---
title: "Replace two wrenches by single wrench"
subject: "Applied Mechanics"
chapter: 3
key: "semester_1/applied_mechanics:two-wrenches-single-wrench"
source: https://www.bctengineer.com/q/semester_1/applied_mechanics/two-wrenches-single-wrench
site: BCT Engineering
---

# Replace two wrenches by single wrench

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

- Asked in: 2074 Asoj, 8 marks
- Video: [Problem related WRENCH | Applied Mechanics | Chapter 3 @EngineeringStudios1](https://www.youtube.com/watch?v=QQbo0wXUmpY) — Engineering Studios
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/applied_mechanics/two-wrenches-single-wrench
- Whole chapter: https://www.bctengineer.com/md/semester_1/applied_mechanics/chapter-3

## Question

Replace the two wrenches as shown in the figure below by a single equivalent wrench and determine the point where its axis intersects the XZ plane. [Figure: at origin O, along the Y axis a 70 N·m couple and a 30 N force; a second wrench with a 60 N force and a 40 N·m couple acts along the line from A (at 5 m on the z side) toward x axis; axes x, y, z shown, A at 5 m from O.]

## Short answer

Add force vectors and moments about O, then resolve the moment along the resultant. Resultant force is 67.08 N with components 60i + 30j N, couple is 138.6 N·m along R, pitch 2.07 m. The wrench axis intersects the XZ plane at (0, 0, 2.8) m.

## Answer

**Reading of the figure:** wrench 1 acts along the y axis through O: force 30 N along +y and a 70 N·m couple whose vector is along y (drawn pointing opposite to the force, so −y). Wrench 2 is at A on the z axis, 5 m from O: force 60 N along +x and a 40 N·m couple vector also along +x. The senses are as read from the arrows in the figure.

#### Step 1: resultant force
R = 30 j + 60 i = (60, 30, 0) N, |R| = √(60² + 30²) = 67.08 N

#### Step 2: resultant moment about O
```text
Couples:        C1 + C2 = (0, −70, 0) + (40, 0, 0) = (40, −70, 0)
Moment of 60 N: r × F = (0,0,5) × (60,0,0) = (0, 300, 0)
M_O = (40, 230, 0) N·m
```
(The 30 N force passes through O, so it gives no moment.)

#### Step 3: components of M_O along R
R·M_O = 60×40 + 30×230 = 9300
M∥ = (R·M_O / |R|) = 9300 / 67.08 = 138.6 N·m (along R)
Pitch p = M∥ / R = R·M_O / R² = 9300 / 4500 = 2.067 m

M∥ vector = (R·M_O / R²) R = 2.067 × (60, 30, 0) = (124, 62, 0) N·m

#### Step 4: axis of the wrench
The axis passes through the point r₀ = (R × M_O) / R².
R × M_O = (30×0 − 0×230, 0×40 − 60×0, 60×230 − 30×40) = (0, 0, 12600)
r₀ = (0, 0, 12600/4500) = (0, 0, 2.8) m

The axis is the line through (0, 0, 2.8) m parallel to R = (60, 30, 0). It meets the XZ plane (y = 0) at this same point, because y = 0 there.

**Answer:** single wrench = force R = 67.08 N (components 60 i + 30 j) with a couple of 138.6 N·m along R (pitch 2.07 m). Its axis cuts the XZ plane at **x = 0, z = 2.8 m**.

## Check yourself

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

1. The two wrenches have forces 30 N along +y and 60 N along +x. What is |R|?
   A. 67.08 N
   B. 90 N
   C. 30 N
   D. 45 N
   Answer: A. Forces add as vectors: R = (60, 30, 0), |R| = sqrt(60^2 + 30^2) = 67.08 N. (Key point: Resultant force)

2. Why is the y component of M_O equal to 230 N·m?
   A. -70 N·m from the couple plus +300 N·m from the 60 N force acting at z = 5 m
   B. 70 + 300 N·m, both positive
   C. The 30 N force gives 150 N·m about O
   D. 40 + 70 + 120 N·m
   Answer: A. The 30 N force passes through O so gives no moment; r × F = (0,0,5) × (60,0,0) = (0,300,0), added to the couple (0,-70,0). (Key point: Moment about O)

3. How is the pitch of the wrench found?
   A. p = (R·M_O)/R^2 = 9300/4500 = 2.067 m
   B. p = |R| / |M_O|
   C. p = |M_O| × |R|
   D. p = M_O / R_y
   Answer: A. The couple component along R is M∥ = (R·M_O/R^2) R, so the pitch is R·M_O/R^2. (Key point: Pitch)

4. Which expression locates the point through which the wrench axis passes?
   A. r0 = (R × M_O)/R^2
   B. r0 = (R · M_O)/R^2
   C. r0 = M_O × R
   D. r0 = R / M_O
   Answer: A. R × M_O/R^2 gives the position of the line of R after removing the perpendicular couple; here it is (0, 0, 2.8) m. (Key point: Axis of the wrench)

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