A particle is said to be in equilibrium if it remains at rest if originally at rest, or has a constant velocity if originally in motion. Most often, however, the term "equilibrium" or, more specifically, "static equilibrium" is used to describe an object at rest.
To maintain equilibrium, it is necessary to satisfy Newton's first law of motion, which requires the resultant force acting on a particle to be equal to zero. This condition is stated by the equation of equilibrium,
$$ \Sigma\mathbf{F} = \mathbf{0} $$
where $\Sigma\mathbf{F}$ is the vector sum of all the forces acting on the particle.
This follows from Newton's second law of motion, which can be written as $\Sigma\mathbf{F} = m\mathbf{a}$. Since the force system satisfies $\Sigma\mathbf{F} = \mathbf{0}$, then $m\mathbf{a} = 0$, and therefore the particle's acceleration $\mathbf{a} = 0$. Consequently, the particle indeed moves with constant velocity or remains at rest.
To apply the equation of equilibrium, we must account for all the known and unknown forces ($\Sigma\mathbf{F}$) which act on the particle. The best way to do this is to think of the particle as isolated and "free" from its surroundings. A drawing that shows the particle with all the forces that act on it is called a free-body diagram (FBD).
Before presenting a formal procedure as to how to draw a free-body diagram, we will first consider three types of supports often encountered in particle equilibrium problems.
If a linearly elastic spring (or cord) of undeformed length $l_0$ is used to support a particle, the length of the spring will change in direct proportion to the force $\mathbf{F}$ acting on it. A characteristic that defines the "elasticity" of a spring is the spring constant or stiffness $k$.
The magnitude of force exerted on a linearly elastic spring which has a stiffness $k$ and is deformed (elongated or compressed) a distance $s = l - l_0$, measured from its unloaded position, is
$$ F = ks $$
If $s$ is positive, causing an elongation, then $\mathbf{F}$ must pull on the spring; whereas if $s$ is negative, causing a shortening, then $\mathbf{F}$ must push on it.
All cables (or cords) will be assumed to have negligible weight and they cannot stretch. Also, a cable can support only a tension or "pulling" force, and this force always acts in the direction of the cable. The tension force developed in a continuous cable which passes over a frictionless pulley must have a constant magnitude to keep the cable in equilibrium. Hence, for any angle $θ$, the cable is subjected to a constant tension $T$ throughout its length.
If an object rests on a smooth surface, then the surface will exert a force on the object that is normal to the surface at the point of contact. In addition to this normal force $\mathbf{N}$, the cylinder is also subjected to its weight $\mathbf{W}$ and the force $\mathbf{T}$ of the cord. Since these three forces are concurrent at the center of the cylinder, we can apply the equation of equilibrium to this "particle," which is the same as applying it to the cylinder.
Since we must account for all the forces acting on the particle when applying the equations of equilibrium, the importance of first drawing a free-body diagram cannot be overemphasized. To construct a free-body diagram, the following three steps are necessary.
