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## If a fly jumps up and down at the resonance frequency of a building, would it fall or seriously shake?

No, not really. The basic idea behind resonance is to apply a small force multiple times at a very accurate frequency - the "resonance frequency". The small forces add up due time in a coherent way, making a total big force (what happens when you push a child on a swing) that could potentially destroy a building.

Practically, there are two main issues which will prevent the building from falling. The first is that the fact that, in order for the resonance to work, the object needs to respond in a linear way to it: a small increase in the force should result in a proportional increase in the amplitude of motion. However, this is only an approximation that is valid only when the (cumulative) forces are very small; when the force become large enough, every object (e.g., a building) will respond to it in a non-linear way.

The fact that the response of an object is linear when the forces are small is a universal fact, which represents, in some way the idea that the physical world is "smooth". However, it is also universal that when the forces become strong enough, the objects respond in a non-linear way (think of the swing when pushed too hard).

The second problem is that even if the objects were responding in a completely linear way, there is still the issue of damping; when a force is applied to an object and forces it to move (or shake), gradually the energy of the motion is dissipated; if you stop pushing the kid on the swing, he will eventually stop moving. The dissipated energy is typically transferred into heat via friction. Now, when we try to make an object resonate by adding forces (which produce waves), the waves that were produced at earlier times gradually damp and lose their energy; thus, they don't add as much as the waves that were produced later. The overall result is that even if everything were to be tuned perfectly, still it is very difficult - virtually impossible, to destroy most objects.

A nice article summarizing this can be found at http://www.askamathematician.com/2013/08/q-can-resonance-destroy-anything-is-the-brown-note-possible/