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Hi, I'm Kate. I'm a first year mathematician at Trinity College.

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Hi, I'm Fareed, I'm a first year mathematician at Trinity College. Hi, I'm in.

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I'm a teacher in applied maths there at Trinity College. OK, so we've got the dynamic sheet and the PD sheet.

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So that's year and that's you. Five, Which was some good work.

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So we're going to spend most of the time on dynamics because I think it's quite interesting.

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Yeah, let's look at some of these questions went right in in the last one was not quite right.

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Some of you don't. So this is all about it's constrained motion and stuff.

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So you've got this particle going around to a cone. Mm-Hmm. And what was the important thing about the normal reaction force?

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Person, that is normal. Yeah. It's normal to the surface.

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So, yeah, I think if I said things which are not quite right here, sir, do you mean?

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I think it's easy to see to zero. Yeah.

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So you've got this thing, which is a cone and then the reaction forces that way you define it this way.

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And is that way and that way, that's got to be two vectors.

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The end is perpendicular to and one of them is easy to reach into the page there.

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And then what do you use? Just the other one? Or what's what else is it perpendicular to?

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The Republican. Yeah, so it comes out of that direction, which is useful to think of it as like the tangent thing to the current of the.

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It's a plane, it's a plane. These are easy to direction and that direction.

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And but what you're often use for these questions is that is that the direction of the

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velocity of gold is perpendicular to end because you know that it's moving on the cone,

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then you know, the velocity must be perpendicular to the normal.

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So you use that statement a lot, and that's where you really get the fact that this force doesn't do any work.

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It's the fact that it's perpendicular to the velocity is why it doesn't do any work.

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And. So and you had to use that when you were showing that the energy was concerned.

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So the first thing that you always do in these questions is to write down this Newton's second law.

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So when you get stuck with these questions, that's always the starting point.

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So in this case, you've got gravity.

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And then you've got this normal reaction, so you would always start with that, and then you should usually be trying to write down what ah is.

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And you want to decide what coordinates it continues and in this case, in your own cylindrical coordinates, so you wouldn't do this.

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R e r Trump said E Z. And then you have to differentiate that and.

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So the first part was fine change that when you got this feature, you get zero because this doesn't have any competitive features,

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actual list of known directly and that tells you the angular momentum is conserved.

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So this. I mean, this is so this is the same as this.

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So this is not true here. That makes sense. Yes, and it is not zero.

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And we get this this one. Whenever you say so, you talked here about saying I was going to equate components in the different directions,

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and that is effectively what you're doing,

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but you should think of it more as taking a product that's taking a product is taking the component in the future direction.

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And I think that's the way that you want to try and think about these things that you always have to differentiate this.

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And then finally, expressions and I tend to just think it's a good idea to write this down straight away.

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Because you need to use them. Also, the question so this one is double minus three seconds squared.

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And then you get one overall independently of R-Squared feature dot.

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I just remember that you can watch it like that. The feature component,

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because it's helpful for them to write it like that because of that fact that it often means that this quantity angular momentum is conserved.

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So, so that bit was fine, then it should explain why the total energy is conserved from the site, as you said,

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because because the thing is smooth, therefore it doesn't do any work and that's really because it's perpendicular to the velocity.

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And I think that you both might. You just argue that that was the case.

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And then you said, I can differentiate us and show zero.

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I tend to think that there's a there's two different ways you can think about all of these problems because you can either say,

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I'm going to start by saying energy is conserved. And then I divide my equation from that, or you can say,

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I'm going to start with Newton's second law and then I can derive that the energy is conserved.

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And they're kind of equivalent, and it's almost a matter of taste as to which one you say is where you're going to say is your starting point.

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Most of these questions, I kind of referred to say, Well, I've I've started writing down in a second, or I should just use that as my.

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That's the thing I know is true. And if I'm going to derive anything else, it should come from there.

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And there's always an easy way to get from the second row to energy.

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Do you know what that is? How can I get to seeing what the energy is from that equation?

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Well, this is equal to force, which is equal to minus Steve by.

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D, r d. Well, it's a conservative force than we could, right, as the gradient of a potential.

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Mm hmm. You've done that. OK. So that's like gravity is can be thought of as the gradient of the potential energy.

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But do you know how I get to that from this equation?

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There's a there's a kind of a trick where it's not really a trick, but it was multiplied by the velocity and becomes a vector equation.

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Multiply totting. So if you don't the equation with an adult, then you go and double dot dot dot.

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There's a lot of dots in a different many different things. Minus M D is dot dot dot dot dot.

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And there is this this thing here? Is the derivative of a half an hour dot squared?

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That's always the case, does that make sense because there are dots squared is all dot dot dot.

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And if you differentiate that you get to dot out of all the law.

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And so if you remember that you can always write that is that that's quite useful.

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This, of course, is a kinetic energy. Yeah. So this is really the rate of change of the kinetic energy is all block times acceleration.

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And then this term here. So that it's just got so you can think of that as DVD of and GZ.

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So that's the major change in potential energy. And then this time we argue for zero.

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Yeah. So that's that's kind of like doing the dot of this with the velocity is a good thing to do because of the fact that we know.

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And not the velocity is zero. And often you don't often you really care what end is kind of what you are limiting, because that's the whole point.

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It's just a constraint that too that we're stuck on the cone. But that you can always derive the energy.

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This just tells us that divide of half an hour dot squared plus and GZ zero energy is conserved, which is exactly what you did just backwards.

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Then you can see. This also would tell you if if you had a friction force and if not pointed in the normal direction,

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then this would tell you what the rate of change of energy is due to that friction force.

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So that should tell you how, how quickly your energy.

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I don't think we have actually have to deal with friction, but that would that would come in so, so often with these things,

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you you can kind of argue that energy is conserved, but often you don't actually need to do that.

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You can just derive whether it is or not from the equations.

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And that's kind of irrelevant to Question three, because that one, you have to do it this way.

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So anyway, so you derived that equation and that was OK. And then there was a showing that the partnership between the two.

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So you managed to show that you got an expression for Z Dot squared?

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And then I wasn't quite convinced about this. I mean, there was on you.

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You did. It's easily manipulated, I guess.

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Yes, it is. I think something went wrong there. Yeah. But what were you trying to argue?

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I was trying to get it. I know that that that's quite an inspiration, zero, so then I was going to got quadratic with that.

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Yeah, that would be. So you can't say it can't be quadratic, you know, cubic.

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Yeah, but I'm going to get three heights. It's got to be a cubic. And so b three heights at which what happens?

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At which? That is right.

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That is after the three free routes of that, Quebec are places where there is zero.

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And then it didn't work out quite right,

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but it turns out that one of those heights is negative and therefore doesn't really make any sense because that's on the bottom of the cone.

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So there are two places where does that go to zero?

