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Say I have a table and I want to produce a vibration in the table. Would I be better off impacting the surface of the table (i.e. smacking the table) or nudging the surface (i.e. leaning on the surface and briefly pushing down hard. Is there a difference in energy transfer between these two actions, even if they're performed with the same force?
This is a question that's been bothering me for a little while. Intuitively, it seems like an impact would deliver more force, but I'm not too sure.
This is a tricky question. If the average force is really the same, what is going to change is impulse, the integral of force over time
$$J = \int_0^{\tau} F(t) dt \simeq F_{av} \tau$$
When you push the surface you interact with it for a longer time ($\tau$), while when you smack it you interact with it for a shorter time. So if the average force $F_{av}$ is the same, by increasing the interaction time you are going to deliver more energy to the system (the impulse will be greater).
You intuitively feel that smacking the table is more effective because it is usually easier to deliver more force by exploiting the kinetic energy of the object we are using to strike. Just think about "pushing" a baseball with the bat instead of hitting it: doesn't feel very effective, right?
Also, if your objective is to damage (or even break) the table, then probably it is best to hit it. This is because if the force is delivered in a relatively long time interval the material will have the time to bend and deform to absorb the energy without damage, while if the force is delivered in a short time it won't be able to do so and will usually suffer damage. Smacking water with your hands is an extreme example of what I'm saying, even though water won't obviously be damaged by the blow. Just think of breaking a small rock with an hammer: to break it by pushing it you will require a lot of force, but if you deliver a nice hit you will be able to smash it easily.
The energy transfer is the work: force times distance. With a strike, the force is bigger and the distance (and time) is shorter. You could tune the two scenarios such that the work is the same.
The big difference: when you cease to lean on the table, the table performs the same work back onto you (assuming reversible, elastic deformation), with zero net energy transfer as a result. If you strike it, your hand is already gone by the time that the table is going back into its original shape, so you will have done a net energy transfer.
