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TL;DR Given the point of centre of mass, width of base and height, is there a way to calculate the angle where the object will fall over?

The TL;DR of this question pretty much sums it up, however I wanted to give an example.

Let us say that we have a cuboid made of uniform weight wood with the dimensions 10 (height) x 5 (width) x 5(breadth). Due to the fact that it is uniform weight, we can tell the CoM (centre of mass) will be at 5 x 2.5 x 2.5.

If we push the object, it will rotate around a turning point, for this let's say the right side (from our perspective) and it's angle will increase compared to the table. When the CoM reaches over the turning point, it will fall over and collapse.

Is there a mathematical way of working this out? Without experimentation, or are there other variables I need.

Note - First question, and I wasn't sure whether to post it on mathematics or physics. Do you think I posted on the wrong stackexchange site?

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  • $\begingroup$ "If we push the object, it will rotate around a turning point, for this let's say the right side (from our perspective) and it's angle will increase compared to the table. When the CoM reaches over the turning point, it will fall over and collapse." this seems to answer your question. $\endgroup$
    – Danu
    Dec 5, 2014 at 8:31
  • $\begingroup$ Ah, but how do you work out when the CoM reaches over the turning point? @Danu $\endgroup$ Dec 5, 2014 at 9:35

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The answer is actually more simple that you think. The tipping point is reached when the x coordinate of the center of mass is the same as the horizontal position of the last contact point. In the case of your cube there is only a contact point. But for instance, if you had a trapezoid at a table, it would tip if the center of mass is beyond the critical point at the end of any of the two sides of the base.

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  • $\begingroup$ Ah, thanks for this! I'll accept this answer in three minutes. It's surprisingly simple =) $\endgroup$ Dec 5, 2014 at 8:15

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