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And then so you then just said it must be the case that it's between those two heights that wasn't kind of completely obvious to me.

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Why if I find places representative zero, how do I know that I haven't just gone to that height and then kept kept going a bit higher?

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I could have, just like a stationary point doesn't necessarily mean it's a maximum or minimum.

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Mm-Hmm. So we need to check. So you don't need to check that because you.

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I guess you could check that you can find out what that antibodies. You don't really want to do that because you have to differentiate again.

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But it is kind of there anyway because you got this inequality. So you're saying a cubic.

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Function has to be greater than or equal to zero.

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And so I think the easiest way to do that is to draw the graph because I think this things going to about something like this.

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As a function of Z. And it has it has one negative ruin, has two positive reviews.

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And this this thing here is, I think it's actually z squared, the dot squared because you have to multiply that z squared.

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Somewhere along the way, the one that's when. But this thing got to be positive.

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And so you must have to be in between those two places to try to zero because you can't

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you can't possibly go through to zero because then that got spread would be negative,

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which doesn't make any sense.

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So I think if you if you draw a little graph and you know, that's clear enough to say that they've got it, you've got to be between those two points.

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Does that make sense? Yeah. So this thing? So actually, what is one of these things must be right?

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I guess it's. This one. Yeah.

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So it was. A.

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So you noticed I think that verdict was as a solution of this. Yeah.

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And sometimes you can forget that the fact is that there is a solution is clear from the initial condition because you set it off going horizontally.

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And that's quite useful. But if it's a big of otherwise, it might be quite difficult to find all the routes.

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So what was the rest of it? It was. Remind us.

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And then. There's a minus Z Times, two G and then plus B squared one minus h spread on z squared.

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So that's two g. That squared a minus, put the whole thing on top of that squared.

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Plus, B squared that squared minus h squared. This is.

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Two gentlemen subsequent. And so that means that to that square that squared is.

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A-minus is average practise out of everything. Times two g.

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That's Fred. And then we factored a minus that we got minus v squared and plus the Z, I think, is the same as what you got faced.

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Yeah, looks like a. And so this thing is definitely positive, so this thing has always got to be positive.

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And this is a negative Chebet, so that's what I've drawn there, and it's got three solutions because it equals a.

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And the other two, which is there's a sequel to the same nice.

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Yeah, it's just a funny expression. Yeah, that's the plus or minus Mr. President B squared plus h.

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That's just the quadratic formula for the solution to those. And then one of those that's bigger than v squared.

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So that's the minus one is definitely negative. So the minus one is definitely that route.

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And then the other one is the positive route.

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So that equals items that equals that one with the plus sign of those two, which is not obvious which one's which.

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Yeah, that depends on if you make v big enough, this one will be big enough.

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You might be small enough. This one will be close to zero. So these two reach.

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And so that means that when you when you take this thing round the circle, it's not dependent on how fast you flip it either.

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It's going to start going up and go to a higher height and then back down to again and it's going to do this kind of thing.

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Or if you flip it slowly enough, it's a it's going to be its highest height and it's going to go down.

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Hmm. And then come back up again. It's going to do this kind of thing.

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That make sense. Yeah, makes sense. And I guess if they're the same when V is equal to.

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The greatest guy. Then there's two things at the same.

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So these collide when you scratch and you.

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What would happen in that case, what what would keep the same height?

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Yeah, we just go around so it would just go around in a circle, so you'd have chosen exactly the right speed so that it can just do second.

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It doesn't have to come down to. OK, good.

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So that's the kind of question comes up a lot in the exam since it's quite fun.

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Well, it is not fun. OK, so then the next question was about this.

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Thing, so, yeah, that's fun. That's really what this thing that's going to roll off the sphere.

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And you just have to come up with some reasonable explanation as to why it leaves on the gates of [INAUDIBLE].

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Kind of obvious. Yeah. Yeah, it's going to be so difficult putting into words that it's one of those things.

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It's really difficult to explain. I mean, it's possible that what they want to do to get out here is about conservation of angular momentum.

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Again, in the sense that if it if it starts rolling off down the sphere, it never has any angular momentum and angry momentum is conserved.

148
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So if it starts off with zero angular momentum, then it can never get any because there is no forces going around.

149
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But it was also fine just to argue that whichever way it goes, it's going to be confined to that plane.

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So if you if you do find your X coordinate to be the way that it starts rolling,

151
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then you've only ever got any forces in the X directly on the Z direction.

152
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So why is it ever going to move anywhere else? And if it's confined to a plane and a sphere, then a great circle, then it's a great circle.

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The the definition. So I bet, anyway, both of your answers that I thought it sounded fine to explain that.

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And then you had to find these equations.

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So this was good.

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So here, Kate, I thought. It was better to try and use vectors rather than you read everything out in Cartesian components, which kind of works,

157
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but usually with these kinds of questions, it's better to try and write things in terms of the vectors and in this case, are any feature very useful.

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And part of the reason for that is that the end end is pointing perpendicular to the sphere.

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So when your thing has fallen to here, you know that the end is going to be going that way.

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And if I do find it all to be going that way. And I think it said it said it when it ceases to be that hangover.

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And so we are. Actually, this one's not totally standard, which way the coordinates are so er, in this case, what's going to be fine, Peter?

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There are constant. And then, Peter, if it comes this way.

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It's going to be costly to a minus sign contracts for particular.

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But I think it's kind of better to work in terms of those,

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and then you and then you can say that the position of that is all is equal to actually because, you know, it's on the on the sphere.

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So it's just a PR. And then it's kind of funny, nice and then and then you can say, well, I've got a constant, so that's just a lot.

167
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And if you do er, you get. Featured not to.

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A you're going to sign, right, because it's not complete, yeah, this is not the normal.

169
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Normally you have air going that way and easy to go in this way, but also features different.

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So if if you, regardless of which we put, would you always get the same derivatives?

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So like, we all don't go too easy to yeah, but do that if I might see to go that way in this case,

172
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I would have got air dotted minus seatbelt, etc. So to see I could always have defined easy to to be going that way.

173
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I mean, it wouldn't really make sense to do that because it would not be in the direction of increasing feature.

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But you might do that and it wouldn't wouldn't go wrong.

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We just have to we just have to keep trying to get it. Yeah, I mean, that's one of the reasons why is that stupid ideas sometimes to write these down?

176
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Yeah. Just because you can check so I can I can do it in my head.

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If I differentiate that with respect to time, I'm going to get costly to feature that I get minus ninety two them.

178
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So I know that I get $50 a feature.

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But if you if you're not, if you're using normal circuit court news, then you should just remember that that's the case.

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I mean, it's quite straightforward to derive their. And then and then you are double dot in this case.

