Equilibrium Analysis for Concurrent Force Systems
If a body is in static equilibrium, then by definition that body is not accelerating. If we know that the body is not accelerating then we know that the sum of the forces acting on that body must be equal to zero. This is the basis for equilibrium analysis for a particle.
We must also remember that forces are vectors, they have both a magnitude and a direction. In order to add together the vectors we will use the vector components in the x, y and sometimes z directions. To ensure that the sum of the vectors is truly equal to zero, the sum of the magnitudes of the components in any one direction must be equal to zero. To solve the equilibrium equations we will break the single vector equation (sum of forces is equal to zero) down into two scalar equations for 2D problems (sum of the x components of the force vectors is equal to zero, and sum of the y components of the force vectors is equal to zero) or down into three scalar equations for 3D problems (sum of the x components of the force vectors is equal to zero, sum of the y components of the force vectors is equal to zero, and sum of the z components of the force vectors is equal to zero).
Once we have written out the equilibrium equations, we can solve the equations for any unknown forces.
Finding the Equilibrium Equations:
The first step in finding the equilibrium equations is to draw a free body diagram of the body being analyzed. This diagram should show all the force vectors acting on the body. In the free body diagram, provide values for any of the know magnitudes or directions for the force vectors and provide variable names for any unknowns (either magnitudes or directions).
Next you will need to chose the x, y, and z axes. These axes do need to be perpendicular to one another, but they do not necessarily have to be horizontal or vertical. If you choose coordinate axes that line up with some of your force vectors you will simplify later analysis.
Once you have chosen axes, you need to break down all of the force vectors into components along the x, y and z directions (see the vector decomposition page for more details on this process). Your first equation will be the sum of the magnitudes of the components in the x direction being equal to zero, the second equation will be the sum of the magnitudes of the components in the y direction being equal to zero, and the third (if you have a 3D problem) will be the sum of the magnitudes in the z direction being equal to zero.
Once you have your equilibrium equations, you can solve these formulas for unknowns. The number of unknowns that you will be able to solve for will be the number or equations that you have.
Particle vs. Extended Body Analysis:
In any body to be in equilibrium, the sum of the forces and the sum of the moments has to be equal to zero. For a particle, or for a concurrent force system, there will no moments acting on the body though. For this reason the moment equations will not be helpful for particles (the equation would end up being zero is equal to zero). Therefore in particle analysis will rely solely on the sum of the forces being equal to zero. When we start analyzing extended bodies with non-concurrent forces we will begin to encounter moments, and we will need to ensure the the sum of the moments as well as the sum of the forces are equal to zero.