Now, let's work on how work and energy relationship could be obtained in 3D rigid body dynamics. To remind you, there was work and energy relationship like by integrating Newton's second law, F equals MA over the displacement, the force applied. You can have the work done or the change equivalent to the change of kinetic energy. When you're handling the rigid body, those work done is being defined by the work done per each particle and its summation over the whole rigid body, and then if you rewrite the force as a particle multiplied by the acceleration, you will end it up having kinetic energy over single-particle integrated over the rigid body, which is the definition of the kinetic energy for the rigid body. Now, if you rewrite the velocity vector as a vector to the center of mass and the displacement from the center of mass to the particle, it will be written as a translational kinetic energy and rotational kinetic energy. In two dimension, this will become an I omega square. However, for 3D, angular momentum is not as simple as just a scalar I. This term, Rho I, that term in 3D is going to be in general expressed by the omega cross Rho, and whenever you have a, A cross-product P dot-product C, it'll be rewritten as A dot-product P cross product C. So this could be changes to this one, and if you arrange that this omega terms out of the integral, then you have is M Rho I cross-product, omega Rho I. So it's R cross product B. This is angular momentum with respect to the center of mass. Or you can have as, kinetic energy can be written as 1.5 of V bar and linear momentum MV bar and 1.5 omega, and the angular momentum with respect to the G. To obtain the kinetic energy, the very first step is figure that out what the angular momentum, and then do the inner product with the angular velocity, and then this may be the general form. You don't have to memorize this. You just have to know the definition of the kinetic energy, the relationship with the kinetic energy in 3D like this, and if necessary just derive the H of G and plug it in. To sum up, the 3D kinetics relationship could be summarized this way. Newton's law F equals MA could hold for a 3D, but moment equals I Alpha is not going to be applied anymore because in 3D it's not simply an Alpha as they apply on the moment of inertia I. We have to derive the angular momentum in this way, and it's derivative consist of its time derivative and its omega cross H term to the rotation. For the kinetic energy for a particle, it was 1.5 MV square, and for a rigid body you should add rotational energy term I omega square. But for the rigid body, angular momentum is not simply the I omega. So now you have the form that kinetic energy for the rigid body in 3D is going to be same for the translational energy, and for the rotational energy you should multiply omega and inner product of angular momentum with respect to the G. Do you remember this example? We previously obtained about what the angular momentum of the desk with respect to point O. Now, what if you are asked to find out the kinetic energy? So H0 is obtained previously this way. It's like you obtain the angular momentum with respect to the center of mass and the angular momentum of the center of mass treated as a particle. Now, since your kinetic energy now could be formulated as 1.5 MV bar square, and 1.5 omega dot H of G, you can plug in the V bar term, the velocity of the center of mass term as what here? Yes, capital omega cross R bar, and your angular momentum with respect to the G. Here, you can use it as omega dot product, and angular momentum with respect to the G. So we briefly go over how you can get the kinetic energy for the 3D rigid body. So later when you're supposed to find the kinetic energy, the very first step is you should figure what's the angular momentum. That's why with the sub chapter 7.4 has been composed by angular momentum first, and use that information. Take a time derivative to obtain the moment equation, and then you can obtain the work and energy in terms of angular momentum in 3D. Thank you for listeningalf.
