Equilibrium

A body is truly in equilibrium when it has no tendency to turn or move.
This means that:

When a body is in equilibrium:


and also:




It is easy to show this using the light loaded beam of length L shown in Figure 1. Notice that F₁ will turn the beam anticlockwise about the pivot (P) while F₂ will turn it clockwise.

Using the principle of moments and taking moments about the pivot:
Balancing the vertical forces:


Notice the minus sign since R acts in the opposite direction from F₁ and F₂.

Further example

A more complex example where there are three forces acting on the rod is shown below (Figure 2).







Using the principle of moments and taking moments about the pivot:
Clockwise moments = F₁d₁ + F₂d₂ = F₃d₃ = Anticlockwise moments

Moments may be taken about any point

(a) If we take moments about end A :

Anticlockwise moments = LR/2 = [L/2 + d₁] F₁+ [L/2 + d₂] F₂ + [L/2 – d₃] F₃ = Clockwise moments

(b) If we take moments about end B :

Anticlockwise moments = [L/2 - d₁] F₁+ [L/2 - d₂] F₂ + [L/2 + d₃] F₃ = LR/2 = Clockwise moments

Combining equations (a) and (b) gives:

Clockwise moments = F₁d₁ + F₂d₂ = F₃d₃ = Anticlockwise moments

This is the identical result that we obtained by taking moments about the pivot.