Or would more power go through the pedals if the rider held the frame still/upright?
Or is there little difference? How would it be measured and what are the forces acting on the bike when rocking while sprinting?
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The rocking motion does not help with power output, but it is the way you maintain balance with least effort.
When one is riding in straight line, the system of bike and rider rotates almost freely around the axis that goes through the points where tires contact the road (there are steering forces to maintain the balance, but these are very small compared to pedaling forces). Because there is nothing to brace against, the torques around this line have to even out. Because pedals are not directly above the line, pushing them down needs some torque to counter it. When sitting down, the most natural support is the combination of handlebars and saddle (if you wondered why saddles extend forward between riders' legs, this is one of the reasons). Standing up to pedal allows using more force on the pedals and loses the support from saddle at the same time. This requires more force handlebars, but also allows another option: Rocking the bike from side to side brings the pedal that is pushed downwards closer to the centerline, which reduces the torque that needs to be countered.
From mechanics standpoint, the arms don't do any work when holding the bike upright. Holding the bike upright by twisting the handlebars wouldn't indeed take any effort if you could just lock your arms like suspension lockout. Unfortunately muscles don't work that way, so maintaining the bike upright while pedaling standing takes more effort than rocking the bike around.
This method could be used to control balance when sitting in saddle, too. The problem is that when sitting, you need to wiggle your lower body to keep up with the saddle and the upper body in other direction to maintain balance. You can do it if you really want, but it is not comfortable or efficient.
It is not the rocking movement that gives you more power. The rocking is a result of applying power in a diferent way.
When you stand up on the pedals you are able to apply all of your wheight onto one of the pedals, but as it is on one side of the bike it will be necesary to compensate. Your arms will instinctively apply force in the oposite direction.
These actions however do not elliminate the fact that your relaitve center of mass will change throghout the movement, so your body and bike oscilate side to side, in order to keep it over the line between your tires.
If while standing on the pedals you also pull up on the handlebars, you will be able to apply more downward force to the pedals than just your weight. However, for it to be affective you must not counteract that force with the other foot (the one going up). You also must keep your torso "aligned" with your legs, otherwhise, part of your efort is also wasted on not bending yourself sideways.
Anecdotally, I've been able to keep up with partners trying to get away by means of pedalling stand up, while I remain on the saddle. My conclusion is that yes, pedalling stand up will help produce more power, but also, It must be performed correctly for it to be effective.
There is another way in wich it helps, and it is on long rides, or long climbs. Stand up pedalling uses a slightly different muscle group, thus can be a relief for some muscles while alowing you to continue moving forward.
I ride a road bike and can rock the bike fine. I also ride a recumbent, where all of the rider's weight is on the seat, and "rocking" the bent is impossible.
The upshot is that rocking the road bike allows the engagement of the core muscles which is everything up the trunk/torso, as well as the shoulders and arms.
A very approximate test showed that I was able to maintain ~250W on a road bike for some minutes, but achieving the same forward speed on my bent was under a half of that power. However I couldn't go a lot faster, because on the bent the core muscles aren't really used.
By comparison, imagine hypothetically riding your road bike hard, but no-hands. How much power could you put down in a no-hands "sprint" ? This would be like riding without engaging any core muscles. note - don't actually do this.
In short, rocking the bike helps engage extra muscles for more power and therefore more sprint speed.
Answer: yes rocking the bike helps in a sprint or when pushing hard on a climb.
I see that the pros rocking the bike on a real steep part of a hill climb. When I look at many less experienced cyclist, a significant number of them keep their bike vertical while climbing, so their bikes don’t rock, but their bums go up and down.
My theory is that the non-rocking cyclists are working harder because they get their body to the top of their power stroke by actually moving their body higher and lower (as can be seen by the bum moving up and down). So for every pedal stroke they are lifting their body. But, the rockers don’t have a bum that’s going up and down, so the rockers are not wasting energy to raise their body (or if they indeed do raise their body, it seems much less than the non-rockers).
Also, I wonder if the rockers hit their power stroke a bit later in the downstroke (say maybe at 2 o’clock), while the non-rockers are climbing up higher over their pedals and getting the start of their power stroke at 12 o’clock. Since the non rockers are at the top of the stroke, they might have a longer power stroke but they probably have to expend more energy to start each power stroke because they are starting it too early. I wonder that if the rockers start pushing the power stroke at an easier position (2 o’clock), then they might be able to maintain a faster cadence (everything else being equal).
Rockers can also have the option use their core muscles and arm muscles (when needing to pull up the handle bar with the opposite arm to the foot that’s pushing) a to get even more torque per pedal stroke. Rockers do not have to pull up with their arms, but if the climb is steep enough, the pulling of the opposite arm helps as well.