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And the derivative of that, so you get a seat in a double dot theatre and then you get seated the citadel,

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which is always going to then be minus $50 E.R. So they give you minus a feature dot squared off, which is what you've got here.

183
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Yeah, it's just it's written that in more competition that there isn't part of the reason why you

184
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do that is then your niche and secondary is going to be more double dot is minus energy.

185
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Let me it this time. OK, right that way.

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Plus end. But you know, the end is in the air direction, so you can immediately write it and scale it.

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And you are. Because then when you take components, this is where I say you're taking components,

188
00:22:49,140 --> 00:22:54,280
you're ready to drop product and you can do products even if you have not written it all out in the coordinate system.

189
00:22:54,280 --> 00:22:59,430
So I've written all doubled up in terms of feature in air and gravity is written in terms of.

190
00:22:59,430 --> 00:23:04,920
So you might say, well, I need you rewrite K in terms of any feature to be consistent, but you don't really need to do that.

191
00:23:04,920 --> 00:23:07,440
You can just say, Well, I'm just going to take the dot product.

192
00:23:07,440 --> 00:23:16,230
So I come and say, let's take the product with the feature and I'm going to get a of double.

193
00:23:16,230 --> 00:23:19,110
Is equal to minus energy.

194
00:23:19,110 --> 00:23:33,330
And then I got Katie Keita to feature resign feature, so this is just Angie Sign Theatre and I had plus and he hardly featured of decision-makers.

195
00:23:33,330 --> 00:23:36,600
That's nothing tequila. So that gives you that aggression.

196
00:23:36,600 --> 00:23:40,200
And then you take the air component, the other one.

197
00:23:40,200 --> 00:23:48,820
But you had to do sort of cross multiply and at the end that if you if you've got your coordinates and directions which are suitable for doing that,

198
00:23:48,820 --> 00:23:59,820
you mentioned so this this gives us give us minus a feature dot squared is minus energy k ah, plus NW.

199
00:23:59,820 --> 00:24:08,350
And that is. Minus energy cost and.

200
00:24:08,350 --> 00:24:12,410
So that gives you a featured object, is Jehovah a science teacher?

201
00:24:12,410 --> 00:24:18,690
You know, that's the equation of the pendulum. Yeah, and.

202
00:24:18,690 --> 00:24:25,500
If you don't depend on them, and then I just. It's like this upside down.

203
00:24:25,500 --> 00:24:32,820
Yeah, that's a good thing. Minus a heated up squared.

204
00:24:32,820 --> 00:24:39,360
So if you only cared about her future changes and you wouldn't worry about this situation because all this equation touches were you.

205
00:24:39,360 --> 00:24:50,530
But if you want to know when's it going to leave the surface, then obviously you live in that equation.

206
00:24:50,530 --> 00:24:56,530
Right. And then for the next fight you, I think you both said this use energy conservation.

207
00:24:56,530 --> 00:25:04,960
Yeah. So this is again, I kind of think it's a bit.

208
00:25:04,960 --> 00:25:06,580
There's so much that you involved in too many things,

209
00:25:06,580 --> 00:25:11,980
if you now say I'm going to have energy conservation because you've already really got that here.

210
00:25:11,980 --> 00:25:15,160
So I would prefer to at this point say, Well, I've got these two requirements.

211
00:25:15,160 --> 00:25:23,770
I just need to solve them because this one really gives you energy conservation if you multiply that one by feature, just kind of the same idea.

212
00:25:23,770 --> 00:25:30,000
If I take this and multiply by future thought.

213
00:25:30,000 --> 00:25:37,670
And it will give me conservation of energy. Procedures, lots, effectively a velocity again.

214
00:25:37,670 --> 00:25:46,900
So this gives me the vitality of the whole future dot squared is equal to Dubai Duty of.

215
00:25:46,900 --> 00:25:53,180
So this is the same treaty, because this is actually the same thing, and that will always give you energy.

216
00:25:53,180 --> 00:25:59,650
Well, in fact, the reason why it is sometimes it's helpful to do is because sometimes you might not be sure whether energy is conserved or not.

217
00:25:59,650 --> 00:26:06,290
And this should tell you in a very quick check. Well, it might not be that quick.

218
00:26:06,290 --> 00:26:13,340
So let's hope that this equation is more complicated. Let's take a look back to make it work.

219
00:26:13,340 --> 00:26:17,420
In this case, it was finally to say I know a news concert because of the Phoenix move,

220
00:26:17,420 --> 00:26:26,450
but it seems to me it feels a little like you're invoking more ingredients than you need because you've already really got it here, too.

221
00:26:26,450 --> 00:26:34,940
So this doesn't stop the half featured dot squared plus g over a cost feature is a constant, and we know that when we start, we start at the top.

222
00:26:34,940 --> 00:26:38,570
The stage doesn't zero and cost each one.

223
00:26:38,570 --> 00:26:43,360
So that's a bet.

224
00:26:43,360 --> 00:26:48,530
You just have to take that and stick to that feature dot squared.

225
00:26:48,530 --> 00:26:54,920
There's two g of a one minus CONSTATER.

226
00:26:54,920 --> 00:27:05,930
And then there was that of a. When you plug that into there and it tells you what.

227
00:27:05,930 --> 00:27:16,250
And. And.

228
00:27:16,250 --> 00:27:22,130
It doesn't tell you this doesn't tell you how long it's taken. So if you wanted to know how long did it take for it to get to that,

229
00:27:22,130 --> 00:27:26,210
then you would need to actually solve this equation, which we have not quite done here.

230
00:27:26,210 --> 00:27:36,620
We found a relationship between Peter, Dutton said, but we've not found what Peter is as a function of time yet.

231
00:27:36,620 --> 00:27:45,400
Right, so and then it falls off when end is. Zero, when apparently goes negative.

232
00:27:45,400 --> 00:27:50,650
What happens after that? For really?

233
00:27:50,650 --> 00:27:59,940
Yeah. So what what kind of path will follow? Falls off.

234
00:27:59,940 --> 00:28:04,530
Did you do this? What did you do to him? And two or three or something?

235
00:28:04,530 --> 00:28:14,270
I did. I do not remember. So we found that it happens at two thirds of Z, apparently, so two thirds of A.

236
00:28:14,270 --> 00:28:20,330
And I think we're measured from here. So. So. Two thirds of age is not here.

237
00:28:20,330 --> 00:28:34,370
So it of. And then it will fall.

238
00:28:34,370 --> 00:28:40,850
And this so which forces are acting on it at that point? Gravity, just grab just gravity.

239
00:28:40,850 --> 00:28:46,340
Yeah. So you just have A. Minus energy.

240
00:28:46,340 --> 00:28:53,100
And so it will follow a parabola. Because of the X component of its velocity will stay constant.

241
00:28:53,100 --> 00:29:01,080
And the Y component of the Z component will be increasing in time.

