Atom Bombs and Dimensional Analysis - Sixty Symbols [EXPLOSION] - The symbol for this video is this, which represents the abstract dimension. So what I'm going to be talking about are the dimensions of physical quantity and why they're used. - Life magazine published photographs of this first explosion-- a big magazine at the time, with a wonderful spread. And the photograph showed the blast from the bomb increasing with time. The photo showed the radius of the cloud. And there was a scale on there in 100 meters. So from that photograph, you could estimate the diameter of the blast. Maybe they didn't realize that they were giving away useful information. Because it seemed as though nobody could work out how big the bomb was from what was seen. But physicists knew lots of cunning techniques to do this. And one of them, GI Taylor, was able to estimate that it was a 20 kiloton bomb. That is equivalent to not 1 kilogram of TNT, but 20,000 tons of TNT. So this is a really big bomb. And it's quite an accurate estimate, it turned out, of the power of the bomb. So they let out this military secret, sufficient information, that somebody was clever enough to work out how much the power of the bomb was And he probably was followed around by the CIA and MI5, thinking that he was a spy. But he did it using basic physical principles. - So what did he do with his findings then? Did he tell people? - No, he did something worse than that. He published it in the proceedings of the Royal Society of London. So it was out in print, for everybody to see. I guess physicists are slightly other worldly in some respects and didn't realize how they could be persecuted for being so slapdash with security. A few months, later President Truman had to release the information. He was up by about 10%, which is remarkable. But then he was a very, very astute person working with hydrodynamics. How did he do it? Well he-- there is a basic idea, when you are doing measurements of every physical quantity, such as mass, that you put it on a scale, such as this scale, which shows you the mass of a few kilograms. And you pull it down, and the spring brings it up. And from reading this, you can get the mass. This is in kilograms or grams. But there is an abstract quantity, which is the mass. And if I put the brackets around this, this no longer is in units. It's how the physics will think, in an abstract way, about what mass is. And this takes it away from any sense of units. So it's not in kilograms. It's not in ounces. It's not in pounds, or whatever. You can do the same thing with length. You can have a meter ruler-- this is a new plastic one, going from 0 to 100 centimeters. Or you can have a foot ruler, or whatever. But the concept of length can be represented with a little bracket round it as length. Now timing is more difficult. But I imagine in Roman times, I had candles like this with little lines on it. So I can have another concept which is time, and I'll put that in brackets. So this is the abstract quantity of time-- not in seconds, years, months, days. This is a thermometer-- not an ordinary thermometer that your mother would shove in your armpit or elsewhere. This is one that goes from 100 degrees centigrade down to 0 degrees centigrade. And now I'm at a loss, because I've used t for time. So I'm going to use theta for temperature. There's one last one, which is shown over here. It's an electrical current. As I move the current up, I can get it to swing around to whatever values because there's a moving coil in there that measures the current. So these are the five basic units, and that's going to be current. Some people use charge, but I prefer current. In units, this will be a meter. This will be a second. This would be a Kelvin. This would be a kilogram. And that would be an ampere. Those are the five basic units. But forget about units. It's nothing to do with units. - Taylor invented a law of physics based on the density of the air, the radius, and the time, and the energy of the bomb-- an equation involving those four parameters. And he put it together in such a way that this law was just a pure number independent of any of these units. And as a result of that, he could write down an equation for the energy with an unknown constant. But he knew what that constant was, because he knew, with ordinary bombs-- tiny little bombs that were in shells and dropped from planes-- what the blast radius would be. So he knew what that number was, and then he scaled it up to a larger scale and then estimated precisely what the tonnage was. - The way of combining these together is called dimensional analysis. And it's a particular technique, mathematical technique, which gives you the first rush at any law of physics that you are unearthing. I think he showed that military intelligence was a bit of an oxymoron. - Fair enough. Fair enough. Very good.