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# 4.12: Couples: Scalar and Vector Formulation

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### 4.12: Couples: Scalar and Vector Formulation

One might wonder how the captain of a large ship can navigate through the ocean with just a turn of the steering wheel. The answer lies in the concept of two parallel forces that are equal in magnitude and opposite sense, creating a couple moment.

A couple moment is a rotational force that tends to rotate the steering wheel. The wheel's rotation can either be in a clockwise or anticlockwise direction. The right-hand rule is a helpful method for determining the direction of a couple moment. To apply this rule, one simply curls the fingers of their right hand along the direction of rotation. The thumb then points in the direction of the moment. It should be noted that the couple moment always acts perpendicular to the plane of the two forces.

In scalar formulation, the magnitude of the couple moment equals the force magnitude multiplied by the perpendicular distance between the forces. This relationship allows for quantifying the rotational effect of the applied forces.

In vector formulation, the couple moment of one of the forces about any point on its line of action equals zero. This is because the force and position vector are parallel, resulting in a zero cross-product. The moment of the other force about this point, however, equals the cross product of the position vector from the line of action of one force to the other and the applied force. This formulation offers a more comprehensive understanding of the couple moment's behavior concerning the applied forces and their positions.