We must not be communicating here. What I'm saying is that the vector component perpendicular to the rear axle is the only part turning the cogs; the component parallel to the rear axle only serves to pull the axle sideways.artemidorus said:I argue that the perpendicular vector does no work, as the sprocket is not accelerated perpendicularly, and is consequently not a power loss, except in so far as it increases hub friction.
Take an extreme example with the chain were running at a 45* angle to the plane of the cogs (also at a 45* angle to the axle). If you were producing 100 lbs of pull on the chain, only the tension x cos 45 deg, or 70.7 lbs would be doing work to turn the cogs. Another 70.7 lbs of force would be pulling the axle to the left, which of course is doing no work; only trying to flex your rear triangle.