Varignon’s theorem (principle of moments) in planar statics: Which statement correctly states Varignon’s theorem for a system of coplanar forces about any chosen point?

Introduction / Context:
Varignon’s theorem, or the principle of moments, is a cornerstone of engineering statics. It simplifies the computation of the moment of a resultant by allowing moments of individual forces to be summed algebraically about the same reference point.

Given Data / Assumptions:

• System of coplanar forces acting on a rigid body.
• Moments taken about an arbitrary point O in the plane.
• Sign convention for moments (clockwise/anticlockwise) must be consistent.

Concept / Approach:
Let R be the vector sum (resultant) of forces F₁, F₂, …, F_n. Varignon’s theorem states: Moment_O(R) = Σ Moment_O(F_i), where the sum is algebraic (with sign). This permits distributing moment calculations before vector addition, which is especially useful in complex systems.

Step-by-Step Solution:

Verification / Alternative check:
Expanding R = ΣF_i and using cross-product linearity shows r × R = Σ(r × F_i), which directly proves the theorem.

Why Other Options Are Wrong:

• a, b restrict to two forces and use “arithmetical” rather than algebraic (sign matters).
• c says “arithmetical sum” but ignores signs; incorrect in general.
• e confuses forces with moments (different physical dimensions).

Common Pitfalls:
Dropping signs; mixing moment arms for different forces; forgetting all forces must be referenced to the same point.

Final Answer:
Algebraic sum of the moments of all forces about any point in their plane equals the moment of their resultant about that point