. When it comes to curves, knowing about acceleration limits intuitively may save your life.

As a final note on this, see that the size of the turn you can make does not double when your speed doubles. Your intuition can be trained to know this. Be brave but also plan ahead before you test your limits.

Feel free to email me with your thoughts and opinions on this matter atÂ andrew@ahogan.org.