In reply to John Gillott:
> However, it is my contention that if there was no friction (between the tray and the ground), this would not be possible, as every time you made use of the internal friction between yourself and the tray to move yourself
you're using the friction force (Ff=uN, unless you're dealing with polymers/gels/tissues which conveniently don't obey Amonton's law) between the tray/you and the floor to do work.
> (try it on wet ice and you'll find it is true, you just thrash around getting nowhere).
the interesting thing though is that without friction acting between the tray and the ground (eg. on wet ice you've got hydrodynamic lubrication - very low friction, but there is some) it takes only a very small force to start you moving. If there was actually zero friction, then theoretically you could just spit and that would be enough to move you in the opposite direction.
> Let's consider just the problem of gaining height once movement has already occured. The boy pushes on the chain say with a given force, this force has a moment about the point of contact with the frame and hence transfers angular momentum to the chain. But... the chain pushes back on the boy in just the same way in the opposite direction, giving him an angular momentum of equal magnitide, but in the opposite direction. Same goes for any use of friction by twitching on the seat etc.
what you are neglecting though is that although there is no external forces doing work on the system (we'll neglect air pressure/drag and friction at the pulley as being negligible for the sake of argument), there is an exchange of energy within the system and this can create motion. In this classic problem (simple harmonic motion) you are basically exchanging potential energy (lost/gained as a result of changing height of the seat) for kinetic energy (lost/gained as the swing passes through the lowest and end points of it's arc of travel). You can show that for an arc half angle of theta, the velocity of the swing at the bottom of the arc is v=sqrt(2gr[1-cos(theta)]) where g=9.81m/s^2 and r is the radius of travel.
Other types of internal energy changes include e.g. springs (elastic), heat transfer, electric/magnetic fields. In reality there are losses in the system (friction, heat, etc) and so theta will decay with time - in a proper system you can use this information to determine what sort of lubrication mode is operating at the pivot point.
As for the climbing question - I reckon sweating in hot weather is the major factor. Depending on the rubber formulation, a temperature drop of 20degC (summer -> winter) might double the stiffness (modulus), so you could well notice a difference in edging, but that's nothing more than a guess really.