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‫Welcome back.

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‫Let's derive the equations for the rotational motion now in the translational motion.

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‫We said that in the most general case, the sum of the forces in the inertia frame equals the change

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‫of linear momentum with respect to time.

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‫And here Gummo, that vector E is the velocity vector in the inertial frame and it's in metres per second.

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‫Now, when we enter into the world of rotation, then there is a similar law.

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‫There is something called angular momentum, which we will call H Vector, and we are in the inertia

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‫frame right now.

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‫And the general definition of it is that it equals the mass movement of inertia in the inertia frame

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‫multiplied by this theta dot vector in the inertial frame here.

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‫The eye superscript e is the inertia tensor.

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‫But remember, now we're working in the inertial frame and therefore all these mass movement of inertia

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‫calculations, they happen about the inertial frame axis.

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‫So this is your inertia frame where you have X, Y and Z, and since it's an inertia frame, it's fixed

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‫to the ground, it doesn't move.

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‫And since we're measuring the mass movements of inertia in the inertia frame, then all these mass moments

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‫of inertia and problems of inertia, they're all measured about these three axes here, capital X,

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‫capital Y and capital Z.

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‫And then the theta dot vector in the inertia frame consists of those time derivatives of the three oilor

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‫angles, five dot, dot, dot and side dot.

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‫And that's also because we are right now in the initial frame.

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‫And so if you multiply them together, then you will get your angular momentum in the inertia frame.

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‫Now, let's look at the two, the case for second, let's look at our car from the top view and let's

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‫just for now make the inertial axis aligned with the car's body frame axis like this.

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‫You can see that in 2D on an X Y plane you only had.

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‫Izzie, as your mass moment of inertia because you were rotating about the inertia frame or body frame

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‫Z axis and your angular velocity was also only about the Z axis.

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‫So that's the only term you cared about in 2D.

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‫However, in 3D, you have this entire matrix to take into account.

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‫And so in 2D, if you multiply isobars Z times side that you will get your angular momentum.

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‫There is also about the Z axis.

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‫And you can see that into the the angular velocity vector and the angular momentum vector are always

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‫in the same direction either towards you or away from you according to the right hand rule.

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‫In the age and Passi that are just scalars and Izi is like a proportionality constant.

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‫That's why they are in the same direction, however, in 3D.

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‫The angular momentum vector is a three by one vector and instead of BPCI that you have this Seeta,

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‫that vector in the inertia frame, which is this one, which is also a three by one vector.

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‫And now you also have a three by three masked moment of inertia matrix, which is an inertia Tenzer,

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‫that means that in 3-D, the age vector and then the angular velocity vector don't have to be in the

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‫same direction.

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‫Let's assume that all the products of inertia are zero KG's times meter squared.

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‫So you only have a diagonal mass moment of inertia matrix without the products of inertia.

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‫And by the way, that is the case in our course.

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‫So this is your angular momentum vector.

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‫This is your mass moment of inertia matrix where all your products of inertia are zeros.

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‫And this is your angular velocity vector.

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‫Let's assume that five dot equals Seeta, dot equals SPSSI, dot equals one radians per second.

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‫So all of them have the same values.

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‫And also let's assume that exact equals I y y equals I z the equals, let's say two kg's times meter

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‫squared.

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‫So all the three mass moments of inertia, they also have the same values.

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‫In that case each X equals I x X times five dot equals two times one equals two kg's times meter squared

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‫per second.

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‫Similarly, you find your H, Y and Z where you multiply ie y, y times Theta dot and I's time, SPSSI

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‫that respectively.

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‫So your angular momentum vector is two, two and two KG's metre squared per second.

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‫In that case, the angular momentum vector and the angular velocity vector, they have the same direction

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‫because you can simply extract a constant there.

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‫So the H vector equals two times one one one which is your angular velocity vector because remember

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‫all of these angular velocities were one.

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‫So you can write it like this.

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‫So you can clearly see that the angular momentum vector has the same direction compared to the angular

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‫velocity vector.