‫And so I also want to give you general formulas for mass moment of inertia.
‫I don't want to stop here for a long time.
‫However, you can read something out of these formulas.
‫In fact, if you think about it, it's a pretty good art and it's pretty good skill to know how to translate
‫mathematical formulas into a human language.
‫So if you look at these formulas, what do you see?
‫Will you see an integral?
‫And that means that you have continuous summation.
‫So in our case, let's focus on this guy here, because this is what we are computing, where computing
‫mass moment of inertia about the Z axis.
‫And so you have this D.M., which is this infinitesimal mass here, and then you have this are Z squared.
‫And that's the radius of that particular D.M. from the Axis.
‫And so essentially what is happening here is that this integral goes through this entire shape, takes
‫each individual D.M. let's say D.M. one here and then later maybe it's D.M. two here.
‫And for each the M, it measures the distance from the Z axis.
‫Now notice that you have squared here, which means that the effect of increasing the distance of that
‫small mass element from the axis is not linear, it's squared.
‫So if you increase the distances of all the DM's from their axis by two times, then the mass movement
‫of inertia about this axis will not increase by two times.
‫It will increase more by four times.
‫And so you go through the entire shape and you sum up each product of an infinitesimal mass element
‫and its distance squared from the axis.
‫All these products you sum up through continuous summation, which is the integral, and that's how
‫you will get your mass moment of inertia.
‫And if this one here was the Z axis like you see here, well, then here you will have the X axis and
‫here you will have the Y axis.
‫And if that's the case, then you can rewrite this R Z squared like this because R Z equals square root
‫of X squared plus Y squared is the Pythagorean theorem.
‫Right.
‫So this distance here can be computed by taking this side here and this side here.
‫So this would be X and this would be Y, and then you compute this length here using the Pythagorean
‫theorem like this.
‫And since you have your R Z squared here, then instead of this, you can write like this and that will
‫give you X squared plus Y squared.
‫And then D.M. can be a constant number.
‫Or if you don't have a density that is constant, then you can rewrite D.M. Like this, the M equals
‫the density times, the V where D.V. is a differential volume, a differential volume essentially is
‫the X Times DUI times.
‫DEASEY So if you imagine a small, infinitesimal cube and then you have your DKs here, then you will
‫have the Y here and then these here, then this entire thing will be your differential volume.
‫And so you might have a case where your density is not constant, but rather it depends, for example,
‫on X and Y.
‫And then instead of D.M., you will just write this one here.
‫And the same concept would apply to this shape as well, even though here it would be more complicated
‫because your answer would change a lot.
‫And the same exact concept would apply to mass movement of inertia about the Y axis and the x axis.
‫Now, one more thing that I want to say.
‫If you take this D.M..
‫And you write it down like this, the density that self depends on X and Y and perhaps even Z and 10
‫times the differential volume where the differential volume is nothing but the product of the DUI and
‫the Z.
‫And you know that this mass moment of inertia is computed by summing up all the products of this differential
‫element and its distance squared from the axis.
‫And now knowing this and this, you can rewrite this equation like this is easy, equals X squared plus
‫Y squared.
‫And now instead of D.M., I'm going to write this Times Road that depends on X, Y and Z, the most
‫general case.
‫And instead of Dve, I'm going to write the X Times DUI Times Dizzie.
‫And now you have three differential elements here, which means that this integral converts into a triple
‫integral.
‫So if you choose this approach, then it's like going through the entire object and taking an infinitesimally
‫small volume that you then multiply by the density at this specific location.
‫And remember now, since the density is not constant, then in every single location, the density might
‫be different.
‫So you take an infinitesimally small volume divi one and then you multiplied by a raw one, which is
‫the density valid only here.
‫And then you multiply it by the squared distance.
‫That is the distance from the axis to this particular point.
‫And you do it for all infinitesimally small volumes in this object.
‫But since you could rewrite DV as DKS Times divides, then you could express this entire equation like
‫this.
‫And from calculus, you know, or you can learn how to compute triple integrals.
‫And so that's how mathematically you can compute your mass moment of inertia.
‫But that's not all, because you also need to know how to compute the product of inertia.