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.

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.

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.

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