What is a couple, what are its properties, and how is a force replaced by a force-couple system?

In one line

A couple is a pair of equal, opposite, parallel, non-collinear forces that produces pure rotation, and any force can be moved to a new point by adding a couple equal to its moment.

Explained simply

When you turn a steering wheel with both hands, one hand pushes up and the other pulls down by the same amount. The car's wheel does not get pushed sideways; it only turns. That pair of hands is a couple.

Technically, two forces F and -F with parallel lines a distance d apart form a couple of moment \( C = F d \). Its net force is zero, so it cannot be balanced by a single force, only by an equal and opposite couple.

How it works

1. Properties: net force is zero; its moment is the same about every point in the plane (a free vector); it can be moved anywhere in its plane or to a parallel plane; couples add algebraically; two couples are equivalent if their moments are equal in size and sense.
2. Force-couple system: to move force F from A to point B, add equal and opposite forces F and -F at B.
3. The original F at A and -F at B form a couple of moment \( F d \), where d is the perpendicular distance from B to the line of F.
4. Result: the same force F acting at B plus a couple \( M_B = F d \).
5. For a whole system: \( R = \) vector sum of forces at B, and \( M_B = \Sigma \) moments about B.

Key formula

Couple moment \( C = F \times d \) (N·m), d = perpendicular distance between the two forces (m). Force-couple at B: \( \vec{R} = \Sigma \vec{F} \), \( M_B = \Sigma M_B \).

Quick example

A 500 N vertical downward force acts at the free end of a 2 m bracket. Moving it to the fixed end gives 500 N downward at the fixed end plus a clockwise couple of \( 500 \times 2 = 1000 \) N·m. This is why a wall bracket must resist both a force and a moment.

Common mistakes

• Thinking a couple can be balanced by one force; only a couple can balance a couple.
• Using the distance along the arm rather than the perpendicular distance between the forces.
• Forgetting to give the sense (clockwise or anticlockwise) of the couple in the force-couple answer.

How MU asks it

Theory 3 to 4 marks: 'Define a couple and state its characteristics'. Numerical 5 to 6 marks: 'Replace the given force system by an equivalent force-couple system at point A'. Give R (magnitude, direction), \( M_A \) with its sense, and a sketch showing both at A.