How did you manage?
I hope you were able to restore the analytical result already, at least you gave it a try and you made
some solid progress.
In any case, maybe you had lots of problems.
It's not a big deal because I'm showing, you know, my solution.
And as always, we are just copying the old code and then we are modifying it.
And I think we are not changing the number of points.
But in any case, let's just copy this here and then let's copy the loop to construct the dataset.
Then we must transpose the list for the plotting.
Then we need the actual plotting, which is this one here.
And last but not least most importantly, we must calculate the rotational energy.
So I think the only thing that we have to change here is the if condition, because previously we have
said that the the norms are basically the radius of the point.
The distance to the center is smaller than the radius of the sphere.
However, now we have a hollow sphere.
So we do not consider every point that is smaller than the radius, but we consider all the points whose
radius is between these two values.
So we write like this, we write, are too small or equal, then this this normally this distance to
the center.
And this one must also be smaller than then compared to our one, which is the outer radius.
And I think now we can run it.
So we transpose and we plot.
And yes, it works.
So you see, maybe it's a bit tricky to see to the projection, but I hope you agree that in the outer
part, to much more points here.
So it is a hollow sphere and now we can calculate the number, which is the rotational energy proportional
to the moment of.
So yeah, the moment of inertia.
And so now we have the value of zero point two 744, which is pretty close to the analytical results.
So I would say this is a pretty solid solution and it works well.
And of course, you could now increase here to number of points.
Let's see.
OK, we have zero point two seven five five.
Yeah.
So it's much closer now, actually.
So when we now compare this to the solid sphere that we see, this is 0.7 and this is 0.2, basically.
So the rotational energy of such a hollow sphere, which has the same mass as the solid sphere, is
higher.
So if we were to assume that both have the same mass, then it would rotate slower if we would provide
the same energy.
So if we would, for example, push them and transfer the same energy to both spheres, then this one
would rotate slower because it costs more energy to rotate at the same velocity.
So the reason for this is that the majority of mass is further away from the rotation axis.
So if you assume to access to be the z axis, then here you have a large proportion of mass, which
is quite far.
So it costs a lot of energy to move because if it rotates here, it has a very large velocity.
So this is why such a hollow sphere rotates slower than the solid sphere.
But it's always important to keep in mind that we assume that both of these spheres have the same mass.
And if you would just take a solid sphere and remove the central part, then this may be different because
the mass will then be reduced.
So you really have to make sure that the two spheres are fabricated out of different materials so that
you can make them have the same mass.
And if you manage to do this, then you will see that the rotation is pretty much different in both
cases.
So this is also a very good methods to figure out whether or not a sphere is hollow or not by just providing
some energy to rotate the whole thing and then to see how fast is actually rotating.
Because you see that the moment of inertia and also the rotational energy differs based on the fact
if it's hollow or not.