Re: [Phys-l] Statics conundrum

[John Denker <jsd@av8n.com>]

On 03/14/2010 02:02 PM, Bob Sciamanda wrote:
> Here is a common elementary statics problem:
>
> A perfectly rigid and uniform beam of given weight and length (W and L) is suported by two men, exerting upward vertical forces F1 and F2, one at each end of the bar.  Determine F1 and F2 in terms of W and L.
>
> This is easily solved by imposing translational and rotational equilibrium:
> F1 + F2 = W
> W*L/2 = F2*L   =>   F2 = W/2  and F1 = W/2  (independent of L)
>
> One can even add other given loads at given positions on the bar, and the problem is still easily solved.
>
> *****************
> But a curious student might uncover the following conundrum:
> If one adds a third man exerting a third upward force F3 at a given location (say L/4 from one end), The two equilibrium equations are insufficient to solve for the values of the three unknons, F1 F2 and F3.
>
> The three man experiment can be performed and the forces measured (even with added loads on the beam). They ARE physically determined.
>  How does one analytically predict this result? 
>
> Please discuss. Is it the perfect rigidity which must be relaxed? Why?
> How explain this to the curious student? (and to me)

I don't accept the claimed result.

Just because the force can be measured doesn't mean the force is
"physically determined" by the statement of the problem.

The problem is ill-posed as stated.  It is underdeterminted.

The experiment has been done millions of times.  It is well
known that N people sharing a load can share it in innumerably
many different ways.

Rather than looking for "The" solution for F1, F2, and F3, 
look for the _solution set_.  This is standard operating
procedure when dealing with underdetermined problems.

The sought-after quantities span a three-dimensional space
F1 x F2 x F3, where x denotes the Cartesian direct product.
The solution set for this simple problem will be a one-dimensional
manifold in this space.  In fact it well be well described as
a straight line in this space.  That is to say, there will be
a linear relationship such that if you know any one of the forces
then the other two are determined.

As a generalization of this problem, with N people, if you know
N-2 of the forces, then the remaining two are determined.  This
works for all N from 2 on up.  The correspondence principle
requires that we check the N=2 case, to see that it reproduces
the familiar simple result ... which it does.