Question

cosider an object is placed in 4 leg rectangular table. if we know the forces acting on 4 legs how can we find the object location in table

Answer 1

Consider the table to be lying in x-y plane and the origin to be at the geometric center of the table (center of mass). The dimensions of table are l=length ,b=breadth. Let mass of object is m.
Let position vector of m is r = x i + y j
The forces N1,N2,N3,N4 are acting in z direction.See the attached file. I have shown the 4 forces acted upon the chair by the ground in the +z direction and the force applied by the object equal to its weight in the -z direction.

Now find the torque applied by each force about O and put total torque equal to zero.
\[\tau _{mg} = -mg \ k \times (x \ i+y \ j)\]

\[\tau _{N1} = N1 \ k \times (-l/2 \ i+b/2 \ j)\]
\[\tau _{N2} = N2 \ k \times (l/2 \ i+b/2 \ j)\]
\[\tau _{N3} = N3 \ k \times (l/2 \ i-b/2 \ j)\]
\[\tau _{N4} = N4 \ k \times (-l/2 \ i-b/2 \ j)\]

Put the sum of all these five torques equal to zero. The eqn will have two components in j and i. Put separately both components equal to zero and solve for x and y -the position of the object on the table.

Answer 2

the net torque acting about any point should be zero since the whole system is in equilibrium.
preferably, any end point on the table can the chosen.