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Hi again, and this lecture is referring to the problem of center of but this some it's referring to

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a negative mass.

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What does it mean?

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Suppose you have a, um, playing surface and you do have some holes inside that surface and you want

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to find out what is the center of mass, uh, of this new shape.

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OK, so in order to solve the problem, we can follow the same, uh, formulation, which is X is equal

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to X.

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I am I divided by summation of, uh, M and also the same for Y, but with the small differences.

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So we can assume that the center of mass of the uh.

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Plane is located at its central point.

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So suppose the dimension of the surface is one by one, I suppose it's a square and the dimensions are

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both equal to each other one and this size one as well.

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So the center of mass is located at the center of this square, half and half, OK?

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And we do have, um, some holes inside that plane.

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So you can see here on this graph, there are three holes.

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And depending on the surface, um, you can see.

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That they can have different masses, so the mass is proportional to the area of each circle.

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OK, so for simplicity, we are assuming that we know the locations of the each hole and also the surface

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of each hole then and in order to solve that, we can easily apply the same formula.

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So let's have a look at the.

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Uh, Pathankot, so, as I said, the mass of the plane is positive because it exists and those hold,

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since they do not exist and we will consider them as negative masses.

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OK, so here you can see the Python code.

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So here, first of all, we import the required packages and then, um, we define them all abstract

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model.

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And also there are some parameters needed.

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The number of, uh, circles does assume, uh, the default value is five and six I which is ranging

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from one to N.

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We do need to, um, a random generator in order to scatter the points on the plane and also the air

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masses.

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And then X location Y locations are also are the parameters, input, parameters and maps as well.

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And the variables, we do have two variables model that X Y, these variables change between zero and

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one, because the center of mass will be inside that plane and the rule is the same, but with a small

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difference this time.

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And the mass of the plane is let's assume it's, um, um, proportional to the area of that, then we

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can say that it's one multiplied by 1/2 minus summation of those.

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Ms.

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And, uh, actually, the area of them is specified here, the M is the radius of those, uh, circles.

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And, uh, divided by the summation of those areas, and you can do the same for why and as I explained,

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as since this example doesn't have any specific objective function, you can choose any of your, uh,

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decision variables is a linear programming approach.

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You don't have any non-linearity if you see, uh, a square here, since M is known there will know

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and non-linearity be created.

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Sorry.

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Um.

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And also we have to choose the solver I chose.

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Go pick and then I will create uh instance so everything will be fed into the model I use the matched

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plots lead to.

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And draw the, uh, graphical shape to show the results for us.

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So for simplicity, I have chosen two points.

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One for the salt solution, the red one you can see here, and the yellow one yellow one is exactly

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at the center of that square.

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And those white ones are representing the, um, holes inside that plate.

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So as far as those random numbers are not overlapping each other, then the solution is correct.

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Otherwise, it this approach doesn't give you a valid solution, OK, because those white ones should

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not overlap each other to be able to represent their weight on the solution.

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I run it again.

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To show you the new results, so every time you run the code, it will give you another set of new solutions

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and here you can see that the center of minds doesn't change that much compared to the center of that

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square.

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I run it again, hoping I can get a better solution, so this time you can see why the solution is not

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correct, because there is an overlap between these two white ones.

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Hmm.

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And so, um, this new calculator at the center of gravity, a center of mass, is not, um, uh, precise

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because those masses are calculated two times.

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I run it again hoping I can get, uh, non overlapping kind of circles.

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OK, so here you actually there are well, there is overlap.

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I run it again.

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If you manually specify the location, you don't need to run it multiple times.

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OK, so here you can see that sees all of those wide circles are these side of the plane, then the

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center of mass will move to the opposite direction.

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OK.

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Thank you very much.