‫Welcome back.
‫So I hope you tried this exercise yourself and let's see which tube will win the race.
‫Well, both of them have the same diameter and both of them have the same mass.
‫Right.
‫The diameter is zero point three meters.
‫In both cases, in the mass is five kilograms.
‫And so that means that both of them experience a force of gravity down.
‫And of course, you can also decompose that force in the direction normal to the slope and parallel
‫to the slope in the same thing here.
‫So both of them will have the same component from the force of gravity that is parallel to the slope.
‫However, because of the fact that in this case, where you have the hollow tube, all this mass is
‫concentrated further away from this central axis.
‫Because of that, it has a higher mass moment of inertia.
‫Then this lower tube that is completely filled.
‫And remember, mass moment of inertia is resistance to rotation.
‫So the higher the mass movement of inertia, the more difficult it is to rotate.
‫Therefore, this tube that is empty in the middle, it will start rolling slower than this tube that
‫is completely filled because this to here that is completely filled, has less mass movement of inertia.
‫So it will roll down faster and it will reach the finish line faster.
‫And you can also look at it from the equations point of view.
‫So you have this force of gravity component parallel to the ramp and you have some kind of distance
‫here.
‫So therefore you will have some kind of moment here and you will have the same moment here as well.
‫So the moments are equal, but the mass moment of inertia here is bigger than it is here.
‫And since the moments are equal, then that means that the angular acceleration here needs to be smaller
‫and here it will be bigger.
‫So this guy here will angular really accelerate down faster.
‫But let's go back to the original point that I want to make mass symmetry.
‫Now, you know that the mass movement of inertia depends on the mass and it depends how far away that
‫mass is from the axis, about which you're measuring your mass movement of inertia.
‫So you need to take into account two things mass and the distance of mass from the axis.
‫In this tube, we said that the mass is distributed equally.
‫So the density here is the same everywhere.
‫But if you look at it, then the radius is also the same everywhere.
‫And we assume that the width of this tube, the width of this wall here is also the same.
‫So let's just put it W here and W here.
‫So even though it doesn't appear like that here in the drawing, actually the width is the same everywhere.
‫That means that this tube has the same radius about the entire Z axis.
‫So that means that if you look at this tube from the side like this and you have the Z axis here and
‫you put two lines here like this, then this entire mass is distributed equally on all sides of these
‫lines.
‫So in this case, this tube is mass symmetric.
‫You can think of it like this.
‫It has equal mass on all sides.
‫And if that's the case about all axis, about the Z axis, x axis and Y axis, then all product of inertia
‫will become zeros and you will have this matrix here.
‫So you will only have these three elements here diagonally.
‫However, if I start changing the densities in some parts of the tube or you have a completely different
‫shape, something like this, and then let's say that this is your axis here and the mass is distributed
‫equally in the walls.
‫However, that this.
‫This is from the Axis are different like this, and if you look at this object from the side and you
‫have again these lines here, then you will see that on this side you will have less mass than, for
‫example, on this side.
‫So this object about this axis is not mass symmetric.
‫So if you want to make this object mass symmetric, then you need to consider another axis somewhere.
‫Maybe here.
‫Who knows?
‫It's a complicated shape.
‫But in our case with this axis, this object is not mass symmetric about this axis.
‫And when that's the case, then you have this matrix here with these products of inertia.
‫That is, of course, if there is no mass symmetry about all the X, Y and Z axis, if that's the case,
‫then there are no elements in this matrix that are zero.
‫And by the way, from mathematics, when you compute products of inertia, you'll find that I x y always
‫equals I y x, then I always equals two, i, x, z and I y, z always equals izi y.
‫There are formulas to compute these products of inertia and when you follow those formulas, then that's
‫the fact that you will find.
‫And by the way, when you compute the mass movement of inertia in the product of inertia, then of course
‫when you computed about the Axis Z, then you don't only look at the H here.
‫You have to go inside the object.
‫And then you also need to measure mass here and here and here and of course, its distance to this Z
‫axis.
‫So let's draw a Z axis here that goes through the tube or through this weird looking object.
‫And so you need to take into account the distance of, for example, this mass to this axis here and
‫here and here.
‫And in fact, what it really comes down to is that you will have infinitesimally small mass elements
‫called D.M. that have their own distance from the axis.
‫And you have to sum up all the DBMS and the distances from the axis.
‫So you can already see that when you calculate mass moments of inertia, products of inertia, then
‫you're going to have to use integrals because integrals are continuous summations.
‫You would be summing up those DBMS combined with the distance to the axis.