Everything treated so far has had no size. The weight hung on a spring, the ball thrown through the air: all of them were handled as points. An object treated this way is called a particle.
A particle has only one condition for balance. The forces acting on it must add to zero. That alone is enough for it never to start moving.
The same holds for an object with size, so long as the forces are gathered at a single point. Two forces of equal size and opposite direction act on the rod in the figure at exactly the same point. Their sum is zero, and the rod does not budge. Strengthen both together or weaken both together, and it still does not move.
An object with size, however, has a freedom that a particle never had. It can rotate.
Take those same two forces, unchanged in size and direction, and apply them at the two ends of the rod instead.
Nothing about the addition changes. They are equal in size and opposite in direction, so the sum is still zero. Written out it is , exactly as in the last scene.
And yet the rod has begun to turn. The same equation, a different outcome. Which means the sum of the forces was not enough.
What was missing is where each force was applied. For a particle the point of application made no difference, but for an object with size it does. This tendency to turn something is called the moment of a force.
So what fixes how strongly a force turns something? Think of pushing a door. Push near the hinge and it is heavy; push at the far edge and it is light. The same force turns the door better from farther away.
What decides it is the size of the force multiplied by the distance from the pivot to the force. That distance is called the arm, and the product is written .
The figure is a balance. A small force acts far out on the left, a larger force acts on the right, and its position is being moved. Bring the right one inward and at some point the beam comes level. That is where . Move past it and the beam tips again.
Forces of different sizes balance because the length of the arm makes up the difference. A small force applied far out turns the beam by as much as a large force applied close in.
To sum up: an object with size needs two conditions to be in balance. The forces must sum to zero, and the moments must sum to zero. Both and have to hold.
The moments may be measured about any point you like. If the object is balanced, the sum is zero wherever it is taken. This is enormously convenient in practice: choose the point well and a force you have no interest in can be made to disappear. The next lecture does exactly that.
In the figure the rod rests on two supports with its weight acting at the middle. Move the right support and the share carried by each support changes. The forces still sum to zero and the moments still sum to zero, so the rod does not move.
The share each support carries is fixed by the balance of moments about a pivot. The farther the right support is from the weight, the less it carries and the more the left one does. It is the same as two people carrying a heavy load, where the one nearer the load feels it more.