242
00:29:01,080 --> 00:29:06,780
So at least in principle, you could kind of work out where it will hit the ground.

243
00:29:06,780 --> 00:29:14,460
By working out what its velocity is, what effect we know what it's velocity is when it leaves that, but we can work that out from feet apart.

244
00:29:14,460 --> 00:29:20,050
It tells us what it's lost in in the feature direction. And then you could carry on.

245
00:29:20,050 --> 00:29:25,750
What I would have to figure out really upset about me, but in principle, it's quite straightforward.

246
00:29:25,750 --> 00:29:31,330
Once it's not on the circle, then it would be a bit crazy to use second coordinates.

247
00:29:31,330 --> 00:29:38,950
So if you actually wanted to do that, then you're best to go back to Cartesian coordinates again and just work in terms of action X and Z.

248
00:29:38,950 --> 00:29:43,720
But if you're writing everything in terms of age and things, that's that's not so complicated.

249
00:29:43,720 --> 00:29:48,880
You can just switch between them if you want to. And.

250
00:29:48,880 --> 00:30:04,650
If the motion was on like an ellipse, Lloyd, would you also use cylindrical coordinates if it was something ellipse or a similar round object?

251
00:30:04,650 --> 00:30:12,020
That's an interesting question. I might do.

252
00:30:12,020 --> 00:30:19,870
Yeah, I guess it's different. It's an irony feature, I'm not going to be so useful if it's deployed,

253
00:30:19,870 --> 00:30:25,840
because because the point of those here is the air is perpendicular to the surface and the feature is tangent.

254
00:30:25,840 --> 00:30:28,480
And if the thing was elliptical,

255
00:30:28,480 --> 00:30:40,190
then it's sort of equivalent of E and E feature are not going to be they're not going to be perpendicular to the surface.

256
00:30:40,190 --> 00:30:47,880
So if you imagine that, it's kind of like that. And then the perpendicular direction.

257
00:30:47,880 --> 00:30:54,270
Coming from some origin. So I think that that's not going to be very useful.

258
00:30:54,270 --> 00:30:58,470
So in that case, would we just imagine that it might be better to go in Cartesian?

259
00:30:58,470 --> 00:31:07,210
Yeah, what you might want to do is write a lot of stuff in terms of the normal tangent at each point.

260
00:31:07,210 --> 00:31:12,060
And you might parameterised those in terms of some feature. Remember,

261
00:31:12,060 --> 00:31:18,600
you can write the equation of the lips as like X is equal to a cost to Y is equal

262
00:31:18,600 --> 00:31:26,490
to being signed into parameters the surface of the thing in terms of feature.

263
00:31:26,490 --> 00:31:28,770
And she might be useful to use some kind of feature,

264
00:31:28,770 --> 00:31:34,410
but then the normal in the tangent direction she would want to write in terms of what you have to wear,

265
00:31:34,410 --> 00:31:38,220
what they are and then you've use those to say, Well, I'm going to write down,

266
00:31:38,220 --> 00:31:44,370
I'm just going to do this because this is true regardless of what coordinate system we then write terms of.

267
00:31:44,370 --> 00:31:48,930
And then we say, let's take a let's take a component of that in the normal direction,

268
00:31:48,930 --> 00:31:52,980
and that will tell us what enters and competitive in the general election.

269
00:31:52,980 --> 00:31:58,000
But the normal and attendant directions are just not going to be given by these four minutes. I can think about something else.

270
00:31:58,000 --> 00:32:04,510
And they're going to change in a different way with position.

271
00:32:04,510 --> 00:32:12,190
So I think, yeah, I think it would be helpful if you're constrained to something you want to find,

272
00:32:12,190 --> 00:32:20,170
what's a nice way of parameters being on that surface. So if you know that you're on the lips,

273
00:32:20,170 --> 00:32:24,520
then using some kind of feature is going to be good because there's a single variable that tells you

274
00:32:24,520 --> 00:32:31,100
where you are on it in the same way that seek to here tells us where we are in this great circle.

275
00:32:31,100 --> 00:32:45,440
I've never seen a question asked, but I mean, in principle, but it's the same theoretically, if the we tried to do this before with my globe.

276
00:32:45,440 --> 00:32:50,210
What's happened to my ball? The ball, this big model,

277
00:32:50,210 --> 00:33:04,670
it would fall off a bit and suddenly that the where do you think we've got these people filming us for a Range Rover and look at the slow motion?

278
00:33:04,670 --> 00:33:12,440
To me, it doesn't look like it falls off the field after two sets of the height.

279
00:33:12,440 --> 00:33:16,930
Try and catch it. And also, how about here?

280
00:33:16,930 --> 00:33:22,670
I figured for I was quite surprised that it was too layoffs.

281
00:33:22,670 --> 00:33:32,360
But the one thing that I did think is that if you if you account for friction and I think you can argue that it must be a bit further down here,

282
00:33:32,360 --> 00:33:51,440
can you see why I can say that? Yes, it's going to lose it some of its energy, so it will have less energy in the horizontal direction.

283
00:33:51,440 --> 00:33:58,190
So what do you mean by energy in the horizontal direction as they will have less velocity in the horizontal direction?

284
00:33:58,190 --> 00:34:03,260
So it's going to stay on for longer? I think that's sort of true, but what does that mean?

285
00:34:03,260 --> 00:34:13,580
It turns down for a longer. You mean, because because it's also going to have less energy and it's going to have less Palestinian actions.

286
00:34:13,580 --> 00:34:22,580
You're thinking because it has a feels that way, its acceleration due to gravity in the vertical direction.

287
00:34:22,580 --> 00:34:26,820
It's like decelerating only. Yeah.

288
00:34:26,820 --> 00:34:36,890
I don't care. Hmm. I think that's true.

289
00:34:36,890 --> 00:34:43,910
I always think it's a way right away with the friction force would come into anything that we wrote down here.

290
00:34:43,910 --> 00:34:51,570
In the. Yeah, so I have another force, which is.

291
00:34:51,570 --> 00:34:56,490
In a theatre director, yeah, obviously frictional force.

292
00:34:56,490 --> 00:35:05,390
And so what difference would it make to this? The energy conservation.

293
00:35:05,390 --> 00:35:12,610
What could we say about the kinetic energy? And we know that was a friction force.

294
00:35:12,610 --> 00:35:25,890
How would it compare with this expression? I would feel it would be less if if if there were some friction than the energy at some later

295
00:35:25,890 --> 00:35:32,110
time must be less than what we started with because we had to do some work against the friction.

296
00:35:32,110 --> 00:35:38,150
So it would mean that featured dot squared is less than what we get from that expression.

297
00:35:38,150 --> 00:35:42,950
And so that would mean. So when you come back and stick that into here, it would mean that.

