Static Equilibrium

The second new concept is Static Equilibrium, that is, the object under study is static - it has no motion. Equilibrium means that everything is balanced - forces in this case. First thing we always do is draw a diagram of the object being acted on and draw all the forces and their directions. We have to know this before we can proceed! Equilibrium means that all the forces balance: the vector sum of all forces adds to zero, but it also means all the moments balance: the moments sum to zero as well. Symbolically:

$$\sum{\vec{F}} = 0$$ (1)

$$\sum{m_O} = 0$$ (2)

where $O$ is the point we take our moments about (like the door hinge above).

Now the first equation really stands for two equations. First we pick axis ($x$ and $y$, or at some angle - whatever is most convenient) and then resolve the forces into their components along these axes. Then we set their sum equal to zero. This gives us 2 equations for finding unknowns.

Next we have to do the moments. Once all our forces are described and draw out, we have to pick a point for our moments. It doesn't matter which point, but since any force acting through the point will give no moment (remember the door!), it's best to pick a spot with the most (unknown) forces passing through it, like an actual hinge. This will makes things easier (i.e. less algebra), but any point will give you the right answer (If the object is in equilibrium somewhere it's in equilibrium everywhere!).

Finally, we need a sign convention for moments. In two dimensional problems we have two senses of rotation: clockwise (cw) and counter-clockwise (ccw). The convention is that if a moment would tend to rotate the object ccw about the axis, then the moment is positive, and if the tendency is to rotate cw the moment is negative. The we add all the moments with the appropriate sign and set equal to zero, and this gives us a third equation for out unknowns.

In Figure 2 above, imagine if we pulled the door in the direction of $\vec{F}$. The door would go up and to the right around the hinge - this is the cw-direction, so in this case the moment is negative.