Balance of a construction

Question

I have a construction as shown in the picture. It is a post (CD) who is supported by another post (EF). You can move the point E on DC, but it is a right angle. My question is when the construction will fall in function $\mid CE \mid$, for a given $\mid EF \mid$ and $\mid DC \mid$. With falling, I mean rotating around point F.

Answer 1

If posts are uniform with the same linear math density, then you only need to care about the centre of mass of the whole construction to be between $F$ and $C$.

Let's say that $|DC|=L$, $|EC|=x$, $|EF|=y$, then $\angle ECF=\arctan(y/x)=\alpha$.

Horizontal coordinate (from point E) of CoM of CD is $(\frac12 L-x)\cos\alpha$, coordinate of CoM of $FE$ is $\frac12 y\sin\alpha$. CoM of the whole system is at: $$ d=\frac L{L+y}\left(\frac12 L-x\right)\cos\alpha+\frac y{L+y}\frac12 y\sin\alpha. $$

If $d\le y\sin\alpha$, then your construction is stable. I leave the algebra to you.