Static is a branch of mechanics that studies the forces acting on objects that are at rest. The static is particularly important in the construction of buildings, bridges, viaducts and other structures that should be at rest (static).
It is important to differentiate a rigid body from a particle.
From Newton's First Law, it is known that: for a particle at rest to remain in this state, it is necessary that the net force on it is zero. Mathematically we have: $$\vec{F}_{net} = \sum_i^n \vec{F}_i = \vec{0},$$ where \(\vec{F}_{net}\) is the resultant force and \(\vec{F}_i\) are the \(n\) forces acting on the system. We can rewrite this explicitly in terms of the horizontal components \(x\) and vertical \(y\) , thus obtained: \begin{array} \\ F_{net,x} = \sum_i^n \pm F_{i,x} =\\ = \pm F_{1,x} \pm F_{2,x} \pm ... \pm F_{n,x} = 0,\\ F_{net,y} = \sum_i^n \pm F_{i,y} =\\ = \pm F_{1,y} \pm F_{2,y} \pm ... \pm F_{n,y} = 0, \end{array} where \(F_{net,x}\) and \(F_{net,y}\) are the algebraic values of the horizontal and vertical components of the net force, that is, the magnitude of the force and the algebraic sign indicating the direction of it. Respectively, \(F_{i,x}\) and \(F_{i,y}\) are the magnitudes of the components of the forces acting on the system, and the signal \((\pm)\) in front of the module must be chosen appropriately, according to the direction of the force component.
The above figure illustrates a problem which can be resolved by considering only a punctual particle, in this case, the node between the ropes. The net force is: $$ \vec{F}_{net} = \sum_i^n \vec{T}_i = 0,$$ or explicitly: \begin{array} \\ F_{net,x} &= -T_{a, x} + T_{b,x} + 0 = 0,\\ F_{net,y} &= +T_{a, y} + T_{b,y} - T_{c,y} = 0, \end{array} where the tractions (\(T_{i,j}\)) are positive numbers.