# Couples and Forces

I have a genral question. I just do not understand two concepts:

1. The idea of a couple being a free vector:
If a couple produces a moment around the midpoint of the distance between the two forces that create a couple then how come if we move it to a different point of a body it creates the same effect. Wouldn't it rotate the body a round new point?

2. How can a couple and a force bew resolved in to a single resultant force. I have seen questions in a text books that have a bunch of forces and couples acting on a body and they ask for the resultant force without a couple. If the couple can be moved to any point and force can not than how come we can find the resultatn force wihtout a couple? As I understand, the moment of a force can not cancel a couple since it is fixed around a point and a couple is not?

Thank you very much for al of you answers. I would really appreciate if someone could explain these two things for me.

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In the analysis of stresses in structures, you are interested in forces and moments at a particular section. It doesn't really matter how the moment arises. It could come from a couple (the simplest method, involving the fewest number of forces), or it could arise from more than two forces. You may not have yet studied shear centre of a section, but if you look it up for a channel section, you will find that the shear centre is outside the section, and is the point of application of an eccentric force that exactly balances the internal Moment in the section. Now I don't really expect you to understand what I have just said, but I can promise you that quite distinguished engineers are still grappling with questions like yours at the end of their lives. I think insights arise from doing a lot of problems thoughtfully, as you are doing. J hope someone else here can satisfy you better, but it will remain a bit of a conceptual mystery, even though you can learn to do the mechanics correctly.

vela
Staff Emeritus
Homework Helper
Your confusion arises because, I think, you're assuming the couple's moment is going to produce a rotation about the point midway between the lines of action of the forces. It'll actually cause a body to rotate about its center of mass. The moment due to a couple is a free vector because you can shift the pair to wherever you want on the body and it will produce the same motion. The net force of a couple is 0, so the object's center of mass won't accelerate regardless of where the forces act. The moment due to the couple is independent of the axis, so the rotational acceleration doesn't depend on the relative displacement of the pair of forces from the center of mass.

When a single force F acts on a body, it will always cause the same linear acceleration of the center of mass regardless of where it acts; however, by moving the force around, you can get different moments. So by moving the force to the right place, you can get the same moment as what the couple would have produced, and the linear acceleration will remain unchanged.

http://en.wikipedia.org/wiki/Couple_(mechanics)

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PhanthomJay
Homework Helper
Gold Member
In response to question 2, I just want to add, in addition to Vela's excellent response, that it is not possible to resolve the couples into a single resultant force when only couples are acting on the system. You can, however, resolve couples and forces acting on the system, when both are present, into a single resultant force without a couple, by summing moments about any point to find the location of that resultant force, per Vela's response.

Thank you very much for all of your replies and especially to vela for his clear explanation. I just do not understand one more thing. In statics, do we always assume that the body is fixed around its centre of mass?

Thank you very much and have a great day

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PhanthomJay
Homework Helper
Gold Member
Thank you very much for all of your replies and especially to vela for his clear explanation. I just do not understand one more thing. In statics, do we always assume that the body is fixed around its centre of mass?

Thank you very much and have a great day
Not in statics. And not in dynamics, either. When you say 'fixed', do you mean 'pivoted' (or 'supported') by a physical restraint? Example: Ignoring friction or gravity, when a force is applied perpendicular to one end of a rigid bar that has no physical restraint from moving or rotating, it will rotate about its center of mass, and its center of mass will translate linearly in the direction of the applied force. If that same bar with the same force applied at the end was attached to a pivot pin at the other end, it would rotate about the far end, not the center of mass. These are both dynamic cases. In statics, there is no net force or torque or couple acting on the system (it would generally have to have more than one support, or a fixed support at one end (like a diving board), per Newton 1, and thus no rotation or translation of the rigid bar.

vela
Staff Emeritus
Homework Helper
No, the center of mass isn't fixed. The concept is that a body's motion can be thought of as a combination of translational motion of its center of mass and rotational motion about its center of mass. In the case of a couple, there's no net force on the body, so the center of mass won't accelerate. If the body was at rest initially, the center of mass remains at rest, and the couple only causes the body to rotate.

Dear Seniors ,

Pls solve this.

A uniform metre rule is pivoted at its midpoint .A weight of 50 gf is suspended at one end of it.Where should a weight of 100 gf be suspended to keep the rule horizontal

Thanks & Regards ,

Sai Vishal

Hey... The moment of a torque is very interesting. However, i am confused as to Couples. Does the moment of a couple depend on the point about which it is rotated??? How do I determine this????

Try it yourself. Take moments about several points and show that the moment of a couple is independent of the point about which you take moments.