Rules to make a box stable, hard to tilt

Question

I have already built a cube-shaped box out of some wood panels I had lying around. It is perfectly solid, I can push it from every side and it will not tilt.

The next box should be not cube-shaped but taller than deep, let's say 20 cm tall, 20 cm wide and Z cm deep. If I make it not deep enough, for example 5 cm deep, it will be easy to tilt it by pushing it from the front.

How small can I make the Z depth while still making the box stable enough? Are there numerical/engineering rules for that?

What if I add more weight to the base? And how much does the weight and thickness of the walls of the box contribute to the stability of the box?

Answer 1

Find the center of gravity, which in a rectangular prism, will be located at x/2, y/2, z/2. The box will tip when the center of gravity moves outside the footprint of the box. (or just look at the intersection of diagonals)

Or, if you like, when the grey line in the diagram below passes vertical.

Adding weight to the bottom will lower the center of gravity, meaning that the grey line is less steep, and will require more of a tip to put it beyond vertical.

Here, with weight added, a lower CG makes the box more stable.

If you know that the box is likely to tip in one direction only, (such as being constrained by a wall, and only able to tip away) you can put all your weight in the corner, to move the CG towards the wall and change the angle even more. But, this will make it tippier in the other direction.