= Static equilibrium
<Static equilibrium> is a state in which a mechanical system remains at rest in a specified <reference frame>. For a <rigid body> in an <inertial frame>, zero resultant external <force> and zero resultant external <torque> are the balance conditions:
$$
\sum_i\mathbf F_i=\mathbf0,\qquad
\sum_i(\mathbf r_i-\mathbf r_O)\times\mathbf F_i=\mathbf0.
$$
The body must also initially be at rest: zero acceleration alone does not exclude uniform translation or steady rotation.<University Physics Volume 1 by William Moebs and others>[Section 12.1, definition and equations 12.2 and 12.5]{ref}
A <free-body diagram> identifies the body being analyzed and includes its external loads and support reactions. Internal action–reaction pairs cancel when both interacting parts belong to the chosen system. Choosing a different point $O$ for the <torques> changes their sum by a term proportional to the resultant <force>. Thus, when the resultant <force> is zero, a vanishing resultant <torque> about one point also vanishes about any other point.<University Physics Volume 1 by William Moebs and others>[Section 12.1, change-of-origin argument following equation 12.6]{ref}
For an original example, take an ideal massless horizontal beam of length $4\,\mathrm m$, supported at its ends, with a downward $120\,\mathrm N$ load $1\,\mathrm m$ from the left support. Let the upward support reactions be $R_L$ and $R_R$. Vertical <force> balance and <torque> balance about the left end give
$$
R_L+R_R=120\,\mathrm N,\qquad
4\,\mathrm m\,R_R-1\,\mathrm m\,(120\,\mathrm N)=0,
$$
so $R_R=30\,\mathrm N$ and $R_L=90\,\mathrm N$. The nearer support carries more load because the farther reaction has the larger moment arm. Two equal and opposite <forces> alone would not guarantee equilibrium: a <force couple> can have zero resultant <force> and a nonzero <torque>.
A simple apparatus test suspends a beam from two <spring balances> and moves a known load along it. Account separately for the beam's own <gravitational weight> and its <center of mass> before comparing the measured reactions with the ideal calculation. The sum of the readings should track total downward load, while their difference changes with load position. Finally, <static equilibrium> does not imply <dynamical stability>: an exactly upright <inverted pendulum> balances at rest but departs from that state after a small disturbance.
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