Saturday, June 22, 2013

Mobile physics

About a year ago Gabi gave me some vintage flying dinosaur toys. Since then, I've been meaning to do something with those. They are not sufficiently nice as to display them by themselves, but they look really good when they are together, that's why we said "let's build a mobile!". It's been more of a child-bedroom mobile than an Alexander Calder's, but hey, it's our first time.

We only have 4 dinosaurs, which is actually the complete collection. We got some wood sticks from the dolar store, and thread.

The objects (dinosaurs) are suspended in a 10 cm string, and the lower level suspends from the double of that (20 cm). Those distances can change depending on the size of the object you hang, the length of the string doesn't affect the balancing, but it's important that the objects don't bump into each other. The rod length can change as wished, that won't affect the balance either (it's only the ratio between each side of the lever what matters, see below.)

It's really great how many configurations such a simple object has, and more importantly, governed by chaotic dynamics!

While I was building the craft, I couldn't help thinking about the physics it envolves. It is a quite simple problem, very easy to model following Archimedes "Give me a place to stand, and I shall move the Earth with it".

We enumerate each level of the mobile as n, the mass of the rod is M, the length of the rod is L, and the mass of the object is m. Here we consider that the strings are massless and that each rod and each object have the same weight.

The small side of the rod (distance from the rod tip to the string that it hangs from) Ls of the nth level will have length:

Therefore it behaves as 1/n. Below there is a Ls/L plot and a scheme of an ideal m=M mobile plotted using Python.

That last scheme made me want to build a very minimal mobile, exactly the same as this one.

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