haruspex said:
You have a typo in (3), h instead of l.
Thank you!
haruspex said:
All equations down to "##(3)\rightarrow##" are ok but should really be inequalities expressing constraints. Otherwise it is unclear which side of that constitutes stability.
I will be more clear, the situation I described in the calculation refers to angles ## \phi ## in the quarter of the positive plane ( ## 0<= \phi < \pi /2 ## )
and ## 0<= \theta < \pi /2 ## .
##\theta _m ## - is the maximal ##\theta## (same with ##\phi ## )
I am not letting ##\theta ## be negative and since the problem is symmetric around the x-axis I calculate ##\phi_m ## only on the quarter positive plane and I use it as ## -\phi _m < \phi < \phi _m ##.
haruspex said:
Beyond that, I do not understand your logic. You took the condition that it is about to tip about AB and combined that with the condition that it is about to tip about BC. So the equation that resulted only applies when it is about to tip about both. Is that interesting? And what does it mean when you turn that into an inequality? That will not be a bound on not tipping. It might be a bound on whether it tips one way rather than another.
Maybe I am wrong about that, but following on from what I wrote above, I guess the rotation axis can be somewhere between AB and BC.
haruspex said:
You should have four separate inequalities, one being the condition that it does not tip about AB, one that it does not tip about BC, then CD and DA likewise. All four must be met for stability.
At that point you can see if you can rewrite them as bounds on ##\phi##.
following on from all that I wrote above, I do not think I need to check it for CD and DA (since the symmetry).
but I think I should calculate ##\theta _m## in the condition where ## \phi =0 ## as well ,
hence:
##(3)\rightarrow## ##cos(\theta_m1 ) = (l/2) *[(mg+2W) / 50W]##
if ##\phi ## is not zero then we can use the expression from # 12 :
## cos(\theta _m2) = (h/l) * [(mg+W)/(50Wsin(\phi _m) )##
now ## \theta _m ## is the max {##\theta _m1##,## \theta _m2##}
please let me know if I was clear enough and if that makes sense to you.