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I was thinking of building a free standing structure out of plumbing pipe that would go flush against the wall to hang jackets and other items.

Attached is a sketch of the general thought of what the structure would look like.

Sketch

Picturing it in my head, it seems like it should be all good. But then I just got curious about that actual math to get a rough idea of what force it would take to tip it over.

This is assuming that the pipe is strong enough to hold.

I started looking into the calculations for levers, but it seems to be like there are a couple levers chained together. At that point it became beyond me.

Note: I plan on taking other safety precautions like attaching to the wall. But I am more curious theoretically now.

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  • $\begingroup$ Check out centre of gravity. $\endgroup$ – Solar Mike Dec 9 '19 at 5:04
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This structure as it is is not going to tip over no matter how much force you apply on the top where you have the arrow at 12 inches. Unless the force is so much that the pipe is going to break or crush under the compression stress.

Because the arrow falls within the footprint of your hanger. Let's do a quick calculation:

Let's say you hang a weight of P lbs from the arrow and let us call the point on the floor at the left corner A and the one on the right corner at the end of 18inch B.

Taking the moments about A and knowing that for equilibrium the sum of moments should be zero, we have:

$$ \Sigma M_a=0 ,\quad P*12+ B_v*18=0 \\ B_v=- \frac{P*12}{18}=- \frac{2}{3}P$$

We note that a reaction vertically pointing up ( by the negative sign) by the amount of 2/3 of any weight we hang on top is sufficient to keep the hanger balanced. And $\ A_v = -\frac{1}{3}P $

In real life though if you hang a really heavy object from the arrow, your vertical pipe can bend forward under the bending moment, or the corner on the bottom may give a bit so the hanging point moves horizontally forward and then the hanger's balance can become shaky.

As a general rule whenever you have the weight's vertical projection fall inside the footprint of the support the structure is balanced. It could be a huge weight of a high-rise building, still, it is vertically balanced and will not tip over.

In the case of the buildings, there are some safety factors that have to be applied depending on the use and type of the building.

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  • $\begingroup$ That makes a lot of sense! $\endgroup$ – Michael Dec 9 '19 at 21:18

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