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?
- 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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