On the definition of force


In setting up our picture of what a force means conceptually, we went to a spring to create a definition.  Although we have an intuitive definition of "force" as a push or a pull from the first time an older sibling or cousin pushed us over, to do science with force, we need some way to quantify it.  In Quantifying impulse and force, we suggested that we could use a spring to create an operational definition of force.  A spring shows when it is exerting a force by changing its length.  The longer it is, the harder it pulls.  We can use its length as a measure of how much force it's exerting.  We could in principle create a careful made standardized spring (or a set at different scales) that would be like our standardize set of masses.

But once we had done enough experiments to decide that a constant force — defined as something produced by a spring stretched by a fixed amount — leads to a constant acceleration (a part of what Newton's 2nd law claims) — we don't have to rely on the spring for our quantification.  We can accept Newton 2, quantify mass, and define force quantitatively through N2 as "1 Newton = the force needed to give a mass of 1 kg and acceleration of 1 m/s2."  We could then use acceleration as a way to infer what forces were acting.

This is a useful definition only if it turns out that Newton's 2nd law is "true" (or rather, "useful for organizing our thinking of the physical world").  What would happen if it weren't?  Well, since we have now defined forces in terms of N2, if N2 weren't "true"  then we would find that every situation needed us to define a different force in order to explain what is happening.  We want to keep the law only if instead of thousands or even millions of different forces, we only have a few.

This illustrates the peculiar logical nature of scientific laws. We assume them as hypotheses.  Those hypotheses lead us to see the world in a particular way — for example to see the slowing down of a sliding block as caused by a frictional force.  If every time a block slowed down we had to invent a different force and could say nothing about it, the law would be useless and we would reject it.  But in the case of N2, a small number of forces turn out to be sufficient to describe (in principle) almost all macroscopic (larger than a molecule) phenomena.

The picture is not really circular (we define forces from looking at acceleration and then predict acceleration from looking at forces).  It starts in one direction — we figure out the properties of force from looking at accelerations (or lack of them).  Then we figure out what those forces are as a function of other observable properties — position, stretch, charge, velocity — anything BUT acceleration.  Now we flip the direction of the process and figure out the acceleration from the forces. 

This turns out to work amazingly well.  At the macroscopic level we only need 8 kinds of forces to describe almost everything from the molecular to the galactic scale.  Even better, when we go to the micro level and understand forces better, we see that 5 of our kinds of forces are really a version of the electric force combined with quantum mechanics. 

Joe Redish 9/24/11

Article 346
Last Modified: August 10, 2018