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 straw walker is driven by a crankshaft having a crank throw radius of $r = 50\text{ mm}$ rotating at $N = 210\text{ rpm}$. The walker throw angle is $\beta = 45^\circ$ to the horizontal. Taking acceleration due to gravity $g = 9.81\text{ m/s}^2$, the vertical acceleration factor (kinematic throw index $K_v = \frac{r \omega^2 \sin\beta}{g}$) of the walker is ________ (round off to 2 decimal places).
Answer Key:
1.74
Step-by-Step Solution & Explanation
```
Straw Walker Crank Kinematics:
Crank radius r = 0.05 m
Rotational speed N = 210 rpm
Throw angle beta = 45 deg
```
**Step-by-step Calculation:**
1. Angular velocity of the walker crankshaft ($\omega$):
$\omega = \frac{2 \pi N}{60} = \frac{2 \times \pi \times 210}{60} = 7\pi = 21.9911\text{ rad/s}$
$\omega^2 = (21.9911)^2 = 483.61\text{ rad}^2/\text{s}^2$
2. Peak centripetal acceleration ($a = r \omega^2$):
$a = 0.050\text{ m} \times 483.61\text{ rad}^2/\text{s}^2 = 24.1805\text{ m/s}^2$
3. Vertical component of peak acceleration ($a_v = a \sin 45^\circ$):
$a_v = 24.1805 \times \sin 45^\circ = 24.1805 \times 0.70711 = 17.098\text{ m/s}^2$
4. Kinematic throw index ($K_v$):
$K_v = \frac{a_v}{g} = \frac{17.098\text{ m/s}^2}{9.81\text{ m/s}^2} = 1.7429 \approx 1.74$
Since $K_v > 1.0$, the straw layer tosses freely in the air, allowing trapped grains to fall through walker racks.