We have started a new topic in our previous post i.e.
engineering mechanics. We have seen thereÂ the basics of engineering
mechanics such asÂ concept
of force system,Â principle
of transmissibility of forcesÂ and its limitations, classification
of force system, body force and surface force and moment of a force in
mechanics.Â

###

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

###
**Varignonâ€™s theorem in mechanicsÂ **

According to the varignonâ€™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.Â

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

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

Force F could be written as mentioned belowÂ

**F**= F

_{x}

**i**+ F

_{y}

**jÂ**

Where,Â

F

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

_{y }= The component of force F in y directionÂ**i**and

**j**are the unit vectors in x and y directions respectively.Â

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

**r**= x

**i**+ y

**jÂ**

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

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

Where,

**k**is the unit vector in z direction.Â
We can also write down the equation for moment of
force about the point O and it will be as mentioned here.Â

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.Â

Therefore, we have seen here theÂ concept and
explanation of Varignonâ€™s theorem with the help of this post.Â Â

DoÂ you have any suggestions? Please write in comment
box and also drop your email id in the given mail box which is given at right
hand side of page for further and continuous update fromÂ www.hkdivedi.com.Â Â

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

### Reference:Â Â

Engineering Mechanics, By Prof K. RameshÂ

Image courtesy: GoogleÂ Â

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