I haven't found much in web search on the physics of an object rotating on a string.
Some of the entries are people writing what they remember of physics class, but just the equations and abbreviations, not the English meaning. There are communities online about the sling and trebuchet, but they often speak about the distance a projectile can be slung, rather than how factors interrelate for total force.
(I didn't do well in physics, because I kept asking "What does this equation mean, can you rephrase it in English?", while my classmates were memorizing it successfully for the test. When I hear F=mA I guess they mean that "force" doesn't quite mean "how it feels if it hits you", because acceleration is only the rate of change, and a faster thing, even if it's slowing, sure has an impact. Joules and Newtons introduced new terms: defined, yet unapplied. Wikipedia doesn't help. Guess I've got special needs with terminology. ;-)
Anyway, assuming you're spinning three feet of rope poi-style in a circle, each revolution covers about 18.85 feet of distance (that's 2piR, 2 * 3.14 * 3). If you're scooping the palms (making the circle larger with wrist movement), then that's about 20 feet of distance per rotation. Three rotations per second gives you a ball travelling 60 feet, or 40 miles per hour. I've read forum discussions where people say they get six rotations per second, which would be 80 mph. Techno music often runs 120-150 beats per minute. I can't measure my own rotations yet. Bottom line: Balls on strings go fast.
They go fast, but what happens when they hit something? I use soft weights of 2-3 ounces, but Jackie Chan in Shanghai Noon used a horseshoe. An iron dart on the end weighs 350-450 grams (12-16 ounces), and a "hammer" type of wushu weight is quoted in a video I've seen as 750 grams (1 3/4 pounds). Shaolin rope darts can cut through an inch of thick wood, and the "meteor hammers" were used to defeat body armor. I think Jackie Chan just knocked someone out, and I didn't see the head fly off, but then again that's Hollywood. ;-)
Anyway, with three feet of rope swinging, you get 13mph for each additional revolution per second. The heavier the weight, the bigger the effect. Beyond that there's body torque... some of the rope dart moves use a 180d body turn to snap the rope forward, in addition to the spin.
I'm pretty sure this packs more of a punch than a punch, but how much, that's not clear. I think I need to go out in the park with a rope-end monkeyfist and aim at leaves and tree trunks, to get a feel for what's going on with the rope when it's swinging around.
One phrase that comes up in a lot of web searches is "the accumulation of energies" -- the ball in circular motion stores energy, and you keep adding to that store. Slingers say they usually swing only a few circles before release, and some techniques release at less than 360d of spin. With the body torque that some rope dart techniques apply, it's like a punch from the hand, but at long distance, and without the follow-through of falling bodyweight behind it.
Rope dart also uses a lot of wrapping techniques, around the elbow, neck, leg. I'm not sure how these suddenly-smaller circles affect momentum upon release. There are some rope dart wrapping techniques where all centrifugal motion is removed, and you end up just flinging a tethered dart outwards. The wrapping techniques might have some power implications, but I suspect they evolved slowly, in response to how swordsmen and others learned to anticipate straightforward circles and dodge the dart -- more for timing and surprise than for power.
The physics of the pendulum, the pulley, the supersonic snap of the whip, all these things are related. I want to gain an intuitive understanding of how these forces play together.