Non-Sinusoidal Oscillation: A Balloon on a String

Picture a buoyant balloon trailing a string with some mass per unit length. As the balloon rises, more string is lifted off the floor and the weight of the balloon-string system increases. As the balloon sinks, string returns to the floor and the weight of the balloon-string system decreases. In effect, the mass of the system depends on its position.


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Snapshot 1: no damping; the displacement curve (blue) might look like a simple harmonic oscillator at first glance, but look at its derivatives and ; note that energy is not conserved even in the absence of damping, as the rope is making inelastic collisions with the floor
Snapshot 2: massless string (and no damping); up, up, and away: in this case, the balloon is under the constant acceleration of its own lift
Snapshot 3: heavy string, small balloon, significant damping; the balloon reaches equilibrium around
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