Couple

Couple

• Couple consists of two equal, opposite and noncollinear forces

• As the two forces are equal in magnitude but opposite in direction, their resultant force is zero but a moment is associated with couple. Hence a couple acting on a rigid body can produce only rotational motion.

• Consider two forces as shown in Fig. 3.4.1. Their combined moment about O is

• As this moment does not depend on d1 and d2, the distances from O, but only depends perpendicular distance between the two forces, it does not depend on the position of point O. i.e., Moment of couple is same about any point in the plane.

• This can be established using vector algebra.

• Consider vectors from O to any point on the two forces as shown in Fig. 3.4.1 (a) (Refer Fig. 3.4.1 (a) on next page).

• As is a vector from any point on one force to any point on the other force, the moment will be same about any point in the plane.

• Thus a couple can be reduced to a moment of magnitude equal to the product of magnitude of any one force and the perpendicular distance between them as shown in Fig. 3.4.1 (b).

Solved Examples for Understanding

Example 3.4.1 

Determine the moment of couple acting on the right square plate shown in Fig. 3.4.2.

Solution: 

Couple due to vertical forces,

Example 3.4.2 

A lug wrench is used to tighten a square head bolt. If 250 N force is applied to wrench as shown in Fig. 3.4.3. Determine the magnitude F of the equal force exerted on the four contact points on the 25 mm bolt head so that their external effect on the bolt is equivalent to that of the two 250 N forces. Assume that the forces are perpendicular to the flats of the bolt head.

Solution: 

The four forces on the bolt head will be as shown in Fig. 3.4.3 (a). The two 250 N forces applied on the wrench are equivalent to a couple producing a moment which must be equal to the moment produced by the four forces which are equivalent to two couples.

Example 3.4.3 

A plate ABCD in the shape of a parallelogram is acted upon by two couples, as shown in the Fig. 3.4.4. Determine the angle ẞ if the resultant couple is 1.8 Nm clockwise.

Solution: 

The moment of couple about any point in the plane is same.

Taking moment about B,