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A solid square table has an object on it, as shown below. There's an object at $M$ on one of its diagonals with $\dfrac{OM}{OC}=k$. Find the support force on table legs.

enter image description here

It is easy to see $N_B=N_D$. We do force analysis on the whole to get $$2N_B+N_A+N_C=G.$$ Then we do torque analysis on $AC$. In this case, $N_B,N_D$ doesn't contribute so $$kG=N_A+N_C.$$

I don't see anything else apart from these.

John Rennie
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youthdoo
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  • With four legs the problem is indeterminate. You would need to know the elastic properties of the table. – mike stone Sep 03 '22 at 15:01
  • @mikestone The legs and table has no mass and never bends. – youthdoo Sep 03 '22 at 15:04
  • If it is compleletly rigid, then there is more than one answer. – mike stone Sep 03 '22 at 15:09
  • The problem is identical to finding the barycentric coordinates of point M given a polygon shape. – John Alexiou Sep 03 '22 at 15:36
  • @JohnAlexiou Why is that? – youthdoo Sep 04 '22 at 05:58
  • @youthdoo - Good question. The math is the same, as to why it is rather interesting. The sum of forces must equal to a fixed value, just as the sum of the baryweights must equal to 1. Take the baryweights a scale them up to sum up to $G$. Now the moment balance on the table, relates to the fraction of area each polygon side makes with point $M$ as a triangle, to the total area of the polygon. Hint the cross product returns the area of the trapezoid. The full explanation deserves more than a passing comment. – John Alexiou Sep 04 '22 at 16:32
  • @youthdoo - Here is another post with the solution to the above problem (with example code) – John Alexiou Sep 04 '22 at 19:13

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Since, the table is in rotational equilibrium (as it does not topple), so torque about any point should be zero.

So, try to balance torque about points where torque due to either Na or Nc = 0 (i.e. - about points A and C) as an attempt to increase our equations with less number of variables in an attempt to simplify the math.

You could have also tried to balance torque about points B or D, but that will worsen the math out there for which you will require some geometry and cosine laws.

Sam
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