# Rigid body, circular motion - centripetal acceleration

Hi all, new user here and straight in with a question to help me understand something. Not sure if I'm in the right sub-forum but oh well.

Regarding circular motion and centripetal acceleration, we're given the classic example of a cyclist leaning into a corner to create a general equilibrium against the friction and the weight in the system, to allow for the resultant centripetal acceleration.

Also there's the example of rolling a coin along a surface. If the coin loses balance, it describes circles until it stops. If the coin were to somehow be going at constant velocity, and then it leant to one side, it would go in circles forever, right?

So my confusion is regarding the rigid body turning in a circle when made to lean. The coin does it, as can be observed quite easily by anyone. But... what about a two-wheel object such as a bicycle? If the front and rear wheel were rigidly locked in a straight line (so it's not really a bicycle but a rigid body with two inline wheels), would it still describe circles if it were made to lean? Or does bicycle turning actually involve a slight element of turning the handlebars?

I'll be honest, I'm just trying to clear something up on a motorcycle forum I use. I reckoned it's possible to make any rigid body turn by leaning it (not using handlebars at all), and so I used the coin analogy as an example, but then someone else said that it wouldn't be possible if there were two wheels rigidly inline with each other, and if you leaned you would just toppled over.

Intuitively it sounds like the guy is right, but then... another niggling bit of intuition tells me the bike should turn?

Which intuition is correct?!

Thanks all

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AlephZero
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But... what about a two-wheel object such as a bicycle? If the front and rear wheel were rigidly locked in a straight line (so it's not really a bicycle but a rigid body with two inline wheels), would it still describe circles if it were made to lean? Or does bicycle turning actually involve a slight element of turning the handlebars?
Because of the geometry of the bike steering, the front wheel will automatically turn the required amount even if you are not holding the handlebars. The wheels on a supermarket trolley "steer" the same way. The point where the front wheel touches the road is slightly behind where the axis of the steering tube would meet the road.

You can show that by holding the bike when it is stationary and leaning it over. The front wheel will turn as if you were leaning into the turn. (Do this standing beside the bike, not sitting on it, so your weight doesn't create too much friction between the front tire and the road and stop the front wheel turning).

You can start the bike leaning by bending your body sideways, so your center of mass is not above the wheels. Or you can start by steering the "wrong" way, so the bike starts to fall over with you leaning out, not in (the bike wheels are steering round a curve but the center of mass of you and the bike carries on a straight line), and then correct the steering to keep your balance.

Ok sure, thanks a lot. So, in short, it's not possible to make a two-wheeled rigid body describe a turn by leaning it. Right?

I was told it's also because of the gyroscopic forces caused by the forward motion? So it prefers to self right itself than to topple sideways. But why does't this happen when you roll a coin? Is it because of the difference in masses?

Thanks