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This question refers to the movie LoTR: The Return of the King.

I'm hoping someone can help me figure out how the Minas Tirith catapults work. I can see how the Uruk-hai catapults function -- they operate by a simple pull-back and release, like flicking a spoonful of pudding at the wall. But how do Gondor's catapults do their job? They seem to have some kind of slingshot-like quality to them, but I cannot for the life of me visualize the mechanics behind their functionality.

How do the catapults of Minas Tirith work?

I'd love to see a drawing, to help me visualize the mechanics of Gondor's catapults. (I don't care about the quality of the drawing if you decide to do one, as long as it makes sense, so don't worry about producing the Mona Lisa!) Of course a verbal explanation is as equally welcome.

They weren't catapults per se, they were trebuchets. Whereas catapults like the ones the Uruk-Hai employed are powered by various sorts of tension, trebuchets use a very heavy counterweight to throw their load.

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.
http://medievalreader.wordpress.com/2012/02/26/trebuchets-a-very-brief-history/
http://medievalreader.wordpress.com/2011/08/01/concerning-trebuchets/
https://web.archive.org/web/20140602103934/http://medievalreader.files.wordpress.com/2011/08/trebuchet_copy.jpg

The Gondorian trebuchets are in appearance fairly standard hanging-counterweight trebuchets. They work by loading energy into the machine by raising a heavy counterweight, loading the projectie into a sling that's attached to the throwing arm (functions as a First Class lever) and releasing the arm.allowing the counterweight to drop which causes the arm to rotate and pull the sling. When the sling reaches a predetermined angle it releases the missile downrange. It's easier to watch than to describe.

A problem with the Gondorian machines is that even though they appear normal they are pictured as throwing huge chunks of masonry that no normal trebuchet could handle. Well, it's fantasy.