# General plane motion and freely floating rigid body

The translational motion of the center of mass (CM) is given by solving the second law: $$Md \vec V/dt = \vec F_{ext}$$ where $$M$$ is the total mass, $$\vec V$$ is the velocity of the CM, and $$\vec F_{ext}$$ is the net external force. This applies for any system of particles, in a rigid body or not.

The following discussion of the rotational motion assumes a rigid body. The rotational motion about the moving center of mass is complicated to evaluate; for example, inertia is a tensor for general 3D rotation. A typical approach is to first find the principal axes for the body; axes for which the products of inertia in the inertia tensor are zero. The principal axes form the body axes, fixed in the body with origin at the CM. The body axes rotate with the body. To evaluate the motion with respect to a fixed set of space axes with origin at the CM (the space axes are fixed and do not rotate), the Eulerian angles can be used. Then, the rotational motion can be modeled with a Lagrangian using the Eulerian angles. This approach is discussed in many intermediate/advanced physics mechanics tests, such as: Symon, Mechanics and Goldstein, Classical Mechanics. I suggest you consult such a textbook for the details, and for examples, such as how to identify the principal axes, the motion of a symmetrical top, and torque-free motion. In general, numerical approaches are necessary, especially for non-symmetrical bodies .

In addition to the information you provide, the density of the plate is also required to set up the equations to evaluate $$T'$$ using the approach summarized above. The principal axes for your plate- assuming constant density- are easy to identify due to symmetry