Related to the guitar project, might have other applications.
Prevailing consensus is that Galileo Galilei made the first attempts at developing a theory of beams, but recent studies argue thatLeonardo da Vinci was the first to make the crucial observations. Da Vinci lacked Hooke’s law and calculus to complete the theory, whereas Galileo was held back by an incorrect assumption he made
So, if Galileo made incorrect assumptions, certainly that should be gone over, and then what the actual correct assumptions are, explained so clearly that it is obvious in retrospect? Because the hard part is the interesting part. Of course not, they just jump to the differential equation.
I think it is rarely gone over how physical models are developed (meaning, like, anywhere — universities, scientific literature, talks, etc). It’s not simply a matter of applying some universal physical law (eg., Newton’s 2nd law, the conservation of energy). Some assumptions need to be made, and those assumptions need to be checked.
I’ll fearlessly do that here. Fearlessly meaning, this might well take us down some blind alleys. Yeah, this is a solved problem, but what the hell.
Let’s start by assuming that a rod is made up of a bunch of little hinges, each of which follows a variant of Hooke’s Law:
So a beam deflecting (due to a weight at the end) might look like this, approximated by these hinges:
Let’s further assume there is a point weight at the end of the beam. The beam will be in equilibrium when the weight has lost potential energy by lowering that is equal to the elastic potential energy stored in the beam due to bending.
For a single hinge beam:
Anyway, the equation can be solved for theta.
For a two hinge beam:
Energy considerations now only gives us one equation, but in two unknowns! So we can no longer solve the system.
Let’s introduce yet another assumption. Clearly, it does not make sense for theta1 = 0 and have theta2 take all the bending. Same for theta2=0. (This is almost an entropy consideration). So, for the heck of it, let’s assume theta1=theta2. Now we again have enough information to solve the system.
With n hinges, we get
where that last step is apparently due to Lagrange
Now let’s see if this result is reasonable. As n tends to infinity, theta must tend to zero. However, for the result to be physically meaningful, the energies must tend to some finite limit (obviously? At least it should be obvious that the weight must ultimately have some final vertical position). So, on the left side, this would suggest that n-theta^2 must tend to a finite limit.
But if n-theta^2 tends to a finite limit, then n-theta must increase to infinity. This means the right side doesn’t seem to make sense. On the one hand, the argument of the second cosine term seems illogical. That might not be so bad if the cos term was multiplied by something that went to zero. Well, actually, it seems the entire right side must tend to zero. This means regardless of the weight or spring constant, the deflection of the beam must be zero.
This does not make sense.
Not, however, one assumption we may have glossed over: That k is a constant. It is independent of the number of hinges. In fact, for normal springs, the spring constant is not (solely) a function of the material the spring is made of, but also depends on its length. A common example might be someone thinking their screen door spring is too long, and rather than cutting it by a conservative amount they cut it in half, and are surprised by how much more forcefully it now slams the door shut.
More specifically, k is inversely proportional to length, and if we used a similar rule here it would be proportional to n, so it would be n-theta, not n-theta^2, that would tend to a fixed constant. We can examine that consequences of that assumption next time.