Subtopic: Functional requirements, principles of working, construction and operation of manual, animal, tractor and renewable energy operated equipment for tillage, sowing, planting, fertilizer application, inter-cultivation, spraying, mowing, chaff cutting, harvesting and threshing
Question Statement:
A combine harvester operates in a wheat field with a reel of radius $R = 0.55\text{ m}$. The combine travels forward at a speed of $v_f = 3.6\text{ km/h}$. For optimal crop feeding and minimal grain shattering, the reel index is set to $\lambda = 1.30$. The rotational speed of the reel in $\text{rpm}$ is ________ (round off to 1 decimal place).
Answer Key:
22.6
Step-by-Step Solution & Explanation
```
Combine Reel Kinematics:
Forward Speed: v_f = 1.0 m/s
Reel Radius: R = 0.55 m
Reel Index: lambda = v_reel / v_f = 1.30
```
**Step-by-step Calculation:**
1. Forward speed of combine:
$v_f = \frac{3.6 \times 1000}{3600} = 1.0\text{ m/s}$
2. Linear peripheral speed of reel bat ($v_r$):
$v_r = \lambda \times v_f = 1.30 \times 1.0\text{ m/s} = 1.30\text{ m/s}$
3. Relationship between peripheral speed and rotational speed $N$:
$v_r = R \omega = R \left(\frac{2 \pi N}{60}\right)$
$N = \frac{60 \times v_r}{2 \pi R} = \frac{60 \times 1.30}{2 \times \pi \times 0.55} = \frac{78}{3.45575} = 22.571\text{ rpm} \approx 22.6\text{ rpm}$