Why Do We Add Mc and Mf in Calculating Resultant Couples?

[Mikejax]

Moments and force couples...confused!

Homework Statement

I have attached 2 pdfs: 1 is the problem, the second is the solution. I need help understanding the solution. Diagrams included.

The gist of the problem:
Need to calculate a dimension "a" such that a resultant couple (from 2 force couples) will equal 480 N-m. I understand how they reduce couple AD, but not how they reduce couple CF. They reduce couple cf in their expression Mc + Mf + forcecouple C and F about C.

I understand why they take C and F about C, but why are Mc and Mf added on top this? I am trying to follow the following principle: The moment induced by 2 equal and opposite forces is equal to the moment of the one force about the point of application of the other".

I don't get how adding Mc and Mf on top this works. (I am referring to the first line of math under the diagram of the solution).

I am so confused, it would be so amazing if someone here could clarify this for me.

Homework Equations

M = M1 + M2

The Attempt at a Solution

see above.

Homework Statement

Homework Equations

The Attempt at a Solution

 

[LowlyPion]

The solution relies on the fact that the geometry of the rollers is such that not only are the horizontally applied forces paired, but they have paired the diagonal forces since they are parallel as well. For instance you can draw the connecting line through D, by moving it out along the parallel lines of action to D. The first term as you note is the resolution of the AD horizontal forces, and the second is the resolution of the diagonal forces that are spaced by the geometry of the common connector that is ⊥ between the two ... 2a√ 2

In the second part you can no longer use the geometry implied by the a = 8, meaning that you cannot couple the 2 diagonal forces so simply. So you need to resolve the diagonal forces into their components, because the dimension a is unknown and since the desired moment isn't the answer from a) is definitely not a = 8.

 

[nlightner]

Hello,

As a fellow scientist, I understand your confusion about moments and force couples. Let me try to clarify this for you.

First, let's define what moments and force couples are. A moment is a measure of the rotational effect of a force around a specific point. It is calculated by multiplying the magnitude of the force by the perpendicular distance from the point to the line of action of the force. A force couple, on the other hand, is a pair of equal and opposite forces acting on a body but not in the same line of action. This creates a rotational effect, or moment, on the body.

In this problem, we are dealing with two force couples, AD and CF, which have equal and opposite forces but are not in the same line of action. The goal is to find the dimension "a" such that the resultant couple created by these two force couples will be equal to 480 N-m.

To understand the solution, we need to first understand the principle of moments, which states that the sum of the moments of all the forces acting on a body is equal to the moment of the resultant force about any point. In other words, the sum of the clockwise moments is equal to the sum of the counterclockwise moments.

Now, let's look at the first line of math in the solution. The expression Mc + Mf + forcecouple C and F about C represents the sum of the moments of all the forces acting on the body about point C. This includes the moments created by the force couple CF, as well as the individual moments of the forces C and F about point C. This is where Mc and Mf come into play. By adding these moments to the expression, we are taking into account the rotational effect of these individual forces on the body.

So, to summarize, the solution is using the principle of moments to calculate the resultant couple created by the two force couples. By including the moments of the individual forces, we are able to accurately calculate the resultant couple and find the dimension "a" that will produce a 480 N-m couple.

I hope this helps clarify things for you. If you have any further questions, please don't hesitate to ask. Keep up the good work in your studies!