A pair of forces with equal magnitudes, opposite directions, and different lines of action is called a “couple.” When a couple acts on a rigid object, the couple produces a torque that does not depend on the location of the axis. The drawing shows a couple acting on a tire wrench, each force being perpendicular to the wrench. Determine an expression for the torque produced by the couple when the axis is perpendicular to the tire and passes through (a) point A, (b) point B, and (c) point C. Express your answers in terms of the magnitude F of the force and the length L of the wrench.
The Correct Answer and Explanation is:
To determine the torque produced by a couple, we use the concept that a couple is formed by two equal and opposite forces whose lines of action do not coincide. Even though they cancel out in terms of net force, they produce a net torque.
General Formula for Torque Due to a Couple:
τ=F⋅d\tau = F \cdot d
Where:
- τ\tau is the net torque produced by the couple,
- FF is the magnitude of one of the forces,
- dd is the perpendicular distance between the lines of action of the two forces (also called the moment arm).
This torque is independent of the axis of rotation because a couple does not produce any net translational force — it only causes rotation.
Let’s analyze this in terms of the given variables:
- Let FF be the magnitude of each force in the couple.
- Let LL be the total length of the wrench (which is also the distance between the two forces, since each force acts at an end of the wrench and is perpendicular to it).
Torque About Different Points:
(a) Torque about point A:
- One force is applied at A, the other at B (a distance LL away).
- Torque = F⋅LF \cdot L
(b) Torque about point B:
- Same setup, just reversing positions.
- One force acts at B, the other at A.
- Torque = F⋅LF \cdot L
(c) Torque about point C (midpoint of the wrench):
- Each force is at a distance L/2L/2 from C, but still pointing in opposite directions.
- Torque from one side = F⋅(L/2)F \cdot (L/2), from the other = F⋅(L/2)F \cdot (L/2)
- Total torque = F⋅(L/2)+F⋅(L/2)=F⋅LF \cdot (L/2) + F \cdot (L/2) = F \cdot L
Final Answers:
In all three cases, the torque produced by the couple is: τ=F⋅L\boxed{\tau = F \cdot L}
Explanation (300+ words):
A couple in physics consists of two forces that are equal in magnitude, opposite in direction, and separated by a perpendicular distance. The key property of a couple is that it generates a pure rotational effect (torque) without causing any net linear acceleration. This is crucial in applications such as turning a steering wheel, opening a jar, or tightening bolts with a wrench.
In the problem, the couple is applied to a tire wrench: one force pushes upward at one end while the other pushes downward at the opposite end. Both forces are perpendicular to the wrench, and their lines of action are separated by the length LL of the wrench.
Despite where the axis of rotation is located — at point A, B, or C — the torque produced by the couple remains constant. That’s because the net torque produced by a couple is independent of the point or axis about which it is calculated. This unique characteristic arises because the sum of the torques from each force, relative to any point, always adds up to the same net torque due to the symmetry and balance of forces in a couple.
When calculating torque, the formula τ=F⋅d\tau = F \cdot d applies, where dd is the perpendicular distance between the two force lines (not necessarily from the axis to the force). Since that distance is LL in all cases, and the forces are always of magnitude FF, the torque remains: τ=F⋅L\boxed{\tau = F \cdot L}
This result illustrates how couples are effective at producing rotation without translating the object, making them extremely useful in tools and machinery.