Couple and Its Torque - Proof of Independence of the Axis of Rotation

The concept explored here is a fundamental property of the moment of a couple (also called the torque of a couple) in rotational dynamics. A couple consists of two equal and opposite forces ($\vec{F}$ and $-\vec{F}$) acting at different locations on an object. The moment of a couple is calculated by considering the individual torques caused by these two forces. This proof demonstrates a very important characteristic: the net torque produced by a couple does not change, even if the fixed axis of rotation is moved. This means the rotational effect of a couple is absolute, regardless of where the object is pivoted.

 (a)

• Setup: The fixed axis of rotation is positioned between the lines of action of the two forces (\[\vec F\] and \[-\vec F\]).
• Distances: Perpendicular distances from the axis to the forces \[\vec F\] and \[-\vec F\] are x and y, respectively.
• Rotation Direction: The rotation due to both forces is anticlockwise (from top view), meaning the directions of the individual torques (τ₊ and τ_-) are the same.
• Total Torque (τ): The total torque is the sum of the individual torques:
  τ = τ₊ + τ_- 
• Substitution: Substitute the torque formula (Torque = Force × Perpendicular Distance):
  τ= x F + y F
• Simplification (Equation 4.21): Factor out the force F and define r = (x + y):
   τ = (x + y)F = r F                                        ---(4.21)

 (b)

• Setup: The lines of action of both forces (\[\vec F\] and \[-\vec F\]) are on the same side of the axis of rotation.
• Distances: Perpendicular distances from the axis to the forces \[\vec F\] and \[-\vec F\] are q and p, respectively.
• Rotation Direction: The rotation due to \[+\vec F\] is anticlockwise, while the rotation due to \[-\vec F\] is clockwise (from top view). As a result, their individual torques are oppositely directed.
• Total Torque (τ): The total torque is the difference of the individual torques (assuming anticlockwise is positive):
  τ = τ₊ - τ_-
• Substitution: Substitute the torque formula:
  τ = q F - p F
• Simplification (Equation 4.22): Factor out the force F and define r = (q - p) (where r is the perpendicular distance between the forces):
  τ = (q - p)F = r F                                      ---(4.22)

• Compare the final results from both scenarios (Eq. 4.21 and Eq. 4.22).
• The final expression for the torque of the couple is τ = rF in both cases, where r is the perpendicular distance between the two forces.
• Conclusion: "...the torque of a couple is independent of the axis of rotation."