Consider a chain of length L that hangs in a U shape with end A fixed to a rigid support and free end E released from rest starting from the same initial height (call it y = 0) as A. Figure 1 sketches the chain after end E has fallen a distance y. Points O and A are assumed to be close enough to each other and the chain flexible enough that the radius of curvature r at the bottom point C can be taken to be negligibly small (compared to the length of the chain).
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