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Varignon’s Theorem


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 about the same point.

Proof. With the help of vector notation of moment)

The resultant of forces P and Q is R as shown in Fig. 3.4(a)

Equation (3.1) states that the moment of R about 0 equals the sum of moment of P  about 0 and moment of Q about 0 which proves the Varignon’s theorem.

M0=-Ref =-P p  + Qq.

Proof of varignon’s theorem : (Without vector notation i.e., Geometrical proof). Refer Fig. 3.4(c)

Let                    θ1 = angle made by p with O l C

θ = angle made by R with O1C

02 = angle made by Q with O l C

O1 O2 =x

Now sum of moments about0 2 is p P +q Q(clockwise)

P .P +q . Q =(x sin θ1) p + (x sin θ2)   Q

= (P sin 01 + Q sin 02) x

= (FB + EA)x

= (CH+CG)x

= (CH+CH+HG)x

= (CH+GD+HG)x

= (CH+HD)x


= R sin θ x

= R (x sin θ)

= Rd. (clockwise)

Rd= pP+qQ

Thus moment of resultant= Algebraic sum of moments of components. Hence proved.