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Originally Posted by yoda
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Actually, the rock is simply trying to move in a straight line. The centripetal string prevents it from doing so, and the apparent force -- from the rock's point of view -- is centrifugal.
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Excellent, we've got to the core of the issue very quickly.
When we speak of the rock's point of view, we are talking (I hope) about what you would see through a tiny camera attached to the rock. Through this camera we would see such interesting sights as a stationary rock (because we are attached to it), the centre of motion (the boy's hand) would be stationary, and everything else in the world would appear to be in the process of being flung from away from the centre of motion. The centrifugal force (away from the centre) will neatly balance the centripetal force (toward the centre), but no net force is acting on the rock. It is not being flung out. It just sits there stationary in the camera shot.
The problem is that this is not a helpful frame of reference because it is not an inertial frame of reference. We have to do all sorts of fudging if we are to explain why all of the objects (grass, trees, etc) are being flung away from the centre.
If we want to apply the usual laws of physics we need to look from an intertial frame of reference. The string pulls in on the rock (centripetal). The string pulls out on the man (centrifugal). The force on the rock is not balanced, and the centrifugal force only affects the string and the boy.