‫Welcome back.
‫In this video, we will discuss the man's moment of inertia and the inertia tensor in 3-D.
‫If we look at the first part of the core series, then over there we only dealt on an X Y plane, which
‫is a to the plane.
‫So our mass moment of inertia was only measured about one axis.
‫And that axis was about the body frame Z axis that popped out of the screen towards you like this.
‫However, things get more complicated when we talk in 3D, in 3D.
‫This is your body frame axis.
‫And so you can have moments and angular accelerations about three axis X, Y and Z.
‫And so you can also measure mass movement of inertia about these three axis.
‫In other words, you measure the resistance to rotation about these axis.
‫Remember, mass movement of inertia was resistance to rotation.
‫For example, you can have this object, here is your mass moment of inertia about this axis, which
‫is the x axis and let's call it like this I, X and X, then this one here, which would be I y y,
‫some mass movement of inertia about the body from y axis and finally mass movement of inertia about
‫the axis.
‫So the body frames the axis.
‫So you see you have three axis and you can rotate about those axis.
‫And then the mass moments of inertia for all these axis might be different.
‫For example, it might be that it is much easier to rotate this object about the Z axis than, for example,
‫about the X or Y axis, because the mass movement of inertia, perhaps about the Z axis is less than
‫I x, x or Y, Y, Y.
‫It means that about the Z axis, you need less movement to angular really accelerate the object compared
‫to the moment that you would need in order to accelerate angular the object about the X and Y axis at
‫the same rate.
‫So in other words, if I want to accelerate this object five radians per second squared above the x
‫axis and y axis and the axis, then if mass movement of inertia for the Z axis is less than the other
‫two, then that means that I need to apply less movement about the Z axis in order to reach that five
‫radians per second squared.
‫So I need to apply more movement if I want to reach five radians per second squared about the x axis
‫and about the Y axis.
‫And you can formulate all that in a vector matrix form like this, you can have a matrix where the diagonal
‫elements are the mass moments of inertia about the X, Y and Z axis, respectively.
‫And then the rest of the elements are zeros.
‫And then you can multiply that matrix by a vector where the first element is P dot and remember P,
‫Q and R, where the angle of velocity is about the body frame X, Y and Z axis respectively, and they
‫were in radians per second.
‫So if I take the time derivative of P then I will get the angular acceleration about the body frame
‫x axis.
‫Then the second element would be Q dot the angular acceleration about the body frame y axis and then
‫R dot, which is the angular acceleration in both the body frame, the axis and all these variables
‫here.
‫They have units radians per second squared because they're angular acceleration square and all that
‫equals moment about the x axis, moment about the Y axis and moment about the Z axis.
‫If you write this system out, then this is what you will get.
‫That's how you connect moments, mass moments of inertia and angular accelerations about their respective
‫axis.