Re: [Phys-l] Statics conundrum

["Bob Sciamanda" <treborsci@verizon.net>]

Talking points for a teaching/learning lesson on this matter:

Problem statement:
A uniform, rigid beam of weight/length W/L is hung from three identical 
strings, each exerting an upward supporting force on the beam.
F1 acts at the left end, F2 acts at L/4 to the right of the left end, and F3 
acts at the right end.
Given W and L, solve for the values of F1, F2. and F3.

> From previous problems, we know that we could choose F2 = 0 and get the 
solution F1 = F2 = W/2 and F2 = 0.  This will work.
But are there other solutions?  From the translational and rotational 
equilibrium equations we have only two equations and three unknowns -
an underdetermined situation - there can be multiple mathematical solutions. 
. .

You may recall some earlier problems where the mathematical model of a 
physical problem admitted of more that one mathematical solution.
For example, we have seen two possibilities occur when the mathematics 
included a quadratic equation to be solved for the relevant unknown 
quantity.
In those cases, although the mathematics allowed two solutions, the physical 
requirements usually allowed only one.
However, in this case, we may have an infinity of solutions!  Let us explore 
the possibilities. . .


Since we know that we can choose F2 = 0 for one solution, let us generalize 
and categorize all solutions by choosing:

F2 = a*W,  where a is a dimensionless, positive parameter chosen to specify 
a particular possible solution. . . .

The equilibrium mathematics will then give the solutions:

F1 = W/2 -3*a*W/4 ,    F2 = a*W,    and   F3 = W/2 - a*W/4

Does every value of a generate a PHYSICAL solution? . . .

If you now impose the PHYSICAL requirement that the forces (string tensions) 
cannot be negative, you will find that a is restricted to values between
zero and 2*W/3 (including the end-points).

EG., the "end point" solution is F1 = 0, F2 = 2*W/3,  and F3 = W/3 .

Note there is no PHYSICALLY valid solution with the choice F3 = 0.

 In a real physical situation, how does "nature" choose among all these 
PHYSICALLY VALID solutions? . . . Blah, Blah, Blah . . .



Bob Sciamanda
Physics, Edinboro Univ of PA (Em)
treborsci@verizon.net
http://mysite.verizon.net/res12merh/ 





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