A Particle Is in Equilibrium When the Forces Sum to Zero
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 perfectly. When the resultant of several forces meeting at a point is exactly zero, that point does not move. This is equilibrium of a particle. Here a particle is a body shrunk to a single point, its size ignored. You need not think about rotation at all; you only weigh the sum of the forces. The single condition that the resultant is zero becomes, in a plane, two clean equations: the horizontal sum is zero and the vertical sum is zero. With these two equations we solve for unknown forces, such as the tension in a cable. Start by moving one force and see when the knot loses its balance.
Look at several forces acting at one point all at once. Let us call this point a particle. As we saw last lesson, these forces combine into a single resultant R. If R is not zero, the particle gets shoved right along that R. One leftover arrow remains and drags the point with it. Turn one force. You will see the resultant R swing this way and that, and the larger R is, the harder the shove. Equilibrium is exactly the moment this leftover arrow vanishes, the moment R becomes zero. We confirm that as a picture in the next scene.
When we chain the forces head to tail, the resultant was the arrow from the first tail to the last head. So what does it mean for the resultant to be zero? It means the last head comes back exactly to the first tail. The arrows close into a gapless polygon. Drag the head of the last force to close the polygon. The gap that stays open is the resultant, and when that gap shrinks to zero you have equilibrium. So forces in equilibrium always draw a closed loop, head chasing tail back to the start. This is the picture version of equilibrium.
Seeing the loop close by eye is intuitive, but for calculation, working in components is exact. For the resultant to be zero, its horizontal component must be zero and its vertical component must be zero too. So the single equilibrium condition splits into two equations: ΣFx = 0 and ΣFy = 0. Add the horizontal components of every force and get zero; add the verticals and get zero. Move the angle and size of F3 to drive both totals to zero at once. When both are zero, and only then, the particle is in equilibrium. This is why a planar problem lets you solve for exactly two unknowns: there are two equations.
Now for the real use of equilibrium. Usually we know a few of the forces and must find one unknown force. The fact of equilibrium pins down the answer. If the known forces add up to a resultant R, then the unknown force must be exactly its opposite, -R, so that the whole sum comes to zero. This -R is called the equilibrant. Look at two known forces F1 and F2. The unknown force F3 that brings equilibrium is drawn in automatically. Turn F1 and the size and direction of F3 follow to match. It is always F3 = -(F1 + F2). In components, F3x = -ΣFx and F3y = -ΣFy. This is how we pick out an unknown force.
Let us finish with a concrete example. A load of weight W hangs from two cables. The knot is the particle, and three forces act there: the weight W pulling straight down, and the tensions T1 and T2 pulling up and out along the two cables. Since the knot stays still it is in equilibrium, so the three forces sum to zero. You set up two equations: horizontally, the horizontal parts of T1 and T2 cancel each other, and vertically, the vertical parts of the two tensions add up to equal the weight W. Drag the angle of the right cable. The shallower the angle, the closer the cable is to horizontal, the more steeply the tension grows. This is why a clothesline pulled tight snaps more easily, and why suspension-bridge cables are so thick.