JR45 I think you have your physics wrong. Centre of gravity is the centre point of the mass of the object and I don't think it matters how the bits are attached to each other. However you could be right in that where the bits are attached effects the leverage. Your trails rider, by standing, has a higher centre of gravity, not lower, because he has moved a large portion of the mass higher. This has given him a longer arm which takes more inertia to move but I think the main point is that it gives him better balance as he can shift his whole body, not just his upper body. He may also have better leverage on either peg.
^ Yep, where you're actually attached to the bike doesn't make a difference to CofG, its the position of the body, high/low, front/back, left/right that makes the difference. A trail rider stands to allow your legs to act as suspension and it also means you can affect the CofG more as you can move more compared with just being within the confines of the seat area, as well as your movements being able to affect CofG far more independently of lean angle.
If you read what I said, I said transition of weight to the rear is important to not break grip at the front. ie, you want a gradual transition, not a sudden. If you watch when they dangle the leg, they are already under full braking - ie, contact patch is maximised at the front. Moving weight to the rear settles the rear of the bike down so that when they get on the throttle again (and move their leg back on the peg), there is less pitching motion as the bike is being leant over. If the rear was less settled, you get a more sudden pitch to the rear tyre as you get on the gas and therefore more risk of unloading the front tyre suddenly when it still requires max grip on turn in. Unload it too quick, and the front will fold. Quick turns are often the aim around a corner to minimise lean time and maximise throttle time, thereby it's the transition time from brake to throttle that you want to minimise but also ensure stability.
If you don't believ my, completely accurate, explanation of how weight transfer / stability / centre of gravity works, then try this one... ride along on your bike sitting on the seat, with your feet on the pegs: go as slowly as you possibly can without wobbling (ie maintaining a straight line ) then take you feet off the pegs and see what happens. The bike will become unstable because the point that the weight is acting on has moved upwards - and therefore the effective centre of gravity has become higher. Do not get confused between centre of gravity and centre of mass. The centre of mass remains the same. The centre of gravity ( ie the point at which the mass effectively acts ) moves if the points upon which it is acting moves. As an example - if you lift a heavy engine off the floor with a crane, the instant that the engine leaves the floor the point at which the weight is acting transfers from the bottom of the engine to the head of the crane. Do this on a ship, or at a long enough extension of the crane arm, and the weight transfer can be violent enough to tip the crane ( or in fact the ship ) over. You may only lift the engine a matter on millimeters, but the point at which the weight is acting can move many meters.
Some confusion here between centre of gravity and moment of force. The centre of gravity is always the centre of mass, in a uniform gravity environment - it has nothing to do with where a force is applied or transmitted.
