Statics – Law of Moments under Equilibrium According to the law of moments, if several coplanar forces act on a particle and the system is in equilibrium, which statement is correct?

Introduction / Context: In engineering statics, the law of moments (Varignon’s theorem) and equilibrium conditions are fundamental. For a system of coplanar forces in equilibrium, both net force and net moment about any arbitrary point in the plane must vanish.

Given Data / Assumptions:

• Forces are coplanar and act on a particle/system.
• The system is in static equilibrium.

Concept / Approach: Equilibrium of coplanar forces requires: ΣFx = 0, ΣFy = 0, and ΣM(any point) = 0. Varignon’s theorem also states that the moment of a resultant equals the sum of moments of the component forces about the same point. If the system is in equilibrium, the resultant is zero; hence the algebraic sum of moments must be zero about any point.

Step-by-Step Solution: Equilibrium implies resultant R = 0. By Varignon: M_R about any point O = ΣM_O(forces). Since R = 0 ⇒ M_R = 0 ⇒ ΣM_O(forces) = 0.

Verification / Alternative check: Choosing different moment centers does not change the conclusion; if equilibrium holds, the algebraic sum of moments about any point is zero.

Why Other Options Are Wrong: 'Their algebraic sum is zero': True for forces (ΣF = 0) but the law of moments specifically concerns moments, not merely the force sum. Equal distances: has no general relevance to equilibrium. 'Sum of moments equals moment of resultant': The theorem is true, but in equilibrium the resultant is zero; the specific equilibrium statement is that the sum of moments is zero. Moment of inertia zero: unrelated to static equilibrium of forces.

Common Pitfalls: Confusing the necessary condition ΣF = 0 with the moment condition ΣM = 0; both are required.

Final Answer: The algebraic sum of their moments about any point in the plane is zero