# Varignon’s Theorem

Varignon’s Theorem states that the moment of a force about any point is equal to the algebraic sum of the moments of its components about that point.

Principal of moments states that the moment of the resultant of a number of forces about any point is equal to the algebraic sum of the moments of all the forces of the system about the same point.

A theorem in mechanics that establishes the dependence between moments of forces of a given system and the moment of their resultant force. This theorem was first formulated and proved by the French scientist P. Varignon.

According to Varignon’s theorem, if a system of forces **F**_{i} has a resultant force **R**, then the moment **M _{0}(R)** of the resultant force relative to any center

*O*(or

*z*-axis) is equal to the sum of the moments

**M**of the component forces relative to the same center

_{0}(F_{i})*O*(or the same

*z*-axis). Mathematically, Varignon’s theorem isexpressed by the formulas

**M _{0}(R) =ΣM_{0}(F_{i})**

or

*M _{z}(R)* =Σ

*M*

_{z}(F_{i})Varignon’s theorem is used for solving a series of problems in mechanics (especially statics), resistance of materials, construction theories,and other areas.

**Method to solve problems;**

The moment of force 100 N about O, can also be determined by using Varignon’s principle. The force 100 N is replaced by its two rectangular components at any convenient point. Here the convenient point is chosen as *C. *The horizontal and vertical components of force 100 N acting at C are shown in Fig.

*(i) *The horizontal component

= 100 × cos 60° = 50 N

But this force is passing through O and hence has no moment about O.

The vertical component

= 100 × sin 60° =-100 × 0.866 = 86.6 N

This force is acting vertically downwards at *C. *Moment of this force about O

=86.6 × OC = 86.6 × 3

= 259.8 N (clockwise). **Ans.**