A body in equilibrium is not just standing still in one sense but in two: it has no tendency to accelerate and no tendency to rotate. That demands the forces balance and the moments balance. From a balanced beam to a sign hung on two wires, the same two conditions decide everything.
A body is in equilibrium when two conditions both hold: the resultant force is zero (in every direction), and the resultant moment about any point is zero. The principle of moments is the second condition applied to a pivot: the total clockwise moment equals the total anticlockwise moment. For exactly three coplanar forces in equilibrium, drawing them nose to tail produces a closed vector triangle.
A weight hangs from two strings. It is in equilibrium, so the three forces close into a triangle. Lower the string angle toward the horizontal and watch the tension climb: the flatter the strings, the larger the pull, which is why no rope is ever perfectly straight when something hangs from it.
| Condition | What it means | How to use it |
|---|---|---|
| no resultant force | forces balance in every direction | ΣF = 0 (resolve into x and y) |
| no resultant moment | no net turning effect | Σ(clockwise) = Σ(anticlockwise) |
A neat trick: you may take moments about any point. Choosing a point that an unknown force passes through removes that force from the moment equation (its moment is zero there), leaving fewer unknowns. For three coplanar forces, the vector triangle and the two conditions are two views of the same fact.
Four quick checks on equilibrium and the principle of moments. Each correct answer earns XP and lights this skill on your star map.
For a body to be in equilibrium, it is necessary that:
The principle of moments states that, for a body in equilibrium, the total clockwise moment about any point equals the:
Three coplanar forces hold an object in equilibrium. Drawn nose to tail, the three force vectors:
In the simulator, as the two strings are made more horizontal, the tension in each string:
Equilibrium needs both conditions. A pair of equal and opposite forces that are not in line (a couple) has no resultant force, yet it still turns the body, so it is not in equilibrium. Conversely, balanced moments about one point do not guarantee equilibrium unless the forces also balance. When taking moments, always use the perpendicular distance, and remember you can choose the most convenient pivot.
Unlocks once the four checks above are done. Worth more XP, written to AS Paper 1 and 2 standard.
A light rod is pivoted at one end. A 30 N force acts downward 0.20 m from the pivot, balanced by an upward force F acting 0.60 m from the pivot. The value of F is:
A uniform beam of weight 40 N and length 4.0 m rests on two supports, one at each end. By taking moments, the upward force from the left support is:
A 12 N weight hangs from two strings that each make 30° with the horizontal. By symmetry the tension in each string is:
A non-uniform plank of length 3.0 m and weight 60 N balances on a single pivot placed 1.2 m from the left end. The distance of the plank's centre of gravity from the left end is:
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