‫However, that's not all, because you also need to know how to compute products of inertia, and these
‫are the formulas for the products of inertia.
‫And you can see that this equals this.
‫This equals this and this equals this.
‫If you compute products of inertia using these formulas, then you will see that this statement holds
‫true.
‫And so essentially, you could do the same thing with D.M. Let's, for example, take this one so D.M.
‫could be written the density.
‫That depends on X, Y and Z times.
‫DEEVEE And you can also see it from the units because the density is kg per cubic meters.
‫Right.
‫And then the volume, it's cubic meters.
‫So if you multiply the density times the volume, then in terms of units it would be kg per meter.
‫Cute times meter cubed.
‫This comes from here.
‫And you can see that the cube meters will cancel out and you will only be left with a kg and KG is the
‫unit for D.M. So this statement here perfectly makes sense.
‫But you can also write it out like this role that depends on X, Y and Z, or as a function of X, Y
‫and Z times.
‫And instead of Dve, I can write it down like this, the X, D, Y and Z.
‫And therefore this formula will become like this.
‫You will have a triple integral X times Z times the density that depends on X, Y and Z.
‫So in each location, the density of the object could be different.
‫And then times the X, the Y in D.C. and if you have studied calculus, then you know how to compute
‫triple integrals.
‫Of course, nowadays you also have a lot of software, a lot of computer programs that do it for you
‫very quickly.
‫But still, it's good to know fundamental formulas behind it.
‫And now if your object is mass symmetric like this one, and in fact it should be mass symmetric about
‫all axis and about the Y axis and about the X axis, because you can also rotate this tube like this
‫and like this, not only like this.
‫So if this object is mass symmetric about all of its axis, then the products of inertia will become
‫zeros.
‫So if you go through this process here, then you will see that it will become zero and that would apply
‫for this and for this.