Couple and Its Torque

Imagine turning the handle of a bicycle or rotating the steering wheel of a car. You usually place both hands on opposite sides and push one way while your other hand pushes the opposite way. The result? The handle/steering turns, but your body (or the axis) doesn’t move sideways.
This type of action—two equal and opposite forces making something rotate—is called a couple.

A couple is a pair of forces that satisfy all of the following:

• They are equal in magnitude (same strength)
• They are opposite in direction
• Their lines of action do not coincide (they act along different straight paths)
  Because of these properties, a couple produces a pure rotation (turning) effect and no net translation (no sliding) of the object.

\[\vec τ\] = \[\vec r_{12}\] × \[\vec F_1\] = \[\vec r_{21}\] × \[\vec F_2\]

Use the right‑hand rule: Curl your fingers from r toward F; your thumb gives the direction of τ. For the couple shown earlier, that direction is perpendicular to the plane of the forces.

Since the two forces are equal and opposite, their vector sum equals zero → there is no resultant force to cause linear (straight‑line) acceleration. Yet, because their lines of action are separated, they create a turning effect: a moment or torque.
In short:

• Net force = 0 → no linear motion
• Net moment ≠ 0 → pure rotation

Fig. 4.8: Torque of a couple.

Vector form:

\[\vec τ\] = \[\vec r_{12}\] × \[\vec F_1\] = \[\vec r_{21}\] × \[\vec F_2\]​
where r_12​ is the position vector from the line of action of \[\vec F_2\] to \[\vec F_1\]​.

Scalar magnitude (for equal forces):

$\backslash [\backslash vec\; \tau \backslash ]=\; \backslash [\backslash vec\; F\backslash ]\times \; \backslash [\backslash vec\; r\backslash ]$

where:

• F = magnitude of one of the forces
• r = perpendicular separation between the lines of action of the two forces

Key point: The torque of a couple is independent of where you “take moments” (i.e., independent of the reference point) — it depends only on the force magnitude and the separation distance.