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 is algebraic sum?
And what are component forces? 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 is equal to F into D, where F is the magnitude of the force, say 4 Newtons, and D is the perpendicular distance between F and P.
The moment of a force about a point P. is the product of the magnitude of the force and the perpendicular distance between the line of action of the force and the point. 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 minus 1 plus 3 is equal to 4. Now, what are 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. Take for 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. 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 theorem, the moment of force R about point O is equal to the moment of P about O plus the moment of Q about O. Let's call the three moments mR, mP and mQ.
mR is equal to mp plus mq. Sometimes mp and mq are both... acting in the same direction, clockwise or anticlockwise. And sometimes mp and mq act in opposite directions.
Let's consider two scenarios. In the first scenario, the location of O is here. Force P causes anti-clockwise rotation about O.
So, Mp is... anti-clockwise and force Q also causes anti-clockwise... rotation about O. So, Mq is also anti-clockwise. Consider the second scenario where the location of O is here.
Force P causes clockwise rotation about O. So, Mp is clockwise. Force 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 i.e. anti-clockwise as positive and clockwise negative.
This is when the concept of algebraic sum comes into picture.