‫Welcome back.
‫Let's derive the equations for the rotational motion now in the translational motion.
‫We said that in the most general case, the sum of the forces in the inertia frame equals the change
‫of linear momentum with respect to time.
‫And here Gummo, that vector E is the velocity vector in the inertial frame and it's in metres per second.
‫Now, when we enter into the world of rotation, then there is a similar law.
‫There is something called angular momentum, which we will call H Vector, and we are in the inertia
‫frame right now.
‫And the general definition of it is that it equals the mass movement of inertia in the inertia frame
‫multiplied by this theta dot vector in the inertial frame here.
‫The eye superscript e is the inertia tensor.
‫But remember, now we're working in the inertial frame and therefore all these mass movement of inertia
‫calculations, they happen about the inertial frame axis.
‫So this is your inertia frame where you have X, Y and Z, and since it's an inertia frame, it's fixed
‫to the ground, it doesn't move.
‫And since we're measuring the mass movements of inertia in the inertia frame, then all these mass moments
‫of inertia and problems of inertia, they're all measured about these three axes here, capital X,
‫capital Y and capital Z.
‫And then the theta dot vector in the inertia frame consists of those time derivatives of the three oilor
‫angles, five dot, dot, dot and side dot.
‫And that's also because we are right now in the initial frame.
‫And so if you multiply them together, then you will get your angular momentum in the inertia frame.
‫Now, let's look at the two, the case for second, let's look at our car from the top view and let's
‫just for now make the inertial axis aligned with the car's body frame axis like this.
‫You can see that in 2D on an X Y plane you only had.
‫Izzie, as your mass moment of inertia because you were rotating about the inertia frame or body frame
‫Z axis and your angular velocity was also only about the Z axis.
‫So that's the only term you cared about in 2D.
‫However, in 3D, you have this entire matrix to take into account.
‫And so in 2D, if you multiply isobars Z times side that you will get your angular momentum.
‫There is also about the Z axis.
‫And you can see that into the the angular velocity vector and the angular momentum vector are always
‫in the same direction either towards you or away from you according to the right hand rule.
‫In the age and Passi that are just scalars and Izi is like a proportionality constant.
‫That's why they are in the same direction, however, in 3D.
‫The angular momentum vector is a three by one vector and instead of BPCI that you have this Seeta,
‫that vector in the inertia frame, which is this one, which is also a three by one vector.
‫And now you also have a three by three masked moment of inertia matrix, which is an inertia Tenzer,
‫that means that in 3-D, the age vector and then the angular velocity vector don't have to be in the
‫same direction.
‫Let's assume that all the products of inertia are zero KG's times meter squared.
‫So you only have a diagonal mass moment of inertia matrix without the products of inertia.
‫And by the way, that is the case in our course.
‫So this is your angular momentum vector.
‫This is your mass moment of inertia matrix where all your products of inertia are zeros.
‫And this is your angular velocity vector.
‫Let's assume that five dot equals Seeta, dot equals SPSSI, dot equals one radians per second.
‫So all of them have the same values.
‫And also let's assume that exact equals I y y equals I z the equals, let's say two kg's times meter
‫squared.
‫So all the three mass moments of inertia, they also have the same values.
‫In that case each X equals I x X times five dot equals two times one equals two kg's times meter squared
‫per second.
‫Similarly, you find your H, Y and Z where you multiply ie y, y times Theta dot and I's time, SPSSI
‫that respectively.
‫So your angular momentum vector is two, two and two KG's metre squared per second.
‫In that case, the angular momentum vector and the angular velocity vector, they have the same direction
‫because you can simply extract a constant there.
‫So the H vector equals two times one one one which is your angular velocity vector because remember
‫all of these angular velocities were one.
‫So you can write it like this.
‫So you can clearly see that the angular momentum vector has the same direction compared to the angular
‫velocity vector.