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So, just to make sure the point on this is too fine, even if the frequency is dialed in perfectly, most things can be destroyed by resonance.
Even if the system is still pretty linear, this puts a cap on how big an oscillation you can get for a given amount of pushing.
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Most physical systems have the “linear on a small scale” property. This is why clock pendulums don’t swing very far, for example.
So this is the first big problem; if you push something hard enough, or if the oscillation gets too large, then the restoring force won’t be linear.
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