A single-bottom mouldboard plough with a cutting width of $b = 30\text{ cm}$ and operating depth of $d = 15\text{ cm}$ experiences a longitudinal draft force of $F_x = 3.6\text{ kN}$, a downward vertical force of $F_z = 0.9\text{ kN}$, and a lateral side force towards the furrow wall of $F_y = 1.2\text{ kN}$. The magnitude of the total resultant soil force acting on the plough bottom in $\text{kN}$ is ________ (round off to 2 decimal places).
Answer Key:
3.90
Step-by-Step Solution & Explanation
```
Coordinate System for Tillage Forces:
+Z (Downwards)
|
| +X (Longitudinal draft along travel)
| /
| /
+-------- +Y (Lateral side force)
```
**Step-by-step Calculation:**
1. The total resultant force $R$ in 3D space is given by vector addition of the orthogonal force components:
$R = \sqrt{F_x^2 + F_y^2 + F_z^2}$
2. Substituting the given values:
$F_x = 3.6\text{ kN}$
$F_y = 1.2\text{ kN}$
$F_z = 0.9\text{ kN}$
$R = \sqrt{(3.6)^2 + (1.2)^2 + (0.9)^2} = \sqrt{12.96 + 1.44 + 0.81} = \sqrt{15.21}$
$R = 3.90\text{ kN}$