Simple Statics Question: Three Legged Table

[meichenl]

Here is the setup to my question:

You have a long table supported by two legs (think of the table as two-dimensional. in the real 3D table each "leg" could be a pair of legs, one behind the other). The tabletop is massless, but has some massive books sitting on it at various places.

You want to know how much force is exerted on the table by each of the two legs. This requires two conditions: that the sum of the forces exerted by the legs is equal and opposite to the total weight of the books, and that the sum of the torques exerted by the legs is equal and opposite to the sum of the torques exerted by the books. If you write out the equations for these conditions and find one the legs has to exert a negative force, the table flips up off the ground.

Here is the question:

What if you have three legs? Now you need another condition. What is the third condition?

Here are my thoughts:

I think there isn't enough information to answer the problem, although I'm not sure. If you assume the table is a perfectly-rigid body, and that the forces on it are all vertical, then you can specify its position by the height of the table off the ground and by the angle its top makes with respect to the ground. The table can't slide sideways or forwards/backwards, and it has no "pitch" or "yaw", only "roll" (if the long side of the table were the wings of an airplane). So the table has only two degrees of freedom, and three legs are superfluous. As long as the total torque and total force on the table are zero, it won't move. There are infinitely many solutions that use three legs to meet these two requirements. So you can't tell how much force each leg needs to exert.

But this assumes the table is a rigid body. If we acknowledge the table to have some bulk modulus, for example, then three legs pushing in it could bend it into different shapes. But it seems to me that we'd have to know a bit about the material properties of the table before we could answer the question.

 

[rcgldr]

If you have a heavy weight placed outside the plane formed by any two of the three legs, the table can still topple. If the table is considered rigid, then this is a "statics" engineering problem. The load on each leg will depend on the relative proximity of the weights. I think you can also just find the center of gravity for all the weights combined, and treat that as a single mass for the load on the legs.

 

[meichenl]

I'm modeling the table as a straight line. The legs don't form a plane.

Think of a regular picnic table. It has four legs in two sets. The question is, what if it had six legs in three sets? Also, think of each set of legs as being just one unit.