Answer in Brief

**Answer the following question.**

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

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#### Solution

- Consider a rectangular sheet free to rotate only about a fixed axis of rotation, perpendicular to the plane.
- A couple of forces `vec"F" and - vec"F"` is acting on the sheet at two different locations.
- Consider the torque of the couple as two torques due to individual forces causing rotation about the axis of rotation.
**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)**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.

Concept: Couple and Its Torque

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