## Three rollers and a ruler

I've seen a lot of discussion recently (e.g. BoingBoing, xkcd) on the Internet about the possibility of a vehicle moving directly downwind faster than the wind. I don't want to go into that debate, but what did caught my attention was the following cute video that demonstrates the behavior of a cart constructed out of three rollers under a moving ruler:

*(Click to watch Under the ruler faster than the ruler video)*

Theatrical skills of the author aside, what I found interesting is that very few people had any doubts about the validity of the experiment and the explanation given in this video. When I watched it for the first time, I was pretty sure the little plush monkey got it right. And after some back-of-the-envelope calculations I still thought he was tricked. The cart shouldn't be able to move at all.

So I did an experiment of my own and it just confirmed my thoughts. The experiment is easily duplicated and I encourage you to try it yourself.

However, just before posting my notes I realized there is one minor, but important difference between the geometry of the cart I analyzed and the geometry used in the video. I'm now confident that the video is genuine and I'll be posting the corrected calculations after I get back from Berlin.

Meanwhile, below is my original analytical solution followed by a video recording of my version of the same experiment (minus the furry spectators). Mind that it's still correct, it just provides an answer to a slightly different question (see if you can spot the difference).

Let's start with a simpler case of a single roller on a flat surface:

Here v_{c} is the velocity of the center of the roller relative to the ground, while v_{r1} and v_{r2} are the velocities relative to the center at different locations along the roller's surface. Remember, the magnitude of v_{r} is constant around the circumference while it's direction changes around the circumference and is always tangential to the surface.

Obviously, the velocity of the roller's surface at the point where it touches the ground must be 0 relative to the ground or the roller would be slipping. So for that point the following equation is true (subtracting magnitudes, since vectors are parallel and pointed in opposite directions):

Now that we know v_{r}, we can calculate the velocity of the top point of the roller (adding magnitudes, since vectors are again parallel, but pointed in the same direction):

So, in this case the top point is moving in the same direction as the center and at twice its speed relative to the ground. And this is of course also the speed of any non-slipping ruler that rests on the top of the roller.

This result is expected: if you're moving a large rock by rolling it on tree trunks, you put trunks in front of the rock and pick them up behind it.

Ok, so let's now go on to the cart. The situation is very similar to the previous example. The centers of all rollers are moving with the same velocity, v_{c}. At the points where the top roller touches the bottom two rollers surfaces must have identical velocities as there is no slipping. From here it follows that the magnitude of v_{r} is equal for all rollers.

Again, we can write equations for the points where the cart touches to top and the bottom surfaces:

And therefore:

So the ruler at the top can not move relative to the ground as long as it is not slipping. The crucial difference here was that the second roller rotates in the opposite direction to the top one. This changed the sign in the second equation, since vectors v_{c} and v_{r} were now pointed in the same direction at the bottom.

As you can see, the radii of the rollers don't even come into play in this calculation. So the final result is identical with arbitrary roller dimensions.

The conclusion therefore is that the ruler is either stationary in respect to the ground or two surfaces are slipping somewhere. It's impossible to move the cart by applying a horizontal force only to the ruler since the bottom rollers will apply exactly the same torque to the top roller, but in the opposite direction.

I've made a series of simple experiments that confirm the theory above. You can see them on video below:

*(Click to watch Experiments with a three-roller cart video)*

You can see that moving the ruler in the 4th experiment didn't move the cart - it only caused the ruler to slip along the top wheel.

The only way to move the cart is to apply the force to it directly as in the 3rd experiment, or as the last experiment in the video shows, by resting the ruler on the cart at an angle, so that the force of the ruler is no longer parallel to the force of the ground. The force and torque diagram in that case is left as an exercise for the reader.

Again, you don't have to believe everything I said, but do try it yourself if you have any doubts. Experiments are fun and this one really just takes some cardboard and a couple of minutes (or seconds if you have Legos handy).