Angular velocity of a box about a corner

[atayne]

I am attempting to find the angular velocity of a 3D box that is balanced on one of its corners and allowed to fall under the force of gravity alone at the instant the box impacts. Take the simplest case for example where the box is square on all sides and falls in a way that results in one of its faces hitting the ground flush (I believe this would result in the maximum instantaneous angular velocity as it has the furthest to fall in this direction). Once I figure out the simple case, I'll need to extrapolate it to other scenarios where the box is rectangular and the center of gravity is not located dead center.

I don't really know where to start on this because the moment is changing as the box falls and finding the mass moment of inertia for a complex shape such as this appears to be very difficult to do by hand. I know I need to calculate the angular velocity about an axis drawn through the CG and then use the Parallel Axis Thereom to project it down to the floor.

If you guys could get me started thinking about this the correct way, I would feel much better!

See sketch

 

[OldEngr63]

Look up the expression for the parallel axis theorem in 3D and then apply it to this situation. The shift is one half a side in each coordinate axis from the original, CM centered coordinates to the corner centered coordinates. For the case of the cube, the rotation will be about two coordinate axes.

 

[atayne]

I ended up calculating the change in potential energy of the CM from its highest point to when the cube impacts the ground. From this I was able to find the angular velocity of the "inverted pendulum" so to speak. Thanks for the response.

 

[OldEngr63]

Be sure you used the correct effective MMOI for your pendulum. Otherwise, your result will be in error. The correct value for the MMOI gets you back to the original question, so it is not clear to me that you have answered your original question at all.