Static Equilibrium

A system is an isolated part or portion of a machine that we want to study. A system, under this definition, may consist of a particle, several particles, a part of a rigid body, an entire rigid body, or even several rigid bodies.

If we assume that the system to be studied is motionless or has constant velocity, then the system has zero acceleration. Under this condition the system is said to be in equilibrium. The phrase static equilibrium is also used to imply that the system is at rest. For equilibrium, the forces and moments acting on the system balance such that:

$$\begin{array}{r}\sum \mathbf{F}=0\\ \sum \mathbf{M}=0\end{array}$$

which states that the sum of all force and moment vectors acting upon a system at equilibrium is zero.

Rigid Body Equilibrium in 3D

Scalar equations:

$$\begin{array}{r}\sum F_x=0,\sum F_y=0,\sum F_z=0\\ \sum M_x=0,\sum M_y=0,\sum M_z=0\end{array}$$

These can also be obtained as vector equations:

$$\sum \vec{F}=0,\sum {\vec{M}}_o=\sum (\vec{r}\times \vec{F})=0$$