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Thursday, 14 November 2019

VARIGNON'S THEOREM STATEMENT AND PROOF

We have started a new topic in our previous post i.e. engineering mechanics. We have seen there the basics of engineering mechanics such as concept of force systemprinciple of transmissibility of forces and its limitations, classification of force system, body force and surface force and moment of a force in mechanics. 

Now, we will be interested to understand here a very important theorem i.e. Varignon’s theorem in mechanics with the help of this post. 

Varignon’s theorem in mechanics 

According to the varignon’s theorem, the moment of a force about a point will be equal to the algebraic sum of the moments of its component forces about that point. 

Let us consider the following figure where a force F is acting at a point P on a body as displayed here. 

Let us consider that the coordinate for point P is (x, y) as displayed in the figure. 

Force F could be written as mentioned below 

F = Fx i + Fy

Where, 
Fx = The component of force F in x direction 
Fy = The component of force F in y direction 
i and j are the unit vectors in x and y directions respectively. 

Let us assume that r is the positional vector and could be written as mentioned below 

r = x i + y

Let us find out and assume that the perpendicular distance of the line of action of force F from the point O is d. 

Moment of the force F about the point O will be given by following equation as mentioned below 

 

Where, k is the unit vector in z direction. 

We can also write down the equation for moment of force about the point O and it will be as mentioned here. 


From above equation, we can say that the magnitude of the moment of a force F about a point O will be equal to the algebraic sum of the magnitudes of the moments of its component forces about that point i.e. about point O. 

Therefore, we have seen here the concept and explanation of Varignon’s theorem with the help of this post.  

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Further we will find out, in our next post, moment of a couple in engineering mechanics.   

Reference:  

Engineering Mechanics, By Prof K. Ramesh 
Image courtesy: Google  

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