---
title: "Couple, wrench, moment of P about points and lines"
subject: "Applied Mechanics"
chapter: 3
key: "semester_1/applied_mechanics:couple-free-vector-wrench-cube-force-p"
source: https://www.bctengineer.com/q/semester_1/applied_mechanics/couple-free-vector-wrench-cube-force-p
site: BCT Engineering
---

# Couple, wrench, moment of P about points and lines

Applied Mechanics · Chapter 3: Forces acting on particle and rigid body

- Asked in: 2073 Chaitra, 2+3+6 marks
- Video: [Couple Moments | Mechanics Statics | (Learn to solve any question)](https://www.youtube.com/watch?v=uV4QDXd0ARM) — Question Solutions
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/applied_mechanics/couple-free-vector-wrench-cube-force-p
- Whole chapter: https://www.bctengineer.com/md/semester_1/applied_mechanics/chapter-3

## Question

Define couple and show that couple is a free vector. How will you reduce a system of forces to a wrench? A cube of side a = 4 m is acted upon by a force P = 20 kN as shown. Determine the moment of force P: a) about A, b) about edge AB, c) about diagonal AG of cube. [Figure: cube with vertices labelled A to G and O, axes x, y, z; force P acts along a diagonal/edge as shown.]

## Short answer

A couple is two equal, opposite, parallel forces; its moment is a free vector since M = (r_A − r_B) × F is independent of point. Reduce a system by finding R and M_O, then the component of M_O parallel to R gives the wrench. For P = (0, 14.14, −14.14) kN: M_A = (56.57, 56.57, 56.57) kN·m, about edge AB = 56.57 kN·m, about diagonal AG = −32.66 kN·m.

## Answer

#### Couple
A couple is a pair of equal, opposite and parallel forces whose lines of action do not coincide. It produces pure rotation, no translation, because the resultant force is zero. Its moment is M = F × d, where d is the perpendicular distance between the forces (the arm of the couple). It is a vector perpendicular to the plane of the couple.

#### A couple is a free vector
Let forces **F** at A and **−F** at B form a couple. Take any point O, with position vectors r_A and r_B.
```text
Moment about O = r_A × F + r_B × (−F)
               = (r_A − r_B) × F
               = r_AB × F
```
The result contains only the vector r_AB between the two points of application. It does not contain the position of O. So the moment of a couple is the same about every point. The couple can be moved anywhere in its plane, to a parallel plane, or turned in its plane, without a change in its effect on a rigid body, provided magnitude and direction of the moment vector remain the same. Hence its moment is a **free vector**.

#### Reduction of a force system to a wrench
1. Reduce the system to a resultant force **R** = ΣF at a chosen point O and a resultant couple **M_O** = Σ(r × F).
2. Resolve M_O into M₁, parallel to R, and M₂, perpendicular to R: M₁ = (R·M_O / R²) R.
3. The pair R and M₂ can be replaced by R alone, moved to a new line of action displaced from O by r = (R × M_O)/R² (distance d = M₂/R).
4. What remains is R along its line of action and the couple M₁ parallel to it. This is the wrench; the ratio p = M₁/R is its pitch. A wrench is the simplest form to which a general 3D system can be reduced.

#### Moments of force P on the cube
**Reading of the figure:** cube side a = 4 m, O at the origin, x right, y up, z toward the viewer. E(0,0,4), F(4,0,4), G(4,0,0), D(0,4,0), A(0,4,4), B(4,4,4), C(4,4,0). P = 20 kN acts along F→C, i.e. direction (0, 1, −1)/√2.

P = 20/√2 × (0, 1, −1) = (0, 14.14, −14.14) kN, applied at F(4, 0, 4).

**a) About A**
r = F − A = (4, −4, 0) m
M_A = r × P = (4, −4, 0) × (0, 14.14, −14.14)
= ((−4)(−14.14) − 0, 0 − 4(−14.14), 4(14.14) − 0) = (56.57, 56.57, 56.57) kN·m
|M_A| = 98.0 kN·m

**b) About edge AB** (A to B is along +x, unit vector i)
M_AB = M_A · i = **56.57 kN·m**

**c) About the diagonal AG**
A(0,4,4) to G(4,0,0): vector (4, −4, −4), unit vector λ = (0.577, −0.577, −0.577)
M_AG = M_A · λ = 0.577 (56.57 − 56.57 − 56.57) = **−32.66 kN·m**
The magnitude is 32.66 kN·m; the negative sign means it acts opposite to the direction A→G.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which describes a couple?
   A. Two equal, opposite, parallel forces with different lines of action; zero resultant force
   B. Two equal forces on the same line of action
   C. Two unequal parallel forces
   D. Two perpendicular equal forces
   Answer: A. A couple produces pure rotation, with moment M = F × d where d is the perpendicular arm. (Key point: Definition of couple)

2. Why is the moment of a couple a free vector?
   A. Moment about any O = r_A × F + r_B × (-F) = r_AB × F, which does not contain O
   B. Because the forces are equal in magnitude
   C. Because the resultant force is zero
   D. Because the arm d can be changed freely
   Answer: A. The position of O cancels out, so the moment is the same about every point. (Key point: Free vector proof)

3. How does a general force system reduce to a wrench?
   A. R acting along a line, plus a couple M1 = (R·M_O/R^2) R parallel to R
   B. R plus a couple perpendicular to R
   C. A couple only, with R = 0
   D. R at O plus M_O unchanged
   Answer: A. The perpendicular part of M_O is removed by shifting R to a new line of action; only the parallel part M1 remains. (Key point: Wrench)

4. How is the moment of P about the edge AB found?
   A. M_A · (unit vector along AB), i.e. the component of the moment about A along the axis
   B. The magnitude |M_A|
   C. P multiplied by the cube side
   D. The cross product of P with the unit vector along AB
   Answer: A. With AB along x, M_AB = M_A · i = 56.57 kN·m. (Key point: Moment about an axis)

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