Subtopic: Measurement of force, stress, torque, speed, displacement and acceleration on machine elements – shafts, couplings, keys, bearings and knuckle joints
Question Statement:
A threshing drum shaft of diameter $d = 50\text{ mm}$ transmits a torque of $T = 600\text{ N}\cdot\text{m}$ to the cylinder hub through a rectangular sunk key. The key has a width of $w = 14\text{ mm}$ and a total height of $h = 9\text{ mm}$. If the allowable shear stress in the key is $\tau_{\text{allow}} = 50\text{ MPa}$, the minimum required length of the key based on shear failure is ________ $\text{mm}$ (round off to 1 decimal place).
Answer Key:
34.3
Step-by-Step Solution & Explanation
```
Sunk Key Cross Section:
+---------------+ <-- w = 14 mm
| |
|---Shaft Line--| <-- h/2 = 4.5 mm
| |
+---------------+ <-- h = 9 mm
```
**Step-by-step Calculation:**
1. Tangential force $F$ acting on the key at the shaft surface:
$F = \frac{T}{r} = \frac{T}{d/2} = \frac{2T}{d} = \frac{2 \times 600\text{ N}\cdot\text{m}}{0.050\text{ m}} = 24000\text{ N}$
2. Shear stress $\tau$ occurs over the longitudinal horizontal area $A_s = w \times L$:
$\tau = \frac{F}{w \times L} \le \tau_{\text{allow}}$
3. Minimum length $L$:
$L = \frac{F}{w \times \tau_{\text{allow}}} = \frac{24000\text{ N}}{0.014\text{ m} \times (50 \times 10^6\text{ N/m}^2)} = \frac{24000}{700000} = 0.034286\text{ m} = 34.29\text{ mm} \approx 34.3\text{ mm}$