# Modeling a stool with three legs pushed on one leg

If a stool with three legs is pushed (horizontally) on one leg in the direction of the axis passing through the other two legs, it starts both moving and rotating.

Is it correct to assume 1/3 of force on each of the legs? This comes a bit in contrast with intuition. If we imagine the legs flexible, then the pushed leg should deform more (i.e. stronger friction force).

What is the correct modeling of the friction forces?

p.s. this seems a simplified version of Push a box in a plane with friction. How to deal with the rotation? but there's no clear answer in that thread.

If you are pushing exactly at ground level you can expect 1/3 of the vertical force on each leg - but if you are pushing any higher, the torque balance will make the pushed leg "lighter".

Key to understanding the problem is that this is an unstable equilibrium - you cannot aim exactly between the other two legs. Even if you did, there will be small random fluctuations in the coefficient of friction between the stool legs and the floor. As soon as a tiny deviation from equilibrium occurs, you impart a rotational torque on the stool; and as soon as the stool begins to rotate, you have moved away from equilibrium and the rotational torque will be amplified because the distance of the two far legs to the line of force wil no longer be equal.

Summary:

1. Push exactly along "line of equilibrium"
2. Fluctuation in coefficient of friction
3. Unequal forces on two legs
4. Net torque rotates stool slightly
5. Distance of legs to line of force no longer equal
6. Rotation established

I will try to draw a picture when I have a bit more time