## Local measurements: the Rosslyn escalator

On my way to (and from) China last month I passed through Rosslyn Metro station, which has the distinction of hosting the third longest escalator in the world. How long is it?

So long, you could… okay, I won’t bore you. But seriously though: suppose I wanted to actually measure how long the escalator is? The obvious method is to pull out a measuring tape and run it along the length of the escalator, but that’s hard without a helper.

What I need is a measurement method which is local: roughly speaking, something I can do using only objects within reach. Fortunately, physics provides such a method. I can (and did) time how long it takes to ride the escalator from one end to the other, then wait at the bottom for that much time and see how many steps passed by. Then the length of the escalator is just

$$\text{length} = \text{length per step}\times\frac{\text{steps}}{\text{second}}\times\text{time of ride}$$

On my way down, I timed the ride at 140 seconds. I didn’t feel like waiting a full 2+ minutes to count steps (people would have thought it kind of weird, and I was running late as it was), but I did pause for 20 seconds and counted 21 steps slipping under the plate at the bottom of the escalator. And finally, I would estimate that each step corresponds to a length of about $$\SI{32}{cm}$$ going diagonally downward, or $$\SI{20}{cm}$$ of height. That makes the length of the escalator

$$\SI{32}{cm}\times \frac{\SI{21}{steps}}{\SI{20}{s}}\times \SI{140}{s} = \SI{47}{m}$$

or its height

$$\SI{32}{cm}\times \frac{\SI{21}{steps}}{\SI{20}{s}}\times \SI{140}{s} = \SI{29}{m}$$

That’s $$\SI{96.5}{ft.}$$, which is quite close to the presumably correct value of $$\SI{97}{ft.}$$ (from Wikipedia). Hooray, it works!

This might seem like I’m making a big deal out of nothing, and in this case I am, but the idea of a local measurement is very important in relativity, where you start out without even having a consistent definition of distance to use. In those situations, local measurements represent what each individual person (or really, observer) measures, in a way that is meaningful without having a globally consistent distance.