it may have escaped your notice that the legs are connected to each other

and you seem to have failed to notice that they have seperate muscle systems each capable of exerting force despite sharing the same body.

Force means to apply energy to accelerate a body – eg kinetic via gravity downhill.Torque means to apply a force around a fulcrum or pivot say like pedalling a bike. Pray exaplain why it matters here ? An increase in force equates to an increase in torque I am fairly certain that when I pedal with more force it increases the torque and I tend to go faster and get up hills rather than when I apply less force and get less torque when I have a tendency to stop. Perhaps there is a relationship at work here between force and torque?
I assume you can create a scenario whereby I increase the force applied to a grank but the torque does not increase? Trackstand?????

Pulling up with one foot whilst pushing down with the other creates exerts a torque on the cranks, but also an equal and opposite one on your body. You need to balance this out to stop you rotating backwards off the bike. You can do this by pulling on the handlebars (or by pushing less).

There's no inherent reason why you can't exert a torque greater than that created by putting all of your weight on one pedal. Whether it's worthwhile is another matter.

I assume you can create a scenario whereby I increase the force applied to a grank but the torque does not increase?

if you read what I said, I didn't comment on the amount of torque in the case you mentioned. You said force so I remarked on that. Inevitably, if you pull up with one foot you'll have to press harder with the other unless your centre of gravity moves with respect to the crank axle.

30% more pedaling efficiency! So that means my 3 hour loop will take 2 if I swap pedals?

My new van's apparently about 30% more fuel efficient than my mates car. Why won't it go 180 mph?

if you read what I said

Must I ? 😉

"or suggesting meaningless concepts like having no weight"
It's not meaningless, it's perfectly sensible assumption if you think of someone sat on a saddle – they have no weight WRT the crank arms, so all torque generated is due to muscle input.

Wrong. As sfb suggests, when pushing hard you have far less than your whole bodyweight resting on your saddle.

They allow the rider to apply more torque as the rising, otherwise idle, leg can also contribute, provided the cardiovascular system can deliver the extra demand.

That's what I mean. I'm a lowly biological chemist though, so my brain doesn't compute in forces and moments and torques. As I have so obviously demonstrated.

Wrong. As sfb suggests, when pushing hard you have far less than your whole bodyweight resting on your saddle.

I'm aware of that, but that's not the situation I used it in, maybe I didn't explain what I meant properly. It is possible to sit on the seat and pedal hard without reducing your force on the saddle, you cause do so by pulling up on the upstroke to match your downstroke. The point was to remove gravity effects from the build-up so it was a little more clear, unfortunately it made it a little less clear.

Any net force in any direction from the two cranks gives you an increase or decrease in apparent saddle reaction force. As long as you can match the downstroke force with your upstroke force you can hold your body still wrt the cranks. If your net force goes positive (pushing harder than pulling) you'll start to unweight your body off the saddle, therefore the maximum net force you can apply is whatever your upstroke can apply+force from your body weight. After that you have nothing holding you down anymore, unless you're also able to stop your rotation about the handlebars, but I'd expect that your grip isnt that strong.

I suspect the reason they say (in that link provided) that net efficiency goes down is that the upstroke is less mechanically efficient (due to the position, size etc of the muscle and limb) and so drags down the overall efficiency even if it shares the loads. But that's in sub-maximal case, in the maximal case where your push muscles are working at full tilt, bringing in a little help from the pull muscles may well extend your range.

is "around a circle" the same thing as a direction ?

to my mind a direction can be expressed as either x,y & z or angle and azimuth, which a circumference cannot.

SFB
It all depends on your co-ordinate system, you essentially have mentioned two possible coordinate systems
1) cartesian (x, y,z)
2) Spherical (or polar) (Radial , polar, tangential) well you said asimuth /angle but its part of the same thing.

But there are other coordinate systems such as cylindrical.

You can express a vector using cylindrical co-ordinates – (radius, tangential and axial)

It is possible to sit on the seat and pedal hard without reducing your force on the saddle, you cause do so by pulling up on the upstroke to match your downstroke.

yes it is, but not a technique I could recommend…

yes it is, but not a technique I could recommend…

Something I do quite a lot if I'm struggling up long hills, it transfers some of the load from my push to my pull and lets them "rest" so I can continue climbing while "easing off" the painful muscles.

but not a technique I could recommend…

particularly to those of us using flat pedals 🙂

Something I do quite a lot if I'm struggling up long hills

I doubt you do. You might pull up, but it's unlikely you get as much pull force as push force.

Pulling up with one foot whilst pushing down with the other creates exerts a torque on the cranks, but also an equal and opposite one on your body. You need to balance this out to stop you rotating backwards off the bike. You can do this by pulling on the handlebars (or by pushing

My legs bend in the middle. And also where they meet the rest of me. And at the ankle come to think of it.

I think the underlying mechanism is fully encompassed here –
http://www.youtube.com/watch?v=RXJKdh1KZ0w