gravity & balance

They will move and try to fall, but since they must always remain the same distance from each other, and one cannot move, they will rotate. There is a paper called Advanced Character Physics which gives an easy, stable, efficient, and flexible way to do this. Despite the paper's title containing the word "Advanced", Thomas Jakobsen does a great job on making it very easy to understand. If you use Verlet integration to apply the force of gravity, but then also apply the rigid constraints of each point, the square will naturally rotate about its upper left corner as the other corners try to fall. I suggest you give it a try. It really is dead simple, and I'd be happy to help if you got stuck.

To simplify, you can find the centre of gravity of your object which should be somewhere near the centre if you have equal density throughout.

As soon as you start to simulate what will happen, you will see that the centre of gravity point will "fall" downward until it lines itself up right below the corner that is pinned.

Listen to MikeTacular. You are not going to find an easier method to do mechanical simulations, and the results are quite acceptable for many situations. It will definitely do the right thing in your simple case.