Originally Posted by
scyclops
I've seen the calculations done by various people who seem to know what they're talking about, and with all other factors being equal - including weight - the difference in energy required to spin up the 29" vs 26" (based solely on diameter) is so miniscule as to be insignificant. Evidently this is due to the fact that the smaller wheel has to turn more RPMs to cover the same distance as the larger wheel - thereby virtually canceling out any effective gearing disadvantage of the larger wheel.
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Ok, before I get jumped all over, I'm not even pretending to know mathematical physics, I'm just relating stuff that seems to be undisputed by any people that do know about such things. And BTW, I have no plans to go 29" any time soon.
You're right, and the physics is not very complicated. Take a bike with total mass M, and two wheels, each with mass m and radius r. The kinetic energy (translational + rotational) of the bike moving at velocity v is
E = (1/2) M v^2 + 2 * (1/2) I w^2,
where "^" indicates exponentiation, I is the rotational moment of inertia of each wheel (approximated as infinitely thin hoops with radius r)
I = m r^2,
and w is the angular velocity of the wheels when the bike is moving with velocity v,
w = v / r.
Plugging the above two expressions into the kinetic energy gives
E = (1/2) M v^2 + m r^2 (v/r)^2
= (1/2) (M + 2 m) v^2
Note that the radius of the wheel cancels out: the kinetic energy of the bike at any given speed is completely independent of the wheel diameter, as long as the larger/smaller wheels have the same mass. However (this is interesting in general) note that adding a gram to the rim has twice as much effect (adds to both "m" and "M") as adding a gram to the frame (adds to just "M").
But, you say, "
real wheels aren't infinitely thin hoops!" Correct. But it makes no difference to the conclusion that wheel size is irrelevant. For any cylindrically symmetric wheel, the moment of inertia will in general be of the form
I = K m r^2,
where K is a constant (always < 1) that depends on the mass distribution in the wheel. Then the radius still cancels out of the energy,
E (1/2) (M + 2 K m) v^2.
http://xkcd.com/54/