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Free Body Diagram

A graphical representation showing all forces acting on a body, simplifying force analysis in statics problems.
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The statement of the theorem

Let SS be a rigid body occupying a region Ω⊂R3\Omega \subset \mathbb{R}^3. The system is subjected to a finite set of external forces {F⃗1,F⃗2,…,F⃗N}\{\vec{F}_1, \vec{F}_2, \dots, \vec{F}_N\} and corresponding points of application {r⃗1,r⃗2,…,r⃗N}∈Ω\{\vec{r}_1, \vec{r}_2, \dots, \vec{r}_N\} \in \Omega. A Free Body Diagram (FBD) is the formal representation of the force vector field F⃗ext:Ω→R3\vec{F}_{ext}: \Omega \to \mathbb{R}^3 defined by the superposition of these external forces. Mathematically, the diagram asserts the existence of a set of force vectors {F⃗i}\{\vec{F}_i\} such that the system is in static equilibrium, which requires the following vector summation conditions to hold:\n\n∑i=1NF⃗i=0⃗\sum_{i=1}^{N} \vec{F}_i = \vec{0} \n\n∑i=1Nτ⃗i=∑i=1N(r⃗i−r⃗0)×F⃗i=0⃗\sum_{i=1}^{N} \vec{\tau}_i = \sum_{i=1}^{N} (\vec{r}_i - \vec{r}_0) \times \vec{F}_i = \vec{0} \n\nwhere r⃗0\vec{r}_0 is an arbitrary origin point. The FBD is the graphical visualization of the force density f⃗(r⃗)=∑i=1NF⃗iδ(r⃗−r⃗i)\vec{f}(\vec{r}) = \sum_{i=1}^{N} \vec{F}_i \delta(\vec{r} - \vec{r}_i) acting on the body, confirming that the net resultant force and net resultant torque are zero.