Balancing a Ruler – Numberphile

Balancing a Ruler – Numberphile


Here’s a ruler. Where’s the center of gravity? Where presumably it’s in the middle, and that’s certainly true. If I put my finger in the middle, and somewhere there, Oops!, It balances. So that’s the center of gravity. But suppose I attach some weight at one end, wher-where is it? Well it’s not so easy to guess right? But there’s a very nice way of finding the center of gravity, which i’ll start by doing it without weight. You simply put two fingers to support the object, and then you bring them toward each other, and automatically, you see that one stops and one starts and one stops-and then they meet automatically at the center of gravity. And I can even do this with my eyes closed ( Opens mouth) This works in general if i attach, for example a mass at one end. I don’t know where the center of gravity is but, my fingers automatically find the center of gravity. (Opens mouth) And, if I put a hefty mass at the other end, well the center of gravity will be really close to that end, and indeed, it finds the center of gravity over here. And if I put both of them at the same time, where is it? Well aaaa-it automatically finds the center of gravity. Again I am not deliberately controlling the movement of my fingers, I’m just bringing slowly towards each other. And automatically one of them starts sliding and the other stops and the other starts sliding and the one stops and they meet always at the center. So how does it work? It works because of a really simple but, widely applicable principle. to start sliding, there is something opposing the slide and that’s called friction It turns out that the friction is proportional to increases with the force that is pressing the two bodies against each-against each other but also, is proportional to some coefficient. The fact of matter, it’s very very widely established observational fact. When you are sliding, the friction is slightly smaller than when you are stopped and wanted to start sliding. We talk about the difference between the dynamic friction ( that’s the friction that’s acting when the bodies already sliding) vs static friction (when the two bodies are stopped against each other and then they wanted to start sliding) Static friction is a little larger than dynamic friction. Okay. Suppose that I start bringing those two fingers toward each other. There will be some random error in the beginning. So first this finger started moving for some reason. Ok. When the-this finger is closer to the center of gravity than this finger, this finger is taking more weight over the ruler than this finger, so there is more friction there. And so because there is friction on this finger than on this finger, naturally the finger that slides is this one. But you see, once this finger starts sliding, it’s sliding with dynamic friction, whereas this is stopped at static friction. So, in principle you see, when these ar-equal distances from the center of-center of gravity. They take equal weights so they should have equal frictions, but thats not true because this is already sliding, so it’s taking the sliding friction (dynamic friction) whereas this is taking static friction. So, this one can keep sliding although it’s taking more and more weight because it’s friction is a little less than this one, so it overshoots towards the center, and when it’s really close, than the other one starts moving and so on. That’s why they alternate. Stick-slip mechanism this is called-slip-stick slip-stick and then they come toward each other, so if you take a weighted version, well for a long time this left finger slides because, left as seen from you, slides because it’s of course it’s far from the center of gravity so it has less weight on it, so it has less friction But, it overshoots and then the other one starts sliding, overshoots overshoots overshoots and then they finally meet in the center. So this mechanism is using the slight difference between the dynamic and static friction and the fact that, well, when you come at equal distances from the center of gravity the one that’s already sliding can overshoot, and then once that stops it stays there for a while and then only when the other sliding finger has stopped, can (the) this one start sliding again and then overshoot and then come to us and that’s how they come to a stand Brady: Is there accepted number of times there will be a slip and a stick, like over the course of thirty centimeters?(tha-te) that seems arbitrary. Tadashi: that is very very-that is an excellent question. I think its umm case by case, and you can do a very very precise calculation using a simple model of stickslip and so forth, but the fact of the matter that it-a is a very unstable process. I mean depending on tiny tiny irregularities of the surface and so on and the number seems to vary quite a lot So, I think we have to resort to just experiments and theoretical calculations does give some picture but I don’t think it gives really as accurate um you know prediction as the numbers suggest paper clips is included in our band but not between themselves. Let’s finish with something that is work in progress. So far, we have been linking paper clips together and sometimes we refer to this as addition

31 thoughts on “Balancing a Ruler – Numberphile

  1. You don’t find the center of gravity by this procedure. If you cut the weighted ruler where your fingers claim the center of gravity to be in two halves and weigh each half, you’ll see that the piece with the weight attached to it weighs more. That’s because the other piece is longer and thus leverage takes effect. Even though its mass is less, it can balance the heavier half because of its longer lever. So you don’t find the point where m_a=m_b, but rather the point where m_a*s_a=m_b*s_b (where a and b are the two halves, m their masses, and s their lengths).

  2. I really need more convincing for this one. I just do not accept that they MUST meet at the center of gravity. Who's to say that in the process of "overshooting" they were both very close to the center of gravity, and one overshoots past the center of gravity. This model is suggesting that it is IMPOSSIBLE for my finger to ever be past the center of gravity, yet it WILL land on the center of gravity. Which in my eyes suggests an infinite number of "slip and slides" nearing the end of the process. This obviously doesn't happen, so there must be some issues with the model.

  3. Knowing the mass of the ruler and Torque = Force * Lever arm, you can even calculate the mass/masses at the end(s) of the ruler.

  4. So if you have a ruler that isn't made out of the same material throughout, it won't work because you have different frictions?
    Like half rubber half plastic.

  5. Tadashi still going strong after getting booted off United airlines lol
    (I know its not him, but he looks like him :p)
    Also I think he should be referred to as Dr. since he has a PhD, had to look this up because I thought he was just a really smart toy enthusiast

  6. Pof. Tadashi is great. the topics he shows here are so interesting and the way he explains them are awesome. One of the best of this channel

  7. I literally stopped watching and tried it, went to go spook my family and they were awed.

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