About things that are `small`...and the butterfly

If you asked yourself: in what sense is that force ‘negligible’...well done! Whenever you come across the claim that something is ‘small’ or ‘big’ or ‘negligible’ you should always stop to ask that question. The point is that these words always presuppose a scale: thus when we say that something is ‘negligible’ we really mean that it is very small compared to ‘something else’ that is important in the context. Obviously the ‘something else’ has to be the same ‘type’ of thing...ie measured in the same units. So here we are really saying that the force you exert on the ball, due to your mass, is small compared to other forces acting on the ball. What are they? Most obviously, the force the cue exerts, briefly; the weight of the ball; perhaps (not so obvious) the frictional forces exerted by air and table.

That is the main point of this little aside. But there is a further twist to this story. If our snooker ball travels far enough to have several collisions (with other balls, or with the cushion) the effects of that very tiny gravitational force you exert on it aggregate up quite remarkably — to the extent that they cannot be safely ignored. This is one illustration of a general principle underlying the science of ‘non-linear dynamics’ or (in popular terminology) ‘chaos’. The long term behaviour of many physical systems is startlingly sensitive to apparently small external influences. In some circles this is known as the ‘butterfly effect’.