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:
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:
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.