298
00:35:42,950 --> 00:35:48,740
And it's going to be. Bigger than what we got from here.

299
00:35:48,740 --> 00:35:56,090
For any given theatre. And so end would be bigger than what it is in the friction this case awaits.

300
00:35:56,090 --> 00:36:04,000
And so presumably that means we have to go for longer before and resistance to zero zero.

301
00:36:04,000 --> 00:36:11,920
So I think you I think you could you can see that from from the fact that you're having to subtract off the kind of energy in that.

302
00:36:11,920 --> 00:36:20,380
So maybe that's partly why it doesn't look like it leaves quite so quickly.

303
00:36:20,380 --> 00:36:27,340
OK. Yeah, the main thing that was that I think it might be trying to use irony feature as much as you can,

304
00:36:27,340 --> 00:36:33,150
even if you got a bit inventive in making up what is the best government.

305
00:36:33,150 --> 00:36:42,270
OK, but then the third one is with this thing on the wire.

306
00:36:42,270 --> 00:36:49,350
And so you made a you made a classical mistake here. Everyone makes.

307
00:36:49,350 --> 00:37:01,230
Which is to say the end has to be perpendicular to our dot. And in this particular question, that's not true of the topics of the rotation.

308
00:37:01,230 --> 00:37:10,540
Yeah. So we've got these beads sitting on this parabolic wire and the whole thing's rotating.

309
00:37:10,540 --> 00:37:18,750
So which direction is don't? This thing, that permission.

310
00:37:18,750 --> 00:37:27,110
It's time to put a. So, yeah, well, well, end is moving around as well.

311
00:37:27,110 --> 00:37:33,810
Yeah, but the component in this plane is tangent to the parabola, but the whole thing is is rotating.

312
00:37:33,810 --> 00:37:39,270
So are doctors actually pointing some? We've got some component along the tender.

313
00:37:39,270 --> 00:37:50,430
It's also got some component into the page. And and that's and the normal in this case is all we know is that it's normal to the wire.

314
00:37:50,430 --> 00:37:58,830
We don't know that it's normal to a plane like the one you're on a cone and you know that this normal forces tangent to the plane.

315
00:37:58,830 --> 00:38:05,470
In this case, it's a wire. So all we know is that the force is perpendicular to the wire, but we don't.

316
00:38:05,470 --> 00:38:12,090
It could be at a point in any direction around the wire and it could. In fact, it has to have some component into the page as well.

317
00:38:12,090 --> 00:38:18,730
And so the DOT product with our DOT in this case is not zero.

318
00:38:18,730 --> 00:38:23,620
Which then means that energy is not conserved, because that's what we saw on this stuff.

319
00:38:23,620 --> 00:38:28,270
And because of the work associated with then Dot got to zero.

320
00:38:28,270 --> 00:38:31,810
And that's the essence why you ended up with a minus sign. It didn't work.

321
00:38:31,810 --> 00:38:36,820
So your expression here is conservation of energy. And where's the expression you're aiming for?

322
00:38:36,820 --> 00:38:43,240
It's actually not conservation of energy. It's something that looks almost like it, except that it's not quite the same.

323
00:38:43,240 --> 00:38:49,180
And that's because in this case, we're forcing the things that we're forcing this way to go round and round around.

324
00:38:49,180 --> 00:39:02,240
And so we're actually we're doing work on the particle in this case.

325
00:39:02,240 --> 00:39:08,530
So you avoided this because you made a new tangent, correct?

326
00:39:08,530 --> 00:39:15,030
Yeah. So the issue here was that so end, it's quite hard to draw this in this if you're like me,

327
00:39:15,030 --> 00:39:22,700
you're going to be drawn to the site and is going in some direction, which is great.

328
00:39:22,700 --> 00:39:31,730
It got a component into the page. So what do you know about RN? In this case, the only thing you know is the end is perpendicular to the wire.

329
00:39:31,730 --> 00:39:38,710
So I would write that is and T is equal to zero, where T?

330
00:39:38,710 --> 00:39:49,290
T is the tangent. To the wire.

331
00:39:49,290 --> 00:39:55,980
So you don't know if it was they or in this case, either. In fact, it is not.

332
00:39:55,980 --> 00:40:04,550
There is a component in that way. And so you have you have to work out something to do with attendant victim for the wire.

333
00:40:04,550 --> 00:40:15,240
Which, you know, is what was the formula for this? That equals. Two today.

334
00:40:15,240 --> 00:40:28,880
So do you know how to work out a tangent to that jury to get to the tension will be that is proportional to one and PR, which is all of a.

335
00:40:28,880 --> 00:40:40,840
In terms of arms, so then I want to write that is the R plus R of a z.

336
00:40:40,840 --> 00:40:49,560
Because we're going to dock with this. We're not we don't really need to make a unit normal and unattended.

337
00:40:49,560 --> 00:40:56,350
We just need something which is in the right direction. So how about air?

338
00:40:56,350 --> 00:41:03,730
I always think it's useful to draw these diagrams and label everything. As much as you can.

339
00:41:03,730 --> 00:41:19,410
Confused about the angles. So this is exactly the right coordinate system to use, and then this stuff is all fine so that you not a movie that fun.

340
00:41:19,410 --> 00:41:20,760
And to this point,

341
00:41:20,760 --> 00:41:28,780
I think you need to say I'm going to start with a tangent and these questions is always just a case of working out what you want to start with.

342
00:41:28,780 --> 00:41:38,960
Get this to work. And so if you do that, so let's just do that because I want to check that we got the right equation.

343
00:41:38,960 --> 00:41:45,520
So with the end double dot dotted with that tangent is minus energy.

344
00:41:45,520 --> 00:41:57,920
Aged. Plus, NN Typekit in attendance and AZT is just minus energy of a.

345
00:41:57,920 --> 00:42:03,900
And up to zero. That's very rare.

346
00:42:03,900 --> 00:42:10,380
And this thing is that's the are dotted with the all component of that.

347
00:42:10,380 --> 00:42:16,530
So it's going to be M Dot minus omega squared from that thing.

348
00:42:16,530 --> 00:42:28,520
And then it said the Z component, so it's going to be plus over a set of about.

349
00:42:28,520 --> 00:42:32,760
And then you had to do a little bit of work to work out where all the that were related to each other,

350
00:42:32,760 --> 00:42:45,560
which I think we could decide which one you're going to get it in terms of.

351
00:42:45,560 --> 00:42:59,820
But I think if we do, all of that then gives us that thing. And it's an it's a bit weird.

352
00:42:59,820 --> 00:43:13,160
The only difference is that sign. But there's that sort of makes sense, especially because when you start with the.

353
00:43:13,160 --> 00:43:24,250
So it's minus and plus, and if I actually did my energy kind of argument here and said, let's do it with our dollars.

