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Swinging ropes are commonly found in platform video games where the player jumps on a rope to reach an adjacent platform or what not.
What math is involved in making this possible. I am thinking of somehow having the end of the rope follow the outline of a circle while having the rope itself create waves via the sine function.
For something physically accurate, I would go for a mass–spring model:
http://nehe.gamedev.net/tutorial/rope_physics/17006/