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an you ride along in the drops and the float your hands off the handlebars and maintain your body position? If not, your cycle fit needs to be adjusted for your current physiological limits.
A commonly circulated bicycle fit falsity. As one pedals harder one's body pushes back. If you can do that it probably means that your bicycle does not fit:-) Seriously... Taking your hands off the bars tends to also mean taking pressure off the pedals and, of course, the body will fall forward. The drops are there on a modern road bicycle not for "just riding along". Given that the currently "standard" riding position is defined by the hoods Zeitgeist is to use bars whose design to ease the transition and provide minimum body movement in the transition from hoods to drops. Drops are lower but provide less reach and more body leverage--- and maybe reduce the frontal area . Moving foward in the saddle during sprints and moving back during climbs is standard.. The main issue with saddle height is injury prevention and less about economy.. If you look at a lot of old pictures you'll see saddle heights and reach all over the place.
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For a more detailed precis of seat set back (which if set correctly will prevent you from toppling forward when you remove your hands from the drops), read this:
http://www.stevehoggbikefitting.com/bikefit/2011/05/seat-set-back-for-road-bikes/
@apc Sounds like your core strength and flexibility may be the issue. Can you ride along in the drops and the float your hands off the handlebars and maintain your body position? If not, your cycle fit needs to be adjusted for your current physiological limits. Possibly your saddle needs to go back a bit to take some weight off your hands.