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When turning the jump rope, inertia (a.k.a Newton's First Law of Motion) causes the rope to "swing outwards". The center of the rope has the least amount of slack and therefore twirls outward the furthest. Gravity wants to pull the rope downwards towards the ground which explains why it stops after you stop turning it.
The jump rope could also be used to model a wave. With one end of the rope securely attached (i.e. a wall) you can move the rope up and down or side to side and view the transfer of energy as the rope moves in "sine wave pattern". The tops of the waves are called "crests" and the bottom are called "troughs".
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