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Does a ruler twang forever?

If a ruler is twanged, the oscillations decay exponentially. The same can be said for capacitor discharge and a number of other functions. My question is this. An exponential decay approaches zero but never actually gets there; it just keeps getting closer. Does this mean that as the ruler is released, the oscillations continue ad infinitum (theoretically) even when submerged in background noise? If so, do we have a graph showing a motion with theoretically infinite energy?
Joe Harding from Gloucestershire (Age 35-44)

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One Response

  1. Theoretically yes, but…
    Keep in mind that a mathematical description of the motion of a ruler you use makes a number of (simplifiying) assumptions, most importantly:
    1- The ruler is made of a continuous, perfectly elastic, material.
    2- The ruler can be completely separated from the environment, your background noise.
    In principle, neither of these is correct, but if we look at relatively short times, where the amplitude of oscillations is large enough, these approximations are quite good, and we can use the simple model you propose. As the motion damps more and more, the atomic nature of the ruler, inhomogenities in the material, coupling to thermal noise, etc., become important and your model fails.

    Can you follow the ruler for longer than the eye can see? Probably yes. Does the motion have infinite energy? No! Energy is not the sum over all time, it is defined at each moment. If the motion is damped the ruler looses energy, and thus remains finite (but goes to zero).

    You may be interested to know that modern research in nanosystem looks at microscopic cantilevers (oscillating rulers) which are excited by quantum effects, thus demonstrating the boundary between the large scale picture and the atomic and thus quantum nature the material is made up from.

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