My last post was wrong. I’m to blame. But in thinking about it and talking about it with with lots of helpful friends I ended up learning a ton. Here’s the upshot: There were kinks in the roller coaster loop that led to integration mistakes on the part of Mathematica. Thanks to a great suggestion from my friend Craig I smoothed those out:

And now the simulation animation looks like this (there’s some extra annotation that I’ll talk about below):

Note first that the system never gets above the dotted green line, which was my (mistaken) idea from the last post. This post will try to talk about what I learned about whether the normal force does any work (which was my mistaken explanation in the last post).

The track exerts a force on the red ball to keep it on the track. Gravity and the connection to the first black ball are yanking on that ball and the track does whatever it has to in order to ensure that the red ball stays on the track. My argument from the last post boils down to this: The normal force is an external force to the system of three balls. That system has a center of mass that I can pretend the external force acts on. If the center of mass is moving perpendicularly to the normal force (as would happen with just the red bead), there would be no work. But if the center of mass is moving at times slightly parallel to the normal force, then there would be some work. It turns out there’s really nothing wrong with that description. However, assuming that changes the kinetic energy of the system is wrong. What it does (as again my friend Craig suggested) is it changes the *translational* kinetic energy of the system (basically the kinetic energy of the system if you replaced it with all the mass being at the center of mass). However, the total kinetic energy of the system is both the translational and rotational kinetic energy. What I intend to discuss here is that the effect on the rotational kinetic energy due to the normal force is *exactly the opposite* of the effect on the translational kinetic energy.

First a quick plot. This shows in blue the time derivative of the translational kinetic energy of the system (subtracting out the effects of gravity) and in orange the work per unit time that the normal force does on the system:

Actually, you don’t see the orange because, to the accuracy of the thickness of the lines, the orange is completely underneath the blue.

Let’s try to understand what’s going on. Consider first the time rate of change of the kinetic energy of the system:

The derivative can come right into the sum, and the vector product rule gives us:

where I’ve used Newton’s second law in the last step. Now the normal force is only “attached” to the red bead, but that’s the one bead that’s guaranteed to be moving perpendicular to the normal force. So the contribution to the time change of kinetic energy due to the normal force is indeed zero! Hence my last post is wrong.

But what about this business with the translational kinetic energy? We tell our students all the time that they can think of all forces as acting on the center of mass. In other words, the change of momentum of the center of mass is due to the collection of all external forces. Those forces will do work if they act, at least partially, in the direction that the center of mass is traveling. In the animation above the purple arrow shows the direction that the center of mass is traveling. You can see that it doesn’t always point along the track. That means that it’s not always perpendicular to the normal force. Hence work is done on the center of mass. But that just affects the translation of that center of mass, not any rotation about it. To see the whole story, let’s redo the last calculation using a coordinate system centered on the center of mass. For those variables, I’ll use primes. First I’ll start with an expression for the kinetic energy:

Now when you do the FOIL of that dot product, two of the terms go to zero (that’s the beauty of using the center of mass, by the way) and you’re left with:

where M is the total mass of the system and V is the velocity of the center of mass. Now, let’s consider doing a time derivative of that. For the first term you’ll get exactly what I was talking about above. In other words you’ll get the dot product of the total external forces and the velocity of the center of mass.

Now here’s a trick. Let’s re-express the velocity back into the normal frame (and use Newton’s second law again) for the second term above:

Here’s where the magic happens. The normal force is only applied to the particle on the track. But it’s velocity is perpendicular to the normal force by definition. So the first term in the parenthesis yields a zero. What we’re left with is:

which is exactly the opposite of the change to the translational energy. In other words, you can either say that, yes, the normal force does some work, but it changes the translational kinetic energy by exactly an amount that is the opposite of how it changes the rotational kinetic energy, or you can just say that the normal force does no work. You decide.

Thoughts? Here are some starters for you:

- Thanks for this, I was totally at a loss for figuring out the mistakes in the last post.
- I’m glad you figured this out for yourself, just know that the rest of us knew this all along and have been laughing at you for your last post for a few days now.
- Wait, it doesn’t work!? I’ve already starting building it in my backyard!
- How did you figure out the normal force? Did you determine the accelerations of all the particles and subtract all known forces, starting with the last black dot and moving up to the red dot. Or did you use Lagrange multipliers to figure out the normal force more directly, and, if so, how did you figure out the constraint equations for the track? (yes, yes, and it’s a long but interesting story involving me jumping out of bed this morning and trying something that worked!)
- So how would you say it? Does the track do work on the system?
- How is it that you were willing to believe that the track could help you violate energy conservation? What, are you some sort of “momentum is king” kind of guy or something?