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.

 

[alphy]

Thank you for your question and for providing a sketch for reference. I can provide some guidance on how to approach this problem.

First, let's define some terms to make sure we are on the same page. Angular velocity is the rate at which an object rotates around an axis, measured in radians per second. In this case, the box is rotating around an axis drawn through the corner that is in contact with the ground.

To calculate the angular velocity of the box, we need to consider the forces acting on it. As you mentioned, the box is falling under the force of gravity alone, so we can ignore any other external forces. The box will have both translational motion (falling) and rotational motion (spinning) as it falls.

To find the angular velocity, we need to calculate the moment of inertia of the box. This is a measure of the object's resistance to rotational motion, and it depends on the mass distribution of the object. As you mentioned, finding the moment of inertia for a complex shape like a box can be challenging. However, there are formulas and tables available for common shapes, such as a rectangular box, that can help you calculate the moment of inertia.

Once you have the moment of inertia, you can use the formula for rotational kinetic energy to find the angular velocity. This formula is:

KE = 1/2 * I * ω^2

where KE is the rotational kinetic energy, I is the moment of inertia, and ω is the angular velocity.

As the box falls, its angular velocity will change due to the changing moment of inertia. To find the angular velocity at the instant the box impacts the ground, you can use the conservation of energy principle. This states that the total energy of a system (in this case, the box) remains constant. So, you can equate the initial potential energy (when the box is at its highest point) to the final kinetic energy (when the box impacts the ground) and solve for the angular velocity.

In terms of extrapolating to other scenarios, you can use the same approach of calculating the moment of inertia and using the conservation of energy principle to find the angular velocity. You may need to adjust the equations slightly for different shapes or locations of the center of gravity, but the overall approach will be the same.

I hope this helps you get started on solving this problem. If you have any further questions, please don't hesitate to ask.