Question

Answer the following question.

Why is the moment of a couple independent of the axis of rotation even if the axis is fixed?

Answer 1

1. Consider a rectangular sheet free to rotate only about a fixed axis of rotation, perpendicular to the plane.
2. A couple of forces `vec"F" and - vec"F"` is acting on the sheet at two different locations.
3. Consider the torque of the couple as two torques due to individual forces causing rotation about the axis of rotation.
4. Case 1: The axis of rotation is between the lines of action of the two forces constituting the couple. Let x and y be the perpendicular distances of the axis of rotation from the forces `vec"F" and -vec"F"` respectively.
   In this case, the pair of forces cause anticlockwise rotation. As a result, the direction of individual torques due to the two forces is the same.
   ∴ τ = τ₊ + τ = xF + yF = (x + y)F = rF      ....(1)
   
5. Case 2: Lines of action of both the forces are on the same side of the axis of rotation. Let q and p be the perpendicular distances of the axis of rotation from the forces `vec"F" and -vec"F"` respectively. In this case, the rotation of `+ vec"F"` is anticlockwise, while that of `- vec"F"` is clockwise (from the top view). As a result, their individual torques are oppositely directed.
   ∴ τ = τ₊ - τ = qF - pF
   = (q - p)F = rF       .....(2)
   From equation (1) and (2), it is clear that that torque of a couple is independent of the axis of rotation.