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Why are the wheels of a bicycle moving in concentric circles with the center O?

I know that the velocity of the back wheel is parallel to the frame of the bicycle and the velocity of the front wheel is parallel to the direction in which the wheel points.

FIXED DIAGRAM

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  • $\begingroup$ The diagram is wrong. The center of the front wheel is supposed to be along the line of the back wheel. $\endgroup$
    – John Alexiou
    Commented Sep 18, 2014 at 16:20
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    $\begingroup$ This is the title puzzle in Which Way Did the Bicycle Go? and a good one. $\endgroup$
    – Ross Millikan
    Commented Sep 18, 2014 at 16:48

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Because that is where the lines normal to the wheels and through their centers intersect. If the center of rotation was at any other point, then there would be a component of velocity perpendicular to each wheel (wheel slip) and ideal wheels roll without slip.

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  • $\begingroup$ And that slip would cause the wheel to move sideways? $\endgroup$
    – user42768
    Commented Sep 18, 2014 at 16:27
  • $\begingroup$ slip is lateral (sideways) movement by definition. Not causality but equivalence. $\endgroup$
    – John Alexiou
    Commented Sep 18, 2014 at 16:43
  • $\begingroup$ I apologize if I am asking too much, but I want to verify if this is correct: if I suppose the back wheel causes no friction, then the motion would be linear, but because the back wheel can't move sideways, now the front wheel has to travel in a circular manner? Is this why the motion is circular in the first place? $\endgroup$
    – user42768
    Commented Sep 18, 2014 at 16:59
  • $\begingroup$ Almost. Not just the back wheel, but both wheels enforce the circular motion through their no side-slip condition. If there is no friction on the front wheel the motion would be linear, as it would be if there was no friction on the back. It is through both conditions that the motion is circular. $\endgroup$
    – John Alexiou
    Commented Sep 18, 2014 at 19:22
  • $\begingroup$ So, assuming for example that (in the conditions of the above diagram) I "push" at the back wheel (in the direction of the back wheel), the front wheel would have to slip, so the frame of the bike rotates. And the same if I "pull" the front wheel (in its direction). Thank you for your help! $\endgroup$
    – user42768
    Commented Sep 18, 2014 at 19:53

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