Varignon's Theorem

Varignon's Theorem, also often called the principle of moments, states that if we have two or more concurrent forces, the sum of the moments that each force creates about a single point will be equal to the moment created by the sum of those forces about the same point.

Varignon's Theroem
If the sum of F1 and F2 is Ftotal, then we can assume that the sum of the moments about point A exerted by F1 and F2 will be equal to the moment exerted about point A by Ftotal

In practice this allows us to do two things when examining moments:

  1. First, if we are interested in finding the overall moment on a body and we have some forces that are concurrent acting on that body, then we can sum those concurrent forces before we take the moments about a point or an axis. This may serve to simplify the problem in some cases.
  2. Second, if we are interested in finding the moment that a single force exerts about a point or axis, we can break this force down into components and find the moment exerted by each component of the force vector. The sum of these moments will be equal to the moment exerted by the original force. This will often simplify the problem of finding the perpendicular distances for scalar moment calculations.

Worked Problems:

Question 1:

Use Varignon's Theorem to find the moment that the forces in the diagram below exert about point A.

Problem 1 Diagram

Solution:



Question 2:

Use Varignon's Theorem to find the moment that the force in the diagram below exert about point B.

Problem 2 Diagram

Solution: