Perhaps the most famous paradox is the one that asks: “What happens when an irresistable force meets an immovable object?” It is generally posed as a question in logic or philosophy but it occurred to me some years ago that with a little application of Newton’s Third Law and some basic Statics one can arrive at an answer.
The short answer to what happens is: nothing. Forces sum to zero so the system is in equilibrium.
For a more detailed answer, let us consider Newton’s Third Law, which states:
Whenever a first body exerts a force F on a second body, the second body exerts a force −F on the first body. F and −F are equal in magnitude and opposite in direction. Or, as most folks learned in school: For every action there is an equal and opposite reaction.
The F in the Law is Force, the equation for which is F=ma, where m = mass and a =acceleration. Given that the force is irresistable and the object immovable, we may assume that the force exerted by both is infinite. Because the paradox does not specify the space in which the problem takes place, for simplicity let us assume that the system occurs in two dimensions, although there is no reason why it couldn’t occur in n-dimensions:
In a two-dimensional system we can see that at any point of contact the forces are coplanar and concurrent. We can use a convenient coordinate system to have the forces acting on the x-axis. Thus two equations are required to describe the system, ∑ Fx to resolve the linear forces and ∑ Fy to resolve the moments.
If we take the object to be suspended in space then there are no moments, and ∑ Fy = 0. Because we have determined that the forces acting on any given point in the system are infinite, by the Third Law we can see that
∑ Fx = ∞ + (-∞) = 0
Thus when an irresistable force meets an immovable object, the system is in equilibrium and no disturbance is observed.
This illustration is for 2-d space. I expect that the general solution for n-dimensions would be similar. If anyone has experience with this problem on a physics exam, either taken as a student or given as an instructor, I’d like to hear about it.
The word for today is lemniscate, which refers to the particular type of closed curve that is used as the symbol for infinity. I had not known that until today, when I went looking for it on the Web.