 Okay, we're now going to work a second example involving the Buckingham Pi method. And for this one, we're going to take a problem that is not within fluid mechanics, but actually within physics. And what we're going to do is examine the period of a pendulum. So here's a schematic of a pendulum. The mass of the bob is weight, or mass, m, I should say. The length of the pendulum is l, and it is at an angle theta with respect to the vertical on the gravity vector is g. And so what we're asked to do is to write an expression for the period of the swing using Buckingham Pi analysis. And here the period of the swing would be the time for a full cycle, and it is given T. So as we go through our analysis, we'll go through the normal steps that we do. And we're first going to assume that the period of the swing of the pendulum is going to be dependent upon the mass of the bob, the length of the cable that is connecting the bob to some ceiling above, the gravitational constant, and the angle that the cable makes with respect to the vertical. So we can see we have n equals 5. And what I'm now going to do is write out the dimensions of all of the main parameters that we have involved, or the main variables. And for theta, the angle, that is dimensionless. So what we're going to do, we're going to choose three of our different parameters for our base group, and we don't want to include T. So what we will do is we will choose m, l, and g. And so according to our analysis technique, we have m is equal to r, which is equal to 3. So with that, we can expect to have n minus m, or n minus r is equal to 2. We should have two pi groups coming out of this analysis. So let's begin with pi 1. So going through, starting with our mass term, we find a is equal to 0, looking at dimension of length. b plus c is equal to 0. And we have to move on to time in order to find c. So we get c is 1 half, and then we can go back up and determine b to be minus 1 half. So those are the coefficients within our pi 1 term, the first non-dimensional group that we're deriving. And so what we find is that pi 1 can be expressed in the following manner. So we have g in the numerator, l in the denominator, and this is to the 1 half. So we have a square root symbol. And then checking the dimensions of that, we have units of time, length time minus 2 divided by length, and then the square root of that. So length goes with length, t to the minus 2 gives us t to the minus 1, t to the minus 1 times t, that gives us t over t, which equals 1. OK, so that is good. So that's pi 1. And what I'll do is I will box that off. So that's our first pi variable. Now let's move on to the second one. And for constructing pi 2, again we have m, l, and g multiplied now by theta, introducing the dimensions of each of these. OK, so again let's go through our process, beginning with the mass. We have an a, and that is equal to a 0. And looking at length on the left hand side for length, we have b plus c, and on the right hand side that's equal to 0. And then finally for time on the left hand side, we have a minus 2c, and that's balanced by 0 on the right. So what are we getting here? We're getting a equals 0, c equals 0, and b equals 0. So this is an interesting one. What we end up with is pi 2 is just equal to theta, and not 0 theta, there we go. And that is dimensionless because we said it was, so that checks out. So what we end up with here, going back to our original expression, pi 1 is equal to g of pi 2. So if we look at this and substitute in our values now the pi groups that we've determined, for pi 1 we have t square root of g over l, and on the right hand side we have g of theta. So what I'm going to do, I'm going to rearrange that now and isolate for the period t, and when we do that, we get an l over g take a square root g of theta. So that becomes the functional relationship for a pendulum. And we did all of that without using any real mathematical analysis. So what that does, it highlights the power of the Buckingham-Pi method. You can just go through using dimensional arguments for the units, and come out with representations. And it does turn out that the period of a pendulum is related to 2 pi l over g square root. And then there's a g of theta here term that would be, if you have larger angles, would come in. So if you have large angle, that would be kind of a correction term. But it's pretty good. When you consider that all we did is we used our dimensional arguments and we came up with a very, very good representation for the actual period of a pendulum. So what we're going to do now, we're going to take a look at a video clip here of a world famous pendulum. And then this was a pendulum experiment that was conducted starting back in 1851. And the person who conducted this experiment was a person who was born in France and his name was Jean Bernard Lyon Foucault. And he lived from 1819 to 1868, I think he was 49 when he passed away. But he started playing around with pendulums, actually he invented a lot of things, self-taught, he invented the gyroscope, but he started playing around with pendulums in his basement in January 1851. And what he noticed was that, and what we'll see in a moment, he first of all noticed that the period of the pendulum was expressed in the relationship that we have derived. But there was a factor of 2pi in there, so that was something that was already known Foucault did not come up with that. But what he did come up with is what we're going to see in this next video clip. So let's take a look at that. And this video clip was taken at my favorite museum in Paris, the museum Arte Metier. And I've sent it before, if you ever get a chance to go to Paris, by all means do go there. So what we're seeing is Foucault's pendulum in the museum Arte Metier. And here we're measuring the period, that's a start, so 45.23 over 30 seconds, and that's the end of the period, 54 seconds, 13, this is using my video frames that I made when we talked about it. So the period of this is around 8.7 seconds. And we're going to use that in a moment when we try to estimate the length of the pendulum. Now in February 3rd, 1851, they put out an invitation saying, you are invited to come and see the earth rotating. And the pendulum was actually started by Napoleon, Napoleon Bonaparte. And what he did is he pulled it back and he started and you can see the presence of the earth rotating, watch this. So they put little pieces of brass in the way of the pendulum and as the earth rotates the pendulum doesn't change orientation and there you go, it hit the piece of brass and there you see evidence that the earth does indeed rotate. And that was quite the thing in the early 1850s in Paris and after that there were pendulums all over the world repeating that experiment. So let's quickly take a look at the pendulum in the museum, Artset Mettier. Now it turns out that the Paris Observatory experiment, the one that Napoleon Bonaparte started, that pendulum had a length of 11 meters and with that we can calculate that the period was 6.65 seconds and the one in Artset Mettier that we just looked at, we said that the period was 8.667 seconds and so we can look at our equation 8.667 equals 2 pi L over G and the interesting thing here in this equation, the mass of the bob is not in there. One of the fascinating things about a pendulum mass is not in the final equation, but when you do this calculation you find out that in Artset Mettier I would estimate that the cable of the pendulum is probably on the order of 18.67 meters. I have never seen anything seeing how long it is, but anyways that gives you an example of the pendulum, some famous world history in terms of an experiment proving that the earth does indeed rotate and the power of dimensional analysis for understanding mathematical physics systems. So that concludes dimensional analysis, we are now going to move into other aspects of dimensional analysis and fluid mechanics, that concludes Buckingham pie I should say, and we are now going to look at similitude, similarity theories and very important non-dimensional numbers that we have and use within fluid mechanics.