I am reading a book on interesting physics problems and demonstrations. One of the problems in the section on buildings, structures and equilibrium talks about a plate with one side attached to the wall. The plate will hang, and the question deals with the amount of work per kilogram (or kinetic energy per kilogram as is written in the problem) on the plate. Now, the information on the dimensions of the plate are given, but I am having difficulty understanding how they would affect the amount of kinetic energy per kilogram in the plate, if the dimensions even affect the plate hanging. So my question is do the dimensions of a plate with one side attached to the wall affect the amount of kinetic energy (or work) per kilogram?
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Yes - if the plate is stiffer, then the deflection will be smaller and so the amount of work done (which is force times distance) will be less. For a typical linear elastic situation, the work done will be $\frac12 F x$ where $F$ is the final force and $x$ is the displacement. The factor $\frac12$ comes about from the fact that the initial force needed for deflection is small and it grows with displacement. If you just applied the full force at once, the plate would accelerate and oscillate (it would overshoot the point of equilibrium).
Because of this, the work done will be smaller if the plate is wider or thicker, and greater when the plate is longer.
