Varignon's Theorem: Definition and Derivation with Proof

In this article, you will learn what is the varignon's theorem with its proof.

So without wasting time let's get started.

Varignon's Theorem 

According to this theorem "If a number of coplanar forces are acting on a body, then the algebraic sum of their moments about a point in their plane is equal to the moment of their resultant about the same point."

This means that the moment of a force about a point is equal to the algebraic sum of the moments of its component forces about that point.

As we know,

Moment =  F × D

Where,

F- Force

D- Perpendicular Distance

Sum means to add a number while algebraic sum means to add positive and negative numbers.

Now, look at the figure for the component of forces where R is the resultant of P and Q while P and Q are the forces component of R.

varignon's theorem

So, if find the moment of R with respect to O.

Then,

Moment Of R About O = (Moment Of P About O + Moment OF Q About O)

MR = MP + MQ

Varignon's Theorem Proof

Let F1, and F2, be the two forces represented by the lines AB and AD. 

See in the figure,

varignon's theorem proof
Where,
  1. O be the point about which moment is to be taken. 
  2. From O draw a line OC parallel to AB meeting AD at D. 
  3. Join BC to complete the parallelogram. 
  4. Now join the diagonal AC which gives the resultant R of the two forces. 
  5. Now Join OA and OB.

From the figure, we can clearly see that,

 (△ABC) = (△ADC) = (△OAB ) 

As we know that the moment of a triangle is twice its area.

So,

Moment of force F1 about O = 2 x Area of △OAB 

Moment of force F2 about O =2 x Area of △OAD 

Moment of resultant force R about O = 2 x area of △OAC


Sum of moments of two forces about O = 
         2 Area of △OAB +  2 x Area of△OAD 


Now we clearly see in the figure ADC = OAB that we can write ADC in place of OAB.

Sum of moments of two forces about O = 
       2 Area of △ADC + 2 x Area of △OAD

Sum of moments of two forces about O = 
      2 (Area of △ADC + Area of △OAD) 

Now you can see well in the figure that
△OAC = △ADC + △OAD

So,

Sum of moments of two forces about O = 
        2 x Area of△OAC 
 

So, proving the theorem that the sum of the moment of two forces about O is equal to the moment of the resultant force R about O.

So, I hope I've covered everything about the "varignon's theorem".  If you still have any doubts or questions about the varignon's theorem, you can mail me or ask via comments, I will definitely reply to you as soon as possible.

FAQ Related to Varignon's Theorem

Varignon's theorem is used to be found?

Varignon's theorem is used to find the location of the resultant force.

What is the application of varignon's theorem?

It is very useful in calculating the scalar moment. In cases where it is difficult to determine the perpendicular distance, the use of this theorem provides an alternative to finding that distance.

What is Varignon's theorem of moments?

The moment of a force about a point is equal to the algebraic sum of the moments of its component forces about that point.

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