[MechEng83]

I'll preface by saying I feel like this topic has strayed too far from the original post and may warrant a new thread over the value of prototyping vs doing theoretical calculations.

Now, my day job involves doing FEA (finite element analysis) all day, every day. So I have a personal stake and I feel it's a disservice to teach the kids that analysis is wasted time and you should just build and try it out. I also recognize that the analysis can become so complex and take too long to solve that you'd be better off getting empirical data from a prototype.
Analysis and calculations are tools. Some tools work better in certain situations than others.

One of my work leaders has a saying that I think provides good insight in to how design/analysis should be conducted: "All models are wrong. Some models are useful." The complexities of the physics involved in most of the systems are well beyond high school, undergraduate, and even some graduate courses. But there are simplified models that have the capability of giving you a ballpark estimate of "will this work?"

I often tell my kids to "Do the math. If it works in theory, it might work in reality. If it doesn't work in theory, it probably won't work in reality"
Before some of you jump on this and provide counter-examples about things that work even though theory says they shouldn't, know that this is useful as sorting tool. It helps in the decision making process.

Running numbers using college level mechanics and playing with the numbers allowed my team to come up with a "perfect" elastic counterbalance for a rotating arm. In implementation, it wasn't exact, but it worked well enough to make a system which was effective for the game challenge that year. Don't forget the I in FIRST. Showing my students what was possible with math inspired some of them to learn about it when they got to college.

Relating this back to the original post, If you pulled out the formula for torsional strength, plugged in the numbers and found out you needed 30 ft-lbs, but your system "theoretically" could only take 10 ft-lbs, then you go explore other options, rather than waste time doing the experiment.