354
00:43:24,250 --> 00:43:31,750
Then we would get half an hour dot squared differentiated.

355
00:43:31,750 --> 00:43:45,580
That's what I get from Dot Double Dot. Is equal to and then this time, which still be minus Z, so we'd still get minus d by D, T and GZ.

356
00:43:45,580 --> 00:43:51,710
But then I would have plus and dot dot there.

357
00:43:51,710 --> 00:44:02,930
Which is no longer there. And that climate is not there, and in fact, you can kind of work out what it is if you relate to this equation and.

358
00:44:02,930 --> 00:44:20,900
So this one is. Not zero in this case.

359
00:44:20,900 --> 00:44:30,530
OK, so you then did what you should do, which is said, let's just ignore the fact that the wrong equation and then go with the equation.

360
00:44:30,530 --> 00:44:30,860
Oh yeah,

361
00:44:30,860 --> 00:44:40,340
you just you were using dots when I just think it just gets a bit confusing if you start putting dots in there and there's so many dots in this,

362
00:44:40,340 --> 00:44:47,900
so they just write numbers next to each other that media notification that sort of.

363
00:44:47,900 --> 00:44:56,000
Or write letters next to each other. So then we had this thing about stability.

364
00:44:56,000 --> 00:45:05,930
So first of all, do you know where this equation comes from, the Taylor expansion, the Taylor expansion of the previous regime?

365
00:45:05,930 --> 00:45:19,710
This one? Can you show me how that works? It comes from.

366
00:45:19,710 --> 00:45:25,300
Retailer expanding the equation for all. Right.

367
00:45:25,300 --> 00:45:30,730
Yeah. The reason I'm asking is because it's not totally that like, it's not the standard for you,

368
00:45:30,730 --> 00:45:34,390
see these things and you're probably used to seeing something like.

369
00:45:34,390 --> 00:45:39,010
But we've got this issue of, Ah. Yeah.

370
00:45:39,010 --> 00:45:43,700
And then so what would be the definition of an equilibrium point in that case?

371
00:45:43,700 --> 00:45:48,900
When? That is zero one zero.

372
00:45:48,900 --> 00:46:00,850
And always a concern. So if we call it a an equilibrium point is just a constant solution of the equation, the R equals a.

373
00:46:00,850 --> 00:46:11,320
It's just any point that satisfies if there is zero and then if you want to analyse the stability, what do you say?

374
00:46:11,320 --> 00:46:16,720
We've seen that be a plus a small perturbation, and for some reason we always call a small patch,

375
00:46:16,720 --> 00:46:23,380
pessimistic sign, even though your hate writing something is very difficult to write.

376
00:46:23,380 --> 00:46:31,810
I think it's easy. You just write a script of it, but you assume that size small that the whole point of.

377
00:46:31,810 --> 00:46:36,700
That's why I called linearity and you plug that in. And so then you get because it's a constant,

378
00:46:36,700 --> 00:46:48,130
you get excitable f a value plus sign and then you take the expand that and that gets you f of a which is zero by definition of a.

379
00:46:48,130 --> 00:46:53,570
Plus, a prime time site and then you ignore the higher order terms.

380
00:46:53,570 --> 00:46:59,770
The terms and so you just end that risk side, everybody is primed of a sigh.

381
00:46:59,770 --> 00:47:07,170
And then that's an equation that has. Either trigonometric or exponential of this,

382
00:47:07,170 --> 00:47:19,300
and I made a comment here that I think you need to spend a little bit more about why the sign of this time then determines the stability of.

383
00:47:19,300 --> 00:47:27,300
So it's unstable for the exponential solution, because, yeah, why, what occurs exponentially?

384
00:47:27,300 --> 00:47:33,790
Yeah, I mean, the question is just if I make a small perturbation like this, does oxide grow or not?

385
00:47:33,790 --> 00:47:38,880
And so it's unstable if the solutions are exponentially growing.

386
00:47:38,880 --> 00:47:45,810
And for this particular equation, you always get solutions which either trigonometric or exponential.

387
00:47:45,810 --> 00:47:51,400
If they're exponential, one of the exponential might decay the other one.

388
00:47:51,400 --> 00:47:55,240
If it's this equation you factor in, one of them will decay. But one of them will grow.

389
00:47:55,240 --> 00:47:59,770
And if there's any component of the solution which grows, then it means it's unstable.

390
00:47:59,770 --> 00:48:05,020
So I think you should always right when you get that kind of thing. So you had side double dot.

391
00:48:05,020 --> 00:48:16,150
Is that a squared minus J? Or do we know we lost the H omega squared minus G of XIV?

392
00:48:16,150 --> 00:48:21,380
What you should write is just the omega squared is bigger than grey means that term is positive.

393
00:48:21,380 --> 00:48:30,120
That means my solutions are trigonometric. That means that exponential that I would say exponential solutions.

394
00:48:30,120 --> 00:48:32,100
And then that means it's unstable.

395
00:48:32,100 --> 00:48:38,970
But I would write that rather than just saying that means it's unstable, that that gives me no explanation as to why.

396
00:48:38,970 --> 00:48:43,350
If you just say this is bigger than this means it's unstable, I don't understand why.

397
00:48:43,350 --> 00:48:51,550
And. Whereas if I'm a square, there's less than a I'd get tricked solutions.

398
00:48:51,550 --> 00:48:57,460
So Clyde is also right. So that means you're right, perturbed by a small amount, it will just stay close by.

399
00:48:57,460 --> 00:49:07,250
And so that means it's it's stable. OK.

400
00:49:07,250 --> 00:49:10,420
But I just drove back to this, so we didn't really quite had this,

401
00:49:10,420 --> 00:49:14,620
we had something that is actually it looks a little bit more complicated because

402
00:49:14,620 --> 00:49:19,870
we've got all of this kind of funny stuff on the left hand side is equal to this.

403
00:49:19,870 --> 00:49:22,810
So there isn't it's not really obvious what is.

404
00:49:22,810 --> 00:49:33,120
In fact, this is not an equation of that form you because you can rearrange it, but you still have double but as a function of all and dots squared.

405
00:49:33,120 --> 00:49:40,160
So it's more complicated, but still you're doing the same, you're basically doing the same thing.

406
00:49:40,160 --> 00:49:50,920
And so in this case, we said R equals zero. Is an equilibrium.

407
00:49:50,920 --> 00:49:56,010
And then so we say that I. There's going to be small.

408
00:49:56,010 --> 00:50:02,940
There's a small perturbation from zero, and then that means that you can ignore anything which is more than linear in sight.

409
00:50:02,940 --> 00:50:12,100
So you're putting our equals sign. You almost could just leave it at all, but you can just then ignore anything which is.

410
00:50:12,100 --> 00:50:14,160
Squared, Ohio squared or.

