‫However, that is not all.
‫Did you notice that there were a lot of zeros in the inertia matrix?
‫Well, in fact, the most general case is this.
‫This inertia matrix is actually called the inertia tensor.
‫All right, the inertia tensor.
‫And so what are these red terms then?
‫They're called products of inertia.
‫The product of inertia is a measure that captures and objects mass symmetry about a certain axis.
‫So what is mass symmetry?
‫Let's take a simple example, let's take a hollow tube here and let's call this axis as the axis.
‫And so this axis goes through the tube and so the tube is empty inside and all the mass is concentrated
‫in the edges here and this tube.
‫Let's assume that it's homogeneous.
‫It means that its density is constant and density was kg per meter cute.
‫It means then mass is evenly distributed along the entire tube.
‫So wherever you have matter, the mass there is evenly distributed.
‫So the density is the same here and here and here everywhere.
‫And so the mass moment of inertia depends on two things.
‫It first of all, depends on how much mass you have in our body.
‫But it also depends how far that mass is from the object.
‫The more mass there is in the object, the more kilograms you have.
‫Then obviously that contributes to more mass movement of inertia about this Z axis.
‫And the less mass you have, in other words, the less kilograms you have, the less mass moment of
‫inertia you have.
‫But then also the further away from the axis this mass is, the higher your mass moment of inertia.
‫And the closer the mass is concentrated to the axis, the less mass movement of inertia you have.
‫So just as a quick side note, if I have another tube here and it's not hello, it's completely filled
‫inside and there masses are equal.
‫Let's say this one is five kilograms and also this one is five kilograms.
‫And both of them have the same diameter.
‫For example, this one has the diameter of zero point three meters.
‫And also this one has the diameter of zero point three meters.
‫And so now I have a small exercise for you.
‫You have two ramps and in the first ramp you have this hollow tube that has five kilograms of mass.
‫And in the second ramp you have this completely filled tube and both of them have the same angle and
‫the same distance from this point to this point.
‫And now your exercise is the following.
‫Which tube will reach the ground faster?
‫Both of them will start rolling at the same time.
‫Time equals, let's say, zero seconds and it doesn't matter.
‫Don't think about time.
‫Just think in terms of concepts.
‫Both of them start rolling at the same time.
‫And which tube will reach the ground sooner, considering that you have the same distance to roll and
‫the angles are the same, which tube will win the race?
‫So try it yourself and let's talk about it in the next video.