As noted, the Gondorian war machines aren't catapults at all, but trebuchets. Basically, a catapult gets its energy from something springy. A trebuchet, on the other hand, gets its energy from gravity. It's generally constructed as an arm on a pivot. On one end of the arm is the counterweight, and on the other is a sling that contains the projectile.
To start, the projectile end is pulled down (via a winch or other mechanical aid) and secured with a trigger of some sort. One end of the sling is permanently attached to the arm, while the other end is held on with a pin that is angled such that it releases the sling when its at the top of its arc. (When firing a trebuchet, the angle of the pin is one of the things you have to constantly adjust to get the thing to release at the proper moment. Too early and you hit the folks standing behind the machine [always fun at a demonstration], too late and your projectile doesn't exactly get far.) You put your projectile in the sling, and then release the trigger. This causes the counterweight to drop, swinging the projectile end of the arm up in an arc, and hopefully releasing the projectile right at that sweet spot so that it continues sailing up and away.
The site I had linked to has gone the way of the dodo. Below are the Wayback Machine links, but unfortunately the pictures are well-hidden. I dug out the diagram - it's the last link below; you're on your own for the rest of them.