THE PRINCIPLE OF MOMENTS

THEPRINCIPLE OF MOMENTS

One of the most powerful theorems of statics is the Principle of Moments, or Varignon’s Theorem.  Many of the concepts you will explore in coming lessons will rely on use of this theorem.  For you to thoroughly understand its concept and fundamental idea is critical.  In words, the Principle of Moments states:
“The moment caused by the resultant force (of some system of forces) about some arbitrary point is equal to the sum of the moments due to all of the component forces of the system.”
Let’s break this down:

• Think about a system of forces.  The simplest equivalent to this system is a single resultant force.
• Now consider some arbitrary point ... any point.  It doesn’t even have to be physically on the system.
• As long as the line of action of the resultant of those forces in the system doesn’t pass through that arbitrary point, it must cause a moment about that point.
• According to the Principle of Moments, if you calculated the moment of each force of the original system about that same point and added them up, you would get the same moment as that calculated due to the resultant force.

So this basically tells us that if we’re attempting to calculate the total moment of forces about some point, we have two ways to do it:

    1.  Calculate each moment (from each force separately) and add them up, keeping in mind the CW and CCW sign convention.
    -- or --
    2.  Calculate the moment caused by the resultant of the system of forces about that point.
The Principle of Moments tells us that they will be the same.
Now there’s a very important “alternate view” of the Principle of Moments.  Let’s say I have a force like the 60 N force you saw in the final example of the Moments module:

You’ll recall that the difficulty with calculating the moment caused by the 60 N force about point A was that you had to calculate the moment arm using trigonometry.  But if the 60 N force is resolved into its x and y components, then isn’t the original  60 N force nothing more than a resultant of those two rectangular components?  If true (which it is ...) then the Principle of Moments tells us that the moment of the 60 N force about point A will be equal to the moment we would get by calculating the moments due to it’s components ... and then simply adding them up!
From the example, the x and y components of the 60 N force are calculated:
    Fx = 60 cos 30 = 51.96 N
    Fy = -60 sin 30 = -30 N   (that's 30 N pointing downward, assuming a typical x-y axis)
So the equivalent system is as shown:

Now, considering only the component forces, look at the moment each one causes about point A (note the sign convention is ONLY due to whether the force causes a CW or CCW moment!)
    M_A/51.96 = -51.96 x 0.8 = -41.6 N-m
    M_A/30 = 30 x 0 = 0 N-m  (the line of action passes through point A, so there is NO moment!)
Note that the moment arms are found very easily from the dimensions given in the problem.  Now, when these moments are added together, the total moment turns out to be:
    M_A = -41.6 + 0 = -41.6 N-m
Which is exactly the same as the moment calculated in the example from the previous module, where the moment arm for the resultant had to be calculated!
IMPORTANT!  You’re going to see that for the vast majority of problems where you have to calculate a moment, the process will be greatly simplified by resolving a force into components and applying the Principle of Moments.  In quite a few problems, you simply will not have enough information to calculate the moment any other way.  Here’s another example:

For this, let's determine the moment that the 150 N force creates about point A.  Now before we do, just mentally think about the amount of effort you would have to put into finding the moment arm if you had to trigonometrically determine the distance between point A and the line of action of the force (remember, that's the distance that is perpendicular from that line of action!).  Fortunately for us, we have Varignon's Theorem to work with.  First, we resolve the 150 N force into x- and y-components (it will be helpful for you to sketch this out on paper, so you can visualize it better):
    Fx = 150 cos(30) = 129.9 N
    Fy = 150 sin(30) = 75 N
Using Varignon's Theorem, we have:
    M_A = (-129.9 N x 120 mm) + (75 N x 350 mm) = 10,662 N-mm or 10.662 N-m
Make sure you see why the 129.9 N component caused a negative moment!  Varignon's Theorem totally eliminated the need to determine a single moment arm, because we were able to use the dimensions given on the figure!