Kinetic energy for a rotating mass is K = 1/2 I w^2, where I is the moment of inertia, and w is the angular velocity. I is equal to the integral over m of r^2 dm.
Let's assume we have two bikes moving at the same speed. They have wheels and tires that are identical apart from the diameter (i.e. same cross section/material/construction). We'll say those wheels have radii ra and rb respectively, ra > rb.
The moment of inertia for bike A's wheels is bigger than that of bike B's, by a factor of (ra-rb)^3, since I is proportional to the radius squared times the mass, and the mass increases linearly in proportion to the radius.
To achieve the same speed, the angular velocity of bike B's wheels is greater than that of bike A's by a factor of (ra-rb).
Therefore the rotational kinetic energy of bike A's wheels is greater by a factor of ra-rb.
The linear kinetic energy (K = 1/2 m v^2) is also greater by a factor of ra-rb.
The hubs are the same, so the above applies to the spokes, rim and tire. If we take the bead seat diameter to be the best indication of actual diameter, 650b wheels require 4.5% more energy to get up to speed than 26" wheels, and 29" wheels require 11.3% more energy than 26", or 6.5% more than 650b.
Let's assume we have two bikes moving at the same speed. They have wheels and tires that are identical apart from the diameter (i.e. same cross section/material/construction). We'll say those wheels have radii ra and rb respectively, ra > rb.
The moment of inertia for bike A's wheels is bigger than that of bike B's, by a factor of (ra-rb)^3, since I is proportional to the radius squared times the mass, and the mass increases linearly in proportion to the radius.
To achieve the same speed, the angular velocity of bike B's wheels is greater than that of bike A's by a factor of (ra-rb).
Therefore the rotational kinetic energy of bike A's wheels is greater by a factor of ra-rb.
The linear kinetic energy (K = 1/2 m v^2) is also greater by a factor of ra-rb.
The hubs are the same, so the above applies to the spokes, rim and tire. If we take the bead seat diameter to be the best indication of actual diameter, 650b wheels require 4.5% more energy to get up to speed than 26" wheels, and 29" wheels require 11.3% more energy than 26", or 6.5% more than 650b.
