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The optimum wheel diameter of cars and bikes appear to be roughly the same, certainly well within an order of magnitude. This is despite very different average speeds and propulsion mechanisms.

Can anyone come up with a dimensional analysis-type argument explaining why this is?

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Because humans need to step onto/into both of them? –  user2963 Feb 13 '12 at 2:16
    
see above comment for explanation as to downvote –  Timtam Feb 13 '12 at 3:12
    
I don't think it's that simple. The land speed record set by a wheeled vehicle (google it) has wheels of approximately the same size as typical bikes and cars. I'm sure whoever paid for that achievement didn't care whether the vehicle was easy to step into. –  BrianC Feb 13 '12 at 3:27
    
Who said that that's the optimum? en.wikipedia.org/wiki/Penny-farthing . WHoever invented the penny-farthing MUST have been looking to optimize the bike, as there's no reason (that I can think of) to make the wheels so dangerously disproportionate. –  Manishearth Feb 13 '12 at 4:37
    
It seems obvious that very small wheels are inefficient. Too much friction when the wheels have to rotate so quickly. So bigger is better - but taking this to the extreme doesn't make sense either. Eventually it takes too much energy just to get the wheels rotating. So there has to be an optimum somewhere in between. I assume this is close to the common adult bike size, otherwise there would be incentive to have racing bikes with ever-bigger wheels. But what's interesting to me is why this apparent optimum is true for gas-powered piston engines as well. –  BrianC Feb 13 '12 at 5:31
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Because they travel on the same surface with the same roughness maintained to the same level. If cycle paths were precision machine flat steel rails you could have very small wheels, but since you must manage the same ruts and potholes you need a wheel of similar diameter.

There are folding bikes with very small (<20cm) wheels - but they are useless on anything other than perfect roads. There is a limit to how big you can make the wheels if you want a person to be able to sit on the bike (remember the penny-farthing. Finally large wheels on an off-road bike would be very heavy to be strong enough and on a high speed bike would have too much wind resistance.

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+ From my mechanical engineering days, where was such a thing as needle bearings, little cylindrical wheels that spin within a hard steel bushing. If roads where also super-smooth hardened steel with lubricant, bikes could have milimeter-sized wheels :-) At the other extreme, I'm reminded of a type of 6-foot diameter wheel with a seat inside it and handlebars, that you coast down a grassy hillside in. –  Mike Dunlavey Feb 13 '12 at 18:28
    
Taking bicycle frame geometry issues into account, bicycle wheels are basically as large as they can be for average height riders. Shorter bicycle riders need smaller wheels or get bicycles with problematic geometries (don't handle well or don't fit well). –  freiheit May 7 '12 at 5:48
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