Rigid-Body Equilibrium: the Forces Sum to Zero and the Moments Do Too
A particle is a point, so it stayed still as long as the forces summed to zero. But a body with size, a rigid body like a bridge, a crane, or a desk, is a different story. Even when the forces are balanced, if they form a couple that twists, the body spins right around. So for a rigid body to truly stay at rest, two things are needed at once: it must not be pushed sideways or up and down, and it must not turn either way. As equations, the sum of the forces is zero, and the sum of the moments is zero too. In a plane these two conditions resolve into exactly three equations. These three are the key to computing invisible reactions, like the force a bridge support pushes up with. Start with the case where the forces are balanced and yet the body still turns.
A rectangular rigid body feels two forces of equal size and opposite direction. Look at the sum of forces: it is exactly zero. For a particle, that would be equilibrium. But press the button to set the two forces off the same line. The resultant is still zero, yet the body begins to spin. A couple has appeared and the moment is no longer zero. The forces are balanced, but the rotation is not. Conversely, place the two forces on the same line of action and the moment becomes zero too, so the body stops completely. This is why a rigid body carries one more condition: not only the sum of forces but the sum of moments must be zero before it is in equilibrium.
So the condition for rigid-body equilibrium is two lines. First, the vector sum of all forces is zero. This means the body does not translate in any direction. Second, the sum of all moments is zero. This means the body does not turn about any point you choose. A central load P is held up by two supports. Adjust the two reactions R1 and R2 to bring the top force gauge and the bottom moment gauge to zero at once. Only one being zero is not enough: if just the force is zero the body tips in place, and if just the moment is zero it slides as a whole. When both are zero, and only then, the rigid body truly comes to rest.
The two conditions written as vectors resolve, in a plane, into three scalar equations by components. The sum of forces being zero splits into horizontal and vertical, giving ΣFx = 0 and ΣFy = 0, and the sum of moments being zero gives ΣM = 0. Together, exactly three equations. Drag the angle of the inclined load, and the three reactions from the pin and roller move to keep all three equations satisfied. That there are three equations matters a great deal: it means you can solve for exactly three unknowns. A particle had two equations and two unknowns, but a rigid body gains one more equation for rotation and can handle three unknowns. You may take any point as the reference for the moment equation, but choosing a point that many unknown forces pass through makes the equation simpler.
Now see the real power of the three equations. A simply supported beam rests on two supports, with a load P placed not in the middle but off to one side. How large are the forces the two supports push up with, the reactions R1 and R2? The equilibrium equations give them at once. Taking moments about the left support and setting the sum to zero, R1, which passes through that point, drops out of the equation, and R2 is solved in one step. Then the sum of vertical forces being zero gives R1. Drag the load position. The closer P comes to a support, the larger that support's reaction. It is exactly the logic of a seesaw: pile a heavy load to one side and that leg carries more.
Finally, note the limit that three equations can solve only three unknowns. You compare the number of unknown reactions the supports create against the number of equations, three. Use the buttons to change the support arrangement. A simply supported beam on a pin and a roller has three unknowns, so it is solved cleanly by the three equations; this is called statically determinate. Put a pin at each end and there are four unknowns, one too many for the equations. This is statically indeterminate: the equilibrium equations alone cannot solve it, and you must also account for how the material deforms. The other way, if there are too few supports and two or fewer unknowns, the body moves: an unstable structure, a mechanism. Matching these counts is the very starting point of sound structural design.