For example in this video at 5:00: They show by turning the gear with little force allows for one to move a VERY HEAVY gate to operate the sea locks. I don't understand how this is possible. Could anyone please explain the physics/theory behind this? Thanks!
Have you read this article? It explains what mechanical advantage is fairly well. http://en.wikipedia.org/wiki/Mechanical_advantage The short version is that the gears amplify the torque because they make each turn of the input gear equal a fraction of a turn on the output gear. So turning one gear 10 turns may only get you 1 turn on the other, which amplifies the torque. The basic mechanism used for mechanical advantage is the Lever. From the linked article: It boils down to the fact that when one end of the lever moves a shorter distance in the same time, the force must increase for the power to remain the same. Same with the gears.
As Drakkith said, gears are basically levers that can operate continuously. See: http://en.wikipedia.org/wiki/Lever They allow you to trade distance for force, or the other way around. For example this simple gear outputs less force, but more velocity than goes in: https://www.youtube.com/watch?v=dvyii6QBLtw Here the lever mechanism is indicated as a red line: https://www.youtube.com/watch?v=Ufk6HVWdSzE
There is a law of "conservation of energy", not "conservation of force" and all "mechanical advantage" laws are based on that. If you have a cog of radius R m and turn it through on complete turn, [itex]2\pi[/itex] radians, by applying force F Newtons, then its surface has moved through a distance of [itex]2\pi R[/itex] m and so you have done [itex]2\pi RF[/itex] Joules work on it. If a chain or other ratcheting mechanism causes another cog, of radius r, to turn through the same distance, by "conservation of energy" you have done [itex]2\pi r f= 2\pi RF[/itex] Joules of work on it also and so must have applied [itex]f= (2\pi RF)/(2\pi r)= (R/r)F[/itex] Newtons force. "R/r" is the "mechanical advantage".
If the other cog, of radius r, turned through the same distance that the cog with radius R turned, then they must have the same work. Since work is F*d, then they both must have the same force? I am kind of confused by how you equated: [itex]2\pi r f= 2\pi RF[/itex] instead of [itex]2\pi R f= 2\pi RF[/itex] since they travel the same distance [itex]2\pi R[/itex]