Subtopic: Measurement of force, stress, torque, speed, displacement and acceleration on machine elements – shafts, couplings, keys, bearings and knuckle joints
Question Statement:
A rectangular sunk key of width $w = 14\text{ mm}$ and height $h = 9\text{ mm}$ connects a $45\text{ mm}$ diameter shaft to a gear hub. The key transmits a steady torque of $T = 750\text{ N}\cdot\text{m}$. If the permissible shear stress of the key material is $\tau = 60\text{ MPa}$, the minimum required length of the key based on shear failure in $\text{mm}$ is ________ (round off to 1 decimal place).
Answer Key:
39.7
Step-by-Step Solution & Explanation
1. Tangential force acting at the surface of the $45\text{ mm}$ shaft ($d = 0.045\text{ m}$):
$F = \frac{2T}{d} = \frac{2 \times 750}{0.045} = 33,333.33\text{ N}$
2. Shearing area of key $A_s = l \times w$:
$F = l \cdot w \cdot \tau$
$l = \frac{F}{w \cdot \tau} = \frac{33,333.33\text{ N}}{0.014\text{ m} \times (60 \times 10^6\text{ N/m}^2)} = \frac{33,333.33}{840,000} = 0.03968\text{ m} \approx 39.7\text{ mm}$