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Equilibrium Condition

The sum of the forces acting on an object is zero, and the sum of the moments about any point is zero for static equilibrium.
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The statement of the theorem

Let BB be a rigid body modeled in an inertial Cartesian coordinate system S\mathcal{S}. Let r⃗i\vec{r}_i be the position vector of the point of application of the ii-th external force F⃗i\vec{F}_i. The system is in equilibrium if and only if the following two vector equations are simultaneously satisfied:\n\n1. **Translational Equilibrium (Force Balance):** The net external force R⃗\vec{R} acting on BB must vanish:\nR⃗=∑i=1NF⃗i=0⃗\vec{R} = \sum_{i=1}^{N} \vec{F}_i = \vec{0} \n\n2. **Rotational Equilibrium (Moment Balance):** The net external moment N⃗\vec{N} about any arbitrary origin OO must vanish:\nN⃗=∑i=1NM⃗i=∑i=1N(r⃗i×F⃗i)=0⃗\vec{N} = \sum_{i=1}^{N} \vec{M}_i = \sum_{i=1}^{N} (\vec{r}_i \times \vec{F}_i) = \vec{0} \n\nThese conditions imply that the linear momentum P⃗\vec{P} and the angular momentum L⃗\vec{L} of the body are constant in time, specifically dP⃗dt=R⃗=0⃗\frac{d\vec{P}}{dt} = \vec{R} = \vec{0} and dL⃗dt=N⃗=0⃗\frac{d\vec{L}}{dt} = \vec{N} = \vec{0}.