torque due to couple is independent of the point about which the moments are taken.prove it

To understand why the torque due to a couple is independent of the point about which the moments are taken, we first need to clarify what a couple is and how torque is defined. A couple consists of two equal and opposite forces whose lines of action do not coincide. This creates a rotational effect without any translational motion.

Defining Torque

Torque (\( \tau \)) is defined as the product of the force (\( F \)) and the perpendicular distance (\( d \)) from the line of action of the force to the point about which we are calculating the torque. Mathematically, it can be expressed as:

Torque Formula: \( \tau = F \times d \)

Understanding the Couple

Consider a couple formed by two equal and opposite forces, \( F_1 \) and \( F_2 \), separated by a distance \( r \). The torque produced by this couple can be calculated about any point. Let's denote the two forces acting at points A and B, where A is at a distance \( r/2 \) to the left of the center and B is at a distance \( r/2 \) to the right of the center.

Calculating Torque about Different Points

Let’s calculate the torque about two different points: point O (the midpoint between A and B) and point P (any arbitrary point). The torque due to the couple about point O is:

• Torque from \( F_1 \) at A: \( \tau_1 = F_1 \times d_1 \) where \( d_1 = r/2 \)
• Torque from \( F_2 \) at B: \( \tau_2 = F_2 \times d_2 \) where \( d_2 = r/2 \)

Since \( F_1 = F \) and \( F_2 = -F \), the total torque about point O is:

Total Torque at O: \( \tau_{O} = F \times \frac{r}{2} + (-F) \times \frac{r}{2} = F \times \frac{r}{2} - F \times \frac{r}{2} = F \times r \)

Now, let’s calculate the torque about point P. The distances from point P to the lines of action of the forces will change, but the net effect remains the same. The distance from point P to the line of action of \( F_1 \) will be \( d_1' \) and for \( F_2 \) it will be \( d_2' \). However, since the forces are equal and opposite, the sum of the torques will still yield:

Total Torque at P: \( \tau_{P} = F \times d_1' + (-F) \times d_2' \)

Independence from Point of Reference

Regardless of where point P is located, the distances \( d_1' \) and \( d_2' \) will always adjust such that the total torque remains equal to \( F \times r \). This is because the couple's effect is purely rotational and does not depend on the specific point chosen for the calculation. The key takeaway is that the torque produced by a couple is always the same, no matter where you calculate it from, as long as the forces and their separation remain unchanged.

Conclusion

This independence from the point of reference is a fundamental characteristic of couples in mechanics. It highlights the unique nature of rotational forces and their ability to produce consistent torque regardless of the observation point. Thus, we can confidently say that the torque due to a couple is indeed independent of the point about which the moments are taken.