Understanding the maximum torque that can be transmitted by a key is crucial in machine design to ensure the reliable operation of mechanical components. A key is used to connect a rotating machine element, like a pulley or gear, to a