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How did you manage?

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I hope you were able to restore the analytical result already, at least you gave it a try and you made

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some solid progress.

4
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In any case, maybe you had lots of problems.

5
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It's not a big deal because I'm showing, you know, my solution.

6
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And as always, we are just copying the old code and then we are modifying it.

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And I think we are not changing the number of points.

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But in any case, let's just copy this here and then let's copy the loop to construct the dataset.

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Then we must transpose the list for the plotting.

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Then we need the actual plotting, which is this one here.

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And last but not least most importantly, we must calculate the rotational energy.

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So I think the only thing that we have to change here is the if condition, because previously we have

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said that the the norms are basically the radius of the point.

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The distance to the center is smaller than the radius of the sphere.

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However, now we have a hollow sphere.

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So we do not consider every point that is smaller than the radius, but we consider all the points whose

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radius is between these two values.

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So we write like this, we write, are too small or equal, then this this normally this distance to

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the center.

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And this one must also be smaller than then compared to our one, which is the outer radius.

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And I think now we can run it.

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So we transpose and we plot.

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And yes, it works.

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So you see, maybe it's a bit tricky to see to the projection, but I hope you agree that in the outer

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part, to much more points here.

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So it is a hollow sphere and now we can calculate the number, which is the rotational energy proportional

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to the moment of.

28
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So yeah, the moment of inertia.

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And so now we have the value of zero point two 744, which is pretty close to the analytical results.

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So I would say this is a pretty solid solution and it works well.

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And of course, you could now increase here to number of points.

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Let's see.

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OK, we have zero point two seven five five.

34
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Yeah.

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So it's much closer now, actually.

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So when we now compare this to the solid sphere that we see, this is 0.7 and this is 0.2, basically.

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So the rotational energy of such a hollow sphere, which has the same mass as the solid sphere, is

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higher.

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So if we were to assume that both have the same mass, then it would rotate slower if we would provide

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the same energy.

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So if we would, for example, push them and transfer the same energy to both spheres, then this one

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would rotate slower because it costs more energy to rotate at the same velocity.

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So the reason for this is that the majority of mass is further away from the rotation axis.

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So if you assume to access to be the z axis, then here you have a large proportion of mass, which

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is quite far.

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So it costs a lot of energy to move because if it rotates here, it has a very large velocity.

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So this is why such a hollow sphere rotates slower than the solid sphere.

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But it's always important to keep in mind that we assume that both of these spheres have the same mass.

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And if you would just take a solid sphere and remove the central part, then this may be different because

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the mass will then be reduced.

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So you really have to make sure that the two spheres are fabricated out of different materials so that

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you can make them have the same mass.

53
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And if you manage to do this, then you will see that the rotation is pretty much different in both

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cases.

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So this is also a very good methods to figure out whether or not a sphere is hollow or not by just providing

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some energy to rotate the whole thing and then to see how fast is actually rotating.

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Because you see that the moment of inertia and also the rotational energy differs based on the fact

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if it's hollow or not.