When moments balance
When the two opposing moments are equal the body does not turn. It is then said to be in equilibrium.
The principle of moments
A body is in equilibrium under moments when the sum of the clockwise moments acting on it equals the sum of the anticlockwise moments.
F1 × d1 = F2 × d2
The resultant moment is then zero.
It is not the forces that are equal
This is the easiest place in the chapter to go wrong. Equilibrium does not require the two forces to be equal — it requires the two products to be equal. So a small force, applied far enough out, can hold a large one.
An example
A weight of 20 N hangs 0.2 m from the pivot on one side, and a weight of 5 N hangs 0.8 m from the pivot on the other. Will the rod turn?
Solution
moment on one side = 20 N × 0.2 m = 4 N m
moment on the other = 5 N × 0.8 m = 4 N m
The two moments are equal, so the rod does not turn — it is in equilibrium.
The forces are nothing like equal: one is four times the other. But the smaller one acts four times as far out, so the two products come to the same thing.
When it is not balanced
If the two products are not equal, the body turns the way the larger moment points. What produces that turning is the difference between the two moments.
Equilibrium is an equality of moments
Looking at the forces is never enough to decide whether a rod balances. Always work out force × perpendicular distance separately for each side, and only then compare the two.