I know that this may sound silly, but I was wondering if there is a proper way to pedal the road bike? I've noticed that you can be pedaling more with the front, or balls of the feet, or by staying flat for the most part, or even having the heel of your foot slightly angled down. Is there a specific way to pedal that will yield the most efficient use of 'foot power'? If so, what makes that specific technique more efficient?
[I will treat this from a purely mechanical point of view, as this is my area of expertise -- the actual body mechanics are much more complex, but the effect is similar.]
From a mechanical point of view, the most efficient way to propel yourself on the bike is to step on the pedal so that the distance between your heel and the pedal is maximized. Ideally this place would be your toes; however, due to the toe joints, your ability to do so is rather limited, as the the flexible joint will change the way power is transmitted through your foot.
Thus, the optimal place to step is the ball of the foot -- it has a good surface area for good contact, and the muscular mass which reduces the reaction from the pedal and dampens somewhat vibrations from the bumpy road.
As to why the most efficient place is as far from your heel as possible:
See this handy force diagram below that I found when writing my answer (image license unknown). Although not exactly what we need here, it will come in handy in the explanation.
How this whole biking deal works:
You step on the pedal with force
Mechanically, the greater the arm
You might be tempted to think that if you make your pedals long enough, you can exert enormous amount of force, and indeed this is so. Archimedes famously said "Give me a place to stand, and I shall move the earth with a lever". The thing is, your feet will need to travel much farther to cover the same distance as with shorter pedals -- but requiring a negligible force. There ain't no such thing as a free lunch in mechanics as well.
But wait, this only takes into account the length of the pedal, not the foot!
Indeed. Your foot only serves as an extension of the pedal, effectively increasing the torque you produce by increasing the lever length. If the length of your foot is
effectively increasing the torque you produce.
The vast majority of your power in a cycling stroke comes from your hip and knee extensors (gluteus maximus and quadriceps pushing your leg/foot down). The amount of power you can get from ankle/foot muscles relative to that is tiny; when cycling those muscles mostly just keep your foot stiff, not provide primary power.
The position of your foot on the pedal should be optimized for comfort. Don't worry about efficiency/power from your foot position relative to the pedal spindle.
With a "normal" shoe having the center of the pedal near the ball of your foot is likely the most comfortable. Your toes can't really handle the forces involved well, and pushing a pedal up into your arch is likely to be uncomfortable.
With special cycling shoes that click onto the pedal and have very stiff soles you can potentially move that contact point towards the arch or towards the toes more. Changing your foot's contact point will also require adjusting the seat and possibly the handlebars. I think that kind of millimeter level fine-tuning is something best done with the help of an experienced bike fitter, since it's easy to get it wrong in ways that can cause knee injury, saddlesores or other problems. What foot position on the cranks will get you the most power/efficiency will depend on the strength of your calves and is probably best done in a bike shop fitting setup with a power meter hooked up to your bike. But those tiny adjustments of foot position are only necessary if you really care about shaving a few seconds off the time of a multi-hour ride.
Any attempts to get angles and lengths just right for leverage are silly on a typical road bike, the cranks are only one of 4 levers and 2 of those levers can be changed in a fraction of a second while riding to fine-tune your overall ratio just right for current conditions (slope, wind, what you're trying to do, how you're feeling).
If you're concerned about efficiency, worry more about your cadence and stroke. People talk about "pedaling in circles" or "squares", but the most essential part of that is to lift your foot up on the back part of the stroke to minimize how much you're resisting the pedaling force from the other foot. Finding the right cadence is important, pushing hard in a lower cadence can wear your muscles out faster than getting the same speed/power from pushing faster in a slightly lower gear.
As I understand it, you want to expand beyond the "power stroke" and consider pedaling in a "full circle". Starting just past the top of the stroke around 1:00 you would drop your heel slightly then gradually extend you foot somewhat during the power stroke, then extend it further at the bottom, sort of "flicking" the pedal across the bottom. Then, you would pull with your hamstrings during the "back" of the stroke, driving your knee up and pulling your toes up slightly as you go to prepare for the next power stroke. At the top you would flick your foot forward then repeat the whole cycle. It is easier to understand if you attempt each of these while pedaling a bicycle. If feels pretty natural after a small amount of practice.
I would not necessarily say this is the "most efficient" way to pedal, but that it is the method that produces the most output. When Mountain Biking I use the full circle to accelerate or climb short hills. On a long hill I alternate emphasis from the power stroke to the back stroke to shift the load around while climbing. On a mostly level road I pedal full circle to maintain the maximum sustainable pace.
You can read a lot of detail here.
I use clipless pedals which places the ball of the foot over the pedal. Given that I've never seen biking shoes with the attachment anywhere but the ball, I'm assuming that there must be some power and/or health advantage there.
Also, keep your feet almost straight with toes pointed slightly out. In the short term, this may provide less power, but if you don't, you'll over develop your outer quads which leads to knee problems. Again, clipless pedals 'enforce' this behaviour.