VARIGNON'S THEOREM

VARIGNON'S THEOREM

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

If we need to understand this theorem, we need to understand:
- What is "moment" of a force about a point?
- What are "component" forces? and
- What is "algebraic" sum?



1: "moment" of a force about a point
Consider a door.
When you open or close the door, you are applying a force on the door.
This force translates into a rotating force on the hinges.
This rotating force is called Moment.

Moment of force F about point P = F x d
where F is the magintude of the force, say 4Newtons
and d is the perpendicular distance between F and P.

The moment of a force about a point is the product of the magnitude of the force and the perpendicular distance between the line of action of the force and the point.



2: "component" forces
Most mechanical engineering problems involve two or more forces.
To make the calculations easier, we often resolve these forces into a single force, called the resultant force.
In this example, there are two forces P and Q.
To make calculations easier, we resolve them into a single force R.
Now, for all our calculations, we can consider R in place of P and Q.
R is called the resultant of P and Q.
P and Q are called the component forces of R.



3: "algebraic" sum
We know that "sum" means adding some numbers.
But what is this "algebraic sum"?
When you add some positive numbers and some negative numbers, it is called algebraic sum.
For example, 2 plus -1 plus 3 is equal to 4.



4: Coming back to the theorem statement,
The moment of a force about a point is equal to the algebraic sum of the moments of its component forces about that point.

Consider forces P and Q whose resultant is R.
Consider point O.

According to Varignon's theoerm, the moment of force R about point O = the moment of P about O + the moment of Q about O
Let us call the three moments Mr, Mp and Mq.
Mr = Mp + Mq

Sometimes, Mp and Mq are both acting in the same direction (clockwise or anti-clockwise)
And sometimes, Mp and Mq act in opposite directions.

If O is here,
P causes anti-clockwise rotation about O. So Mp is anti-clockwise.
Q also causes anti-clockwise rotation about O. So Mq is also anti-clockwise.

However, if O is here,
P causes clockwise rotation about O. So Mp is clockwise.
Q causes anti-clockwise rotation about O. So Mq is anti-clockwise.

In such cases, where Mp and Mq act in opposite directions, you must consider clockwise as positive and anti-clockwise as negative or vice-versa (that is anti-clockwise as positive and clockwise as negative). This is when, the concept of algebraic sum comes in to picture.


[GE6253 - ENGINEERING MECHANICS - EQUILIBRIUM OF RIGID BODIES]