The Characteristics of a Couple

A couple has been defined as a system of two parallel forces equal in magnitude but opposite in direction. We shall now show that the sum of the moments of the forces of a couple is independent of the point about which the moments are taken.

Consider two equal and opposite forces ${𝐅}_{\mathbf{1}}$ and ${𝐅}_{\mathbf{2}}$; ${𝐅}_{\mathbf{1}}+{𝐅}_{\mathbf{2}}=0$ (Fig. 1). Let ${𝐫}_{\mathbf{1}}$ and ${𝐫}_{\mathbf{2}}$ be vectors drawn from any point $O$ to points on the lines of action of ${𝐅}_{\mathbf{1}}$ and ${𝐅}_{\mathbf{2}}$. $𝐚$ is the vector $({𝐫}_{\mathbf{2}}-{𝐫}_{\mathbf{1}})$. The sum of the moments of the two forces of the couple will be called the moment of the couple and will be represented by the vector $𝐂$.

$$𝐂={𝐫}_{\mathbf{1}}\times {𝐅}_{\mathbf{1}}+{𝐫}_{\mathbf{2}}\times {𝐅}_{\mathbf{2}}$$ Since ${𝐅}_{\mathbf{1}}=-{𝐅}_{\mathbf{2}}$, this becomes:

$$𝐂=({𝐫}_{\mathbf{2}}-{𝐫}_{\mathbf{1}})\times {𝐅}_{\mathbf{2}}=𝐚\times {𝐅}_{\mathbf{2}}$$ Since $O$ can be taken as any point without altering the vector $𝐚$, the moment of the couple is independent of the center about which the moments are taken. The moment is thus a property of the couple itself and can be taken as a quantitative measure of the magnitude of the couple. A couple, therefore, is specified completely by the couple vector $𝐂=𝐚\times 𝐅$. The magnitude of the couple vector is equal to $(𝐅)(𝐚\sin \theta )=(𝐅)(d)$, where $d$ is the perpendicular distance between the forces of the couple. The direction is normal to the plane containing the two forces and the sense corresponds to the direction of advance of a right-handed screw rotated by the forces of the couple.

Since $𝐂$ is independent of any particular point, the couple vector does not have a unique line of action as does a force vector.

From the above definition of a couple as a vector, a number of characteristics follow directly:

1. Two couples are equivalent (have the same motional effect upon a body) if their moments and directions are the same. The particular values of the force and the moment arm are not significant; it is only the product of the two which determines the action of the couple.
2. The forces of a couple can be rotated through any angle in their plane, or translated to any position in the plane, without changing the motional effect of the couple upon a body.
3. The forces of a couple can be translated into any parallel plane without changing the motional effect of the couple upon a body. 

PROBLEMS

1. Given the two couples as shown in the diagram, find their sum:
(a) By adding the couple forces directly and thus forming a new couple.
(b) By adding the couple vectors.

answer

$54.0\mathrm{ft}\cdot \mathrm{lb}$

2. Referring to the diagram in the preceding problem, find the sum of the moments of the forces of the system about the line $OP$ which lies in the $xz$ plane. Do this by first taking moments directly about the line $OP$, and then by finding the component of the resultant couple vector, as found in the preceding problem, along the line $OP$.

answer

$-40.1\mathrm{ft}\cdot \mathrm{lb}$

3. Find the resultant of the system of three couples shown in the diagram.

answer

$68𝐣+39𝐤$

4. Find the resultant couple of the three space couples shown in the figure.

answer

$28.3𝐢-104𝐣-585𝐤$

5. Six forces of equal magnitude act along the edges of a cube as shown in the diagram. Find the resultant of the system.