Under the Ruler Faster than the Ruler

19 07 2009

I was just talking to Michael Weisberg, who is also visiting ANU currently, and he pointed me to this video showing a counterintuitive physics demonstration. I had seen the video before, so we started discussing how it might work. He pointed me to the explanation videos that the author of that video made, but they don’t really clarify things very much. When we tried to work it out ourselves, we came to the conclusion that it had to be impossible (unless there was slippage between the wheels or one of the surfaces) – until I realized one feature of the cart that I hadn’t noticed before.

It turns out that the relevant feature of the setup is the fact that the spools that serve as the bottom wheels have a different radius at the point of contact with the upper wheel than they do at the point of contact with the ground. The speed of the cart relative to the ground has to be the radius of the spool times the angular velocity of the spool. If the radius at the point of contact with the big wheel was the same, then the radius times angular velocity of the spool would have to be the same as the radius times angular velocity of the wheel, which in turn must equal the speed of the cart relative to the ruler. So if the two radii were equal, then the cart would have to have the same speed with respect to both the ground and the ruler – which means that the ruler couldn’t move relative to the ground without slippage.

Some quick calculations will show that the ratio of the inner and outer radius of the spool has to be the same as the ratio of the speed of the cart with respect to the ruler and with respect to the ground. If we treat the ground as fixed, then we can calculate that the speed of the cart must be the speed of the ruler times 1 minus this ratio. The cart moves about twice as fast as the ruler because the outer radius of the spool is about twice the radius of the part of the spool that contacts the big wheel. We could of course get whatever ratios of speeds we want (including ones going the other direction) if we changed the ratio of the two radii, or used something like a spool for the other wheel, or otherwise made the wheels behave like multiple different-sized cogs on a single axis, rather than as simple cylinders.