 Now it's good to have an understanding of constructing a mathematical model for a physical problem and we've looked at oscillations now and how to use second-order ordinary differential equations to solve for some of these systems. We had the spring, we have the pendulum and we have an LC circuit. But we've got to learn a bit more from that and let's have a look at this. So yeah, I've constructed our simple harmonic motion. In other words, we just had x or x double prime plus k over mx equal to zero. So it's just free harmonic, simple harmonic motion. And we saw that we got an answer that suggested to us that it was c sub one times the cosine of omega t plus c sub two times the sine of omega t. Now we remember that we said omega squared equal to k over m, in other words, omega was k over, a square root of k over m. So that's what I've done here. Now I'm using this program so I can't use the independent, maybe I can, but here I've just used it as x, but remember this is x of t now written as the f of x. So x is now my independent variable here, which you should just read with the problems that we've done before as t. My a is now my c sub one and b is my c sub two. I still have omega t now written as square root of k over m times x and square root of k over m times x for the co-bother sign and the cosine. But just if you look at the work that we've done up till now, it's this t that I've changed for an x and then the c sub one as a and c sub two as b. So there we have it and if we look at it, there is our sine curve, it's not really a sine curve for the values that I've put in now, but let's just move that up a bit. So we're just going to look at this as our time axis now and this is the position on the y axis where we're going to find this particle of our say on the spring mass system. So I've set k equal to one, so our spring constant here is one and m, now look what happens when I change the mass. As I change the mass, you can see that the period and the frequency will change. The period and the frequency will change, that's what the mass does. For instance, I increase the mass, say for instance, I hung a larger mass on my spring. It means that the period of oscillation is going to get longer and longer and longer and longer. That is what happens. If I use a different spring, I see the opposite as the spring constant goes up, my period is going to get shorter. So if you play around with this, you can nearly start to see. You're thinking your mind is mass hanging on a spring, you pull it down and you let it go and it goes up and down as the mass increases, well think of this one then. I'm just hanging a mass from a spring and I'm putting it to the side and letting it swing back and forth, a pendulum. So as the mass gets larger, what is going to happen? Well, the period is going to get longer. The frequency is going to get shorter, a smaller, less oscillations per second as the mass increases. Now I can rewrite this in another form. Let's have a look at this. Remember our alternate form, our alternate form said the x of t, again I'm using x now as my independent variable, equals now c sub 1 squared and c sub 2 squared, I forgot here, which gives me the amplitude. Remember it was a times the sine of omega t plus phi. Remember with alternate form we got phi as the arc tangent of c sub 1 over c sub 2. Omega is still the square root of k over m and x. Now look what happens if I, let's go back to the graph, just to show you that these two are equal, let me put this one on. It's now, it's in purple. I wonder, I've forgotten how to change the color on this. Let's go there, let's change that color to orange. Now look at that, I've got both the purple and the blue and the orange and they're exactly the same, they are exactly the same curve. So this alternate form is exactly the same. Let's make life a lot more interesting though, let's shut these two off. Let's put them on, I just want you to make sure to look at them, they stay the same as the mass increases, the period gets longer, the frequency gets smaller, but let me show you that. Let me show you that, let's put these two graphs off and let's just concentrate on the period which is 2 pi over omega and the frequency which is omega over 2 pi because frequency is just 1 over the period. Let's just look at the period, I set you now look closely as I increase the mass, look at the period it goes up, the frequency gets longer as I increase the mass, period goes up. Now for the frequency we just have to increase the scale a bit otherwise we're not going to see much difference. As I change the mass the frequency comes, the frequency gets lower, less per second, the frequency goes up. What's not so intuitive here though is that the amplitude, remember if I, what would happen, I have the same mass, let's leave the mass at 3.18, I leave the spring constant at 1, spring constant, and what would happen to the amplitude, in other words I take that same mass and I just pull it slightly away from the equilibrium as it hangs down under the force of gravity, it will swing back and forth, if I pull it slightly further, my amplitude now it will swing back and forth a bit further, so the amplitude changes, and remember the amplitude is this a squared plus b squared, c sub 1 squared plus c sub 2 squared. So what would happen if I change the amplitude, will the frequency or the period change at all? Of course no it won't, I can do anything to this, to the amplitude, so if I have a mass on a string, it's the same mass, no matter how far I pull it out, even if it had to then move a longer way as the amplitude gets bigger, the frequency and the period, let's just change back to the, so you can see the period as well, look at that period, as I change, as I change the amplitude by letting c sub 1 and c sub 2 or a squared plus b squared change, I'm changing values of a, I'm changing values of b, nothing happens to the period. So that's the same mass I say again and I pull it, now remember we are using small angle approximation, so you can't have it up by 90 degrees a minute ago. Small angle approximation doesn't matter how far I pull it out, what I do to the amplitude, the period of oscillation and the frequency of oscillation will stay a constant and that is set by the mass, if I change the mass, now I'm changing frequency and period, the frequency, orange line, you don't see it move because we're not zoomed in enough, you have to look really close, so many radians per second, I'm just changing mass, of course if I change the spring constant it's the same, it's the same thing that happens, but nothing will happen if I change the amplitude. So we can use these equations to give us some answer as to what is going to happen, it will predict what will happen in the real world, so take a mass, but the mass has got to be a lot more than the mass of the spring, so the mass or the string that you hang it from, the mass has got to be more of the object more than the mass of the string, the mass of the string has got to be negligible and pull it out at different amounts and you'll see it'll move faster because it now has to cover the same distance in the same time, but try to time it, the frequency and the period will stay exactly the same.