Today I was looking at some examples of balancing torques with my first year class and on the spur of the moment decided to consider a bicycle going in a circle. That's what the diagram shows by the way, viewed from the back, must be one of those penny farthing bikes. First step is to show that the torques can't balance if the bike is vertical. so the bike must lean inwards so the torques about the centre can balance, but what if you consider torques about the bottom, they can't balance. OK so it's not in equilibrium but how do you explain why it doesn't fall over? I looked for explanations on the web and all used the centrifugal force to balance torques e.g. Wikipedia. Another source, ask the physicist said you can only balance torques about the centre without introducing centrifugal force. Surely there is an explanation without mentioning the dreaded centrifugal.

First lets get rid of the circular motion and consider the bike on an accelerating road, the same forces apply. Friction at the tyre accelerates the bike so it must lean in the direction of acceleration. The whole difficulty here is that the bike is accelerating so the forces aren't balanced, the bikes frame of reference is accelerating, a ball dropped in this frame of reference will not fall vertically, it would actually fall along the line of the bike. So it is as if gravity were acting along the bike. Aha, so if that was the case the weight would act through the bottom of the tyre and there would be no torque. It's all to do with the principle of equivalence but probably best not to bring Einstein into it.

Another way of looking at it is to consider an equivalent situation. Instead of the road accelerating imagine the road is tilted. The bike would then be balanced at an angle to the road so that the weight is directly above the tyre. Not obvious why this is equivalent but it is.

On reflection, maybe not the best example to use.

Here is an animation showing how the gravity acts in an accelerating frame of reference. The first part is viewed from the point of view of an observer outside the accelerating frame, the second is in the accelerating frame of reference.