On the other hand, I think this is an outstanding problem
if you are willing to let students come up with a model, discuss creative ways to check the model and the reasons it doesn’t work, and lead them through something like what Frank did.

Frank did not solve the fan problem with numerical analysis, he employed numerical analysis in the solution. He started with physics, a kinematics problem involving rotational inertia and an opposing torque. He further refined the opposing torque as a combination of a constant element (the motor/bearings) and a non constant element (the air resistance). He chose the v^2 version of the drag equation because in problems such as fan blades and air that makes sense (there are other drag equations depending on the circumstances). Through all this he has been using physics and algebra to mathematically (quantitively) rationalize and describe the kinematics of the fan, with a guiding purpose, to determine how long it will spin after it is turned off. Finally, after the physics and algebra he applies calculus (because of the time element) and obtains a differential equation, a (small) cliff that he circumvents using a step wise approximation, that requires him to refactor that situation algebraically into a program that must also correctly approximate the solution of the differential equation. He also employed some data analysis to determine the friction coefficients.

To answer your question, there is no “or” between numerical analysis and algebra (and calculus). They are woven together in the same cloth. I think what you are asking is whether we “solve” differential equations analytically or numerically. We solve them the same as we did 100 years ago, if a solution exists and we need that we use that and if not then we use numerical methods (which have existed for as long as differential equations existed). It also depends on the situation. This fan differential equation might be part of a bigger problem and thus there would be no purpose for a numerical result at this point. We would still have to tread through more physics, algebra and calculus and then (likely) at the end of all that numerical methods might be employed.

“However, with the use of computers and software, we can numerically solve them. If you look at the actual equations that Frank used in his program, there wasn’t any one that was too complex (most were linear or simple quadratic). Yet we ask kids to solve equations that are sometimes needlessly complex.”

There is some confusion here about the meanings of “solve” and “numerically solve”. When we solve an expression we generally mean to put it in a closed form with the dependent variable on the left of the equals. For example, to solve the expression “y – x^2 = 0” we get “y = +/-sqrt(x)”. But that isn’t a numerical solution because, except for special cases, we cannot calculate the sqrt(x) exactly, we can only approximate it using numerical methods, either by hand or with a computer. So, when a numerical result is the goal, numerical methods are involved regardless. But numerical results are not always the goal and generally they are almost never the intermediate goal. It takes a considerable amount of familiarity with the math (and in this case the physics) and the problem to know where and when the problem favors numerical or analytical methods. On top of that it also takes a considerable familiarity with numerical methods (algorithm and programming) to employ them, especially to do a simulation as Frank just did.

I hear what you are saying, but it isn’t as easy as Frank makes it appear in a post. Like watching a musician perform, it isn’t as easy as they make it look. If you analyze what Frank did you will realize that there are a number of proficiencies involved and as far as I can tell there is no easy and quick formula for teaching all of that in one fell swoop. It boils down to interest, coaching and a lot of practice. In the beginning it is unrecognizable as what Frank just did, but at some point the fruit of all of that work starts to become recognizable and then it matures and then, after even more work, it becomes Frank.

Here is a case where the complexity of the problem rules out a theoretical analysis and favors an empirical one. In order to find out what makes engineers engineers, go find successful engineers and ask them. They are generally going to tell you “not any one thing in particular, just all of it, even the stuff they didn’t like.”

PS: My point to Garcia was that without the mechanical (physics) analysis of the kinematics of the fan, Frank wouldn’t have a mathematical model at all. This was not a problem to be solved by data analysis and modeling (because there wasn’t enough data). This was a physics problem from the get go. Physics starts with how and why and derives the math from that. Regardless of its simplicity, this was a very realistic engineering problem requiring theory (physics), mathematical analysis (algebra calculus), data analysis (the coefficients) and numerical methods (the simulation).

Completing my recent comment, may be the quantity that should be used to fit the data is the “Rotational Kinetic Energy” Er=Iw^2/2. Where I is the moment of inertia (a constant value, equivalent to the mass in linear velocity) and w is the angular velocity.

That energy should be dissipated completely before the fan is stopped.

I guess that considering a constant angular acceleration from the initial time will result in a wrong value for the friction (higher), and thus a shorter stopping time will be obtained.

Friction losses from the axis of the fan would be constant in time I suppose, but drag on the blades will be dependent on the angular velocity (w^2 dependence). This will result in a large losses of energy in the beginning, and very small losses of energy at the end. This effect can be misleading about the value for the deceleration, and thus it can result in wrong answer.

It seems nobody has considered the mass distribution of the fan…

I guess it will take more time to stop it if the mass would be only in the end of the blades than if it would be distributed homogeneously. Dan, could you put for me an extra mass at the end of the blades and try again measuring the stopping time? If that is different then there would be the explanation for the large stopping time you recorded.

PS. By the way, let me tell you I am a fan of your approach to teach.

David Garcia wrote…

“Because, outside of physics teachers, who solves mechanics problems anymore today?”

I work at a company with 5000 engineers, this is exactly the stuff we do. Buildings, bridges, airplanes, cars, even iPads and iPhones are not designed hit or miss. Speaking of fans, they exist in turbo pumps, jet engines and power plants. You don’t just take an educated guess. It takes solid theory and experimentation to get these things to perform.

Teacher school should include a year of interning at companies that rely on these subjects. I realize that a full analysis of a ceiling fan looks intimidating, but that is exactly how fans are designed, albeit not the ones you buy at Lowes that are made in China.

I fit 4 parabolas to Frank’s data using: 1st six values, 2nd six values, 3rd six values, last six values. The predicted stopping times for the models are 19 sec, 30 sec, 36 sec, and 47 sec. This sort of analysis is easily accesible to students and clearly shows the danger of extrapolating with a model for a situation you do not understand very well.

It is not so hard to build a spreadsheet for Frank’s model. It is interesting to play around with the choices for the values for the two sources of friction. Constant friction alone does not produce a good fit. Using wind resistance friction alone fits the data very well but produces a model where the fan never stops.