411
00:50:14,160 --> 00:50:21,920
And in fact, there isn't anything squared, but with two cubic times you can all squared off double dot and you've got R dot squared.

412
00:50:21,920 --> 00:50:27,960
And you're always when you say when you take science, where you always assuming it's derivatives are small as well.

413
00:50:27,960 --> 00:50:33,140
So then so you then you say a linear rise, meaning not just keep.

414
00:50:33,140 --> 00:50:44,320
Turns up to what extent, and then so you just end up with a squared side of adult age and scratch them with the scrap minus sign, which is.

415
00:50:44,320 --> 00:50:51,640
Why can we assume that it's derivatives also as well? So it's an assumption that you would then you then check.

416
00:50:51,640 --> 00:51:00,260
OK, so you were as long as PSI itself had sufficiently small?

417
00:51:00,260 --> 00:51:10,120
Then the solutions of this at for small time will be small and their derivatives will also be small.

418
00:51:10,120 --> 00:51:16,580
So you do this, find the solution and then you can check that it's consistent to say derivatives of

419
00:51:16,580 --> 00:51:22,050
that solution were small and therefore you were OK to assume that they were small.

420
00:51:22,050 --> 00:51:31,800
The first was like a consistent. It's a bit like they often do that when we're solving differential equations where we don't,

421
00:51:31,800 --> 00:51:38,010
we don't deduce the solution, we say, let me guess the solution and then I find that it works.

422
00:51:38,010 --> 00:51:46,650
And then I rely upon someone else who's proven that there is any competition that allows me to do that.

423
00:51:46,650 --> 00:51:56,620
And. For discussing the stability, we also talk about the case where they are equal.

424
00:51:56,620 --> 00:51:59,770
I think you don't need to talk about that.

425
00:51:59,770 --> 00:52:05,500
I don't think you ever expected to damage that because because it because you then need to look at the terms.

426
00:52:05,500 --> 00:52:10,360
So that then becomes a non-linear calculation for you. We're not at that level.

427
00:52:10,360 --> 00:52:17,380
So you just do the two cases. I decide and don't worry about the intermediate one.

428
00:52:17,380 --> 00:52:20,860
Well, what we say, what happens in this case, if it's unstable?

429
00:52:20,860 --> 00:52:33,520
So that means it's rotating fast enough and it is big enough that that means the thing's rotating fast enough for what would happen.

430
00:52:33,520 --> 00:52:38,560
Beats me up. Yeah, we keep going up for a while.

431
00:52:38,560 --> 00:52:44,700
Yeah, yeah, it's because there's actually only that's the only equilibrium solution here.

432
00:52:44,700 --> 00:52:52,890
If it's unstable, then it's just going to keep going out and then we get faster and faster and get higher and higher.

433
00:52:52,890 --> 00:52:56,400
And that's kind of showing you that the energy can't be conserved because

434
00:52:56,400 --> 00:53:04,860
because actually the limit of this is that the thing would go off to infinity. And that's because you're pumping energy into the system all the time.

435
00:53:04,860 --> 00:53:12,010
And that's that's associated with the fact that this energy got this component in the direction of things moving.

436
00:53:12,010 --> 00:53:22,400
And so you definitely we're not going to get that here with the fact that energy concert means it can't get it, can't get that high.

437
00:53:22,400 --> 00:53:39,530
OK, so we've done all of that, so I just spend a few minutes on the PDF, so.

438
00:53:39,530 --> 00:53:51,730
The first question I was driving away you, Cleveland, and I think I just wanted to say something about that because I mean,

439
00:53:51,730 --> 00:53:56,550
it was about this dimension, so this bit was not quite like it. You you.

440
00:53:56,550 --> 00:54:01,030
I think you just forgot to get rid of that when you differentiated at the time.

441
00:54:01,030 --> 00:54:05,820
So because this this equation would certainly alarmed me because you've got all

442
00:54:05,820 --> 00:54:09,390
these components of generation and you've got an isolated one in the other.

443
00:54:09,390 --> 00:54:15,030
So you were fine to do that, but it would be an issue that shouldn't, shouldn't be there.

444
00:54:15,030 --> 00:54:22,910
But yeah, this non dimensional ization that you had to do was lost.

445
00:54:22,910 --> 00:54:41,240
The. Senator Richard.

446
00:54:41,240 --> 00:54:49,010
These alpha and beta that you had to come up with, which the non dimensional equivalence of gravity and areas and it says what,

447
00:54:49,010 --> 00:54:53,810
what the conditions under which there are negligible.

448
00:54:53,810 --> 00:55:01,360
And so what do you want to use to say there was an alpha and beta would need to be small, then you could ignore them.

449
00:55:01,360 --> 00:55:09,630
So did you have to go into detail about the physical? Implications of alpha and beta things won't fly.

450
00:55:09,630 --> 00:55:17,340
No, no, it just wants to just be careful because you then sort of try to convert that back into saying something else.

451
00:55:17,340 --> 00:55:20,340
And this is precisely why you do it on demonstration because you've now written something.

452
00:55:20,340 --> 00:55:29,850
It doesn't make sense because this is an acceleration and this is a speed and this has got some other funny units.

453
00:55:29,850 --> 00:55:36,720
So, yeah, Gamera, but it doesn't make sense to and that celebration is less than the speed because I mean,

454
00:55:36,720 --> 00:55:43,080
it's like saying it's like it's like saying I weigh less than your height.

455
00:55:43,080 --> 00:55:51,180
It doesn't make any sense. So so what you actually mean there is to say gravity is small,

456
00:55:51,180 --> 00:55:59,410
and this alpha is kind of telling you what that what it would mean to say that gravity is small because you're comparing gravity to something else.

457
00:55:59,410 --> 00:56:05,790
So in this case, you compare it to some other force, which is due the tension.

458
00:56:05,790 --> 00:56:13,850
So how that happens could be in a number of different ways, but no alpha versus alpha is.

459
00:56:13,850 --> 00:56:22,220
G L squared, h. C. C squared, but that c c where this is T on the road,

460
00:56:22,220 --> 00:56:29,060
rogue l spread over h t is just sort of the dimension this measure of how much gravity so

461
00:56:29,060 --> 00:56:33,170
you could you could have gravity not being important because the string is very short,

462
00:56:33,170 --> 00:56:42,480
having will be very small. Or you could have it not be important because tension is really large, set up already a big long string.

463
00:56:42,480 --> 00:56:48,060
It might weigh quite a lot. But if I if I make the tension large enough, I pull it really taught,

464
00:56:48,060 --> 00:56:55,560
then gravity becomes irrelevant, even though it might wake the rogue element be quite big.

465
00:56:55,560 --> 00:57:01,170
That's the way to the string, but the potential is big enough. That doesn't matter for the way that the equation works.

