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 system, principle
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**= F

_{x}

**i**+ F

_{y}

**j**

Where,

F

_{x}= The component of force F in x direction
F

_{y }= 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

**j**

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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