particle equilibrium statics
In a tug of war, when both sides pull equally the knot stays put. Not because there is no force, but because the forces cancel, and if the sum of the forces at a point is exactly 0 the point does not move. That is the equilibrium of a particle, and a particle here is an object seen as a point, its size ignored. Rotation is set aside and only the sum of forces matters, and the condition that the sum is 0 becomes, in the plane, a horizontal sum of 0 and a vertical sum of 0. Those two equations solve an unknown force, the pull of a cable. Move one force and see when the knot loses its balance.
Several forces act together at one point. Call the point a particle, and the forces join into one resultant R. If that sum is not 0 the particle is pushed in the direction of that R, one arrow left dragging the point. Turn one force and the resultant swings, and the larger it is the harder the push. Equilibrium is the moment the leftover arrow vanishes, the moment the sum becomes 0. The next picture checks that.
Join the forces head to tail and the resultant was the arrow from the first tail to the last head. A resultant of 0 means the last head returns exactly to the first tail, so the arrows form a polygon closed without a gap. Drag the last head to close the polygon and the gap that stays open is the resultant, and when that gap shrinks to 0 there is equilibrium. Forces in equilibrium always draw a closed loop that returns to its start, and that is the picture of equilibrium.
Seeing whether the picture closed is intuitive, but the calculation is more exact in components. For the sum to be 0 the horizontal component must be 0 and the vertical component must be 0, so one condition splits into the two equations ΣFx = 0 and ΣFy = 0. Add every horizontal component and get 0, add every vertical component and get 0. Move the angle and the size of the third force F3 until both sums are 0 at once, and the particle is in equilibrium only when both are 0. The reason a plane problem can solve two unknowns is that there are two equations.
The use of equilibrium is finding an unknown force. Some forces are known and one force is not, and the fact of equilibrium fixes the answer. For the sum of the known forces to be 0 the unknown force must be their opposite, and that opposite is called the equilibrant. Place the two known forces F1, F2 and the unknown force F3 that restores equilibrium is drawn, and turning F1 makes the size and direction of F3 follow. It is always F3 = -(F1 + F2), and in components F3x = the opposite, F3y = the opposite. That is how an unknown force is picked out.
A load of weight W hangs from two cables and the knot is the particle. The weight W pulls down, and the tensions T1, T2 pull up along the two cables, three forces. The knot is still, so there is equilibrium, the sum of the three forces is 0. Set up the two equations: horizontally the horizontal pieces of T1 and T2 cancel, and vertically the vertical pieces of the two tensions must add up to the weight. Lower the angle of a cable, bring it closer to horizontal, and the tension grows sharply. That is why a line pulled tight snaps more easily.