466
00:57:01,170 --> 00:57:09,320
So this is kind of telling you how to how to say that gravity's. Not important.

467
00:57:09,320 --> 00:57:20,260
I've just completely take it all from when you lost the gamma, but then I and how that could work because I don't think any government that.

468
00:57:20,260 --> 00:57:26,110
But actually of this, this doesn't work. So karma is whatever makes the work.

469
00:57:26,110 --> 00:57:32,910
How did you work out the dimensions of government? So they said that.

470
00:57:32,910 --> 00:57:42,390
This is force per unit length. I worked out the dimensions of that and then we know what the dimension of that is.

471
00:57:42,390 --> 00:57:45,930
So then we must have that for the dimension of coming back to you.

472
00:57:45,930 --> 00:57:49,620
This has got to be a force that can work out what needs to be.

473
00:57:49,620 --> 00:57:51,240
Does that make sense for you? Yeah.

474
00:57:51,240 --> 00:57:59,340
And yeah, I mean, even if it not said that, you can look at this equation and say, Well, I know I'm putting this 10 together at this time.

475
00:57:59,340 --> 00:58:07,940
So these terms must have the same dimensions, so at least have the same dimensions as argue about changed.

476
00:58:07,940 --> 00:58:19,730
So it's just rogue. So if you don't know the units of something, that would be kind of how you can work them out.

477
00:58:19,730 --> 00:58:35,860
He's been making everything work. Then I think, yes, are you plotting the wrong this is dropped, so if you are told that if it is zero for Model X?

478
00:58:35,860 --> 00:58:42,560
Bigger than L. It doesn't matter what F is that for.

479
00:58:42,560 --> 00:58:47,390
Between Al and minus L. It's definitely zero.

480
00:58:47,390 --> 00:58:52,400
Then what can you say about motive of X minus C T?

481
00:58:52,400 --> 00:58:56,980
You know, this was all originally just translated it, and then I think parents convinced otherwise.

482
00:58:56,980 --> 00:59:05,260
I think it was because of this thing. They say, you know that f of X, Y and Z, that just means put in its mind to just translate into F.

483
00:59:05,260 --> 00:59:10,270
So you know that this is going to be zero for X minus c t bigger than that.

484
00:59:10,270 --> 00:59:13,000
So this is the argument to the functions of it goes in there.

485
00:59:13,000 --> 00:59:27,730
And so when you plot this as a function of X, then it's the region between C T, C T plus L and C T minus L is the region where it might not be zero.

486
00:59:27,730 --> 00:59:35,740
And it's got to be zero everywhere else. Yeah. And so that question you had this thing, which was this funny shape.

487
00:59:35,740 --> 00:59:42,660
That fits in that. Yeah, I can see you up to that.

488
00:59:42,660 --> 00:59:48,310
Have you changed your mind? But yeah, sometimes these things, you just have to be almost like a computer.

489
00:59:48,310 --> 00:59:51,010
You've got to say, well, f is a function which takes this argument.

490
00:59:51,010 --> 01:00:00,600
So if I put it in a different argument, I've just got to replace that argument everywhere in my expression and to be really systematic about.

491
01:00:00,600 --> 01:00:05,690
Right. I think that. You may be out of time.

492
01:00:05,690 --> 01:00:14,930
And the last person was, OK, I made some comments about food and part of it.

493
01:00:14,930 --> 01:00:25,270
And we calculated the criticism. Not quite right. Yeah, but I'm not too fussed about going to each of those wrong and.

494
01:00:25,270 --> 01:00:36,370
Oh, yeah, there was just this last part, so let's say it said it, so can you break down this function into two functions?

495
01:00:36,370 --> 01:00:43,360
Can you tell me what these functions are and you found what they were in terms of some very attractive issues?

496
01:00:43,360 --> 01:00:51,770
Can you tell me what those functions are? What what was the definition of the end?

497
01:00:51,770 --> 01:01:01,100
In terms of a.

498
01:01:01,100 --> 01:01:07,750
Very long question. It was a long questionnaire. Did you remember where the three of coefficients came from?

499
01:01:07,750 --> 01:01:14,080
And the thing? OK.

500
01:01:14,080 --> 01:01:25,210
So H and XS iPhone X, I think it's almost almost effortless, whereas f of X defined.

501
01:01:25,210 --> 01:01:29,900
For X within L minus of or something.

502
01:01:29,900 --> 01:01:39,400
This your lessons, just as much as we we're only ever given anything on this fixed domain between zero and L.

503
01:01:39,400 --> 01:01:44,820
And so. At this point when you've tried to apply the initial conditions to the problem,

504
01:01:44,820 --> 01:01:51,690
you said, I need to try and find some and such that F of X is the sum of the aims sign.

505
01:01:51,690 --> 01:01:58,080
And do you thinking, is that going to say, well, I know because we were taught this at the start of term.

506
01:01:58,080 --> 01:02:06,840
I know how to choose and such that that is the case. Those aliens are going to be the free sign coefficients of F.

507
01:02:06,840 --> 01:02:12,950
And then you do the same thing with this one, you say, Well, I need the derivative to be G and the derivative is this.

508
01:02:12,950 --> 01:02:19,790
So I need to choose these coefficients to be the free assign coefficients of the G.

509
01:02:19,790 --> 01:02:27,410
And the function that you're then creating when you do the some of the and scientists and an extension of the F.

510
01:02:27,410 --> 01:02:35,360
Because you made it periodic, which extension is if you make, it seems there is.

511
01:02:35,360 --> 01:02:46,640
I thought extension, which is the old extension of that, that's a you're on minus Al Jazeera, you flipped it over.

512
01:02:46,640 --> 01:03:01,280
So these functions that you ended up with at the end is HMX and you have there for h of X is the odd extension of F and U events is.

513
01:03:01,280 --> 01:03:10,470
The odd extension of G. They're related to the initial conditions,

514
01:03:10,470 --> 01:03:14,250
and then that actually kind of goes back to the question to that because the way that you

515
01:03:14,250 --> 01:03:21,900
can think of these things is that the initial shape splits in two and then moves sideways.

516
01:03:21,900 --> 01:03:30,690
And in this case, we have boundaries, but the boundaries are kind of reflecting and they're allowing the same wave to come back in again upside down.

517
01:03:30,690 --> 01:03:32,550
And so this is saying, actually,

518
01:03:32,550 --> 01:03:42,040
you can think of the solution that you end up with by this method is kind of in the same framework as the solution that you had here.

519
01:03:42,040 --> 01:03:50,800
Even though it looks very different. OK, and that's enough.

520
01:03:50,800 --> 01:03:57,310
So next week, we might concentrate a bit more on this stuff and not the dynamic.

521
01:03:57,310 --> 01:04:15,978
OK.