On Monday, I posted a short video and asked about the underlying physics. Here’s the clip again, showing SteelyKid and then me going down a slide made up of a whole bunch of rollers at a local playground:
The notable thing about this is that SteelyKid takes a much, much longer time to get down the slide than I do. This is very different than an ordinary smooth slide, where elementary physics says we should go down the slide at the same rate, and empirically I tend to be a little slower than she is.
So what’s the difference?
First of all, let’s be a little more quantitative about this. Here’s a graph of the distance moved by each of us as a function of time, obtained from the Tracker video analysis program (stealing Rhett’s shtick, here):
The black points are the position of SteelyKid’s feet along the slide relative to a fairly arbitrary starting position, which the red points show the position of my feet relative to my starting position. (I used feet because they’re easier to mark than the center of mass, and neither of us really changes our orientation during the sliding.) The lines are fits to the data.
As you can see, I’m clearly accelerating as I go down the slide, while SteelyKid is accelerating only a tiny bit (the fits give accelerations of 1.32 +/- 0.03 m/s/s and 0.065 +/- 0.008 m/s/s, respectively). What causes this?
A lot of people pointed to the way SteelyKid’s feet hit each of the rollers as a possible cause, but I don’t think that’s the dominant effect. It probably does play a role, but her acceleration would be smaller than mine even without that.
The issue here is best identified by SimonW in comment #10. The rollers making up the slide have some mass, and after we pass each roller, it continues spinning– in fact, it takes several seconds for them to stop afterwards. This means that the rollers have acquired some kinetic energy, which has to have come from the potential energy SteelyKid and I had at the start.
On an ideal smooth slide, of the sort you might encounter on a physics exam, the potential energy we start with is completely converted to kinetic energy. Both gravitational potential energy and kinetic energy increase with the mass in the same way, though, so the final speed does not depend on the mass. SteelyKid and I each end up moving at exactly the same speed. This is true even if the slide has some friction, at least in the standard intro physics assumption that friction only depends on the force between the sliding surfaces, and not the size and shape of the objects, because the frictional force also increases with the mass in the same way. On a slide with friction, some of the initial potential energy gets converted to heat energy warming up the slide and the slider a tiny bit, but this does not have any effect on the final speed. The same fraction of the initial energy ends up as kinetic energy for both of us, which means we have the same speed.
(The assumption that friction only depends on the mass is not a great one, and as many commenters noted, my surface area is much greater than SteelyKid’s, so friction has a bigger effect on me, which is why empirically, I tend to be slower than she is on a smooth slide.)
The roller slide, though, introduces another energy sink, one that does not depend on the slider’s mass in any simple way. That means that the fraction of the initial energy that ends up as kinetic energy of the slider is different for the two of us. The rate of spin of the rollers will be tied to the speed of the slider (assuming we roll without slipping), making the exact numbers a little tricky to figure out, but in the end, I keep more of my initial energy as kinetic energy than she does.
(You could try to explain the whole thing with forces, but it’s even messier, because you need to think in terms of the torque on the rollers. Energy is the cleanest way to think about it.)
Can we do anything quantitative with this? Sadly, not without more information than I have (the roller diameter and mass would be useful). I could probably get that information, but I don’t really have the time for it at the moment.
(What I really ought to do is go back there with a camera on a tripod and roll a bunch of different masses down the slide on a sled of some kind (so the surface area is constant). Given that data, I bet we could come up with something quantitative. But again, I have finite time available. I bet this could be turned into a great class exercise, but I really don’t want to explain this to the elementary school where the slide is located…)
Anyway, that’s my explanation of the roller slide mystery. Further elaborations or arguments with my phrasing are welcome in the comments.