EDIT: SEE THE NEXT POST THAT FIXES A MAJOR MISTAKE IN THIS ONE
I’ve been doing a lot of modeling of beads on wires lately, but today I discovered something that really surprised me. The surprise came when I found a bead/wire system that seemed to violate conservation of energy. Now, it turns out that I was just thinking about it wrong, but still, it’s interesting. Here’s the animation that got me thinking:
Double pendulum on a roller coaster
Take a look at how high the whole system gets compared with the original height. It sure looks like it ends up a little higher. Where does that energy come from? Why I was wrong: I think I had it in my head that the track which provides the constraint force never does any work. That’s certainly true for the case of a single bead on a wire as the normal force is always perpendicular to the direction of travel, hence no work. But in this case that’s not true! Instead, the normal force is at times not perfectly perpendicular to the direction of travel of the center of mass of the system. You can see that in this annotated version of the animation where I track the center of mass in green and show the initial height as a dotted line:
Annotated version with the center of mass in green. The green dotted line is at the original height of the center of mass.
Cool, huh? Your thoughts? Here are some starters for you:
- Thanks, this is very cool. Can you find the same thing with a single pendulum on a track?
- This is old news. Designers of hang-under coasters take this into account all the time.
- Can you post the Mathematica code? (yes: here you go)
- Can you figure out the normal force of the track using Lagrange multipliers? (note: I can’t figure out a way to parametrize the loop as a constraint that looks like g(x,y)=0)
- Why do you always post animated gifs. Don’t you know people hate those?
- I think your code is wrong. There’s no way the center of mass can get up higher than it starts.
SuperFly Physics 
The total energy plot worries me because it’s piecewise constant. I wonder if this is an artifact of interpolation. Mathematica’s default interpolation method (Hermite polys) doesn’t guarantee continuous derivatives at the joins.
Try:
{xint,yint} = Interpolation[#, Method -> “Spline”] & /@ Transpose[pts]
The “Matter and Interactions” approach to analyzing the behavior might also provide some insight. Modeling the system as a point particle at the CM, the constraint force does work on this system and changes the value of KE+PE for exactly the reason you give. But in this model, the KE is only CM translation; point particles can’t rotate, so the rotational KE is just “internal energy” and therefore hidden.
Looking at the “real system” with all three masses, M&I would say that the constraint force does no work, because the differential displacement at the point of application is always perpendicular to the force. The constraint force in this model isn’t being applied to the CM.
This Mathematica notebook is a lot of fun, btw.
Bruce Sherwood (of M&I fame) made basically the same argument over on google plus. I think the kinks in the interpolation could explain it, but is that then an explanation or an integration mistake?
I would think an integration mistake, judging by the stepped KE+PE plot. Smooth out the interpolation kinks, and then KE+PE is constant within integration precision. It seems like the discontinuous derivatives are imparting “spurious” delta function constraint forces.
I did your suggesting with the Spline and it certainly does stay below the line. I’m curious, though, whether Mathematica is modeling the kinks correctly. It reminds me of how you get yourself going on a swing without touching the ground
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Andy, this pair of gifs really got my office going this afternoon. It was only after I read the comments here that I saw your note about adjusting the program caused the CG to stay below the line. You should probably add that note into the article as an EDIT FYI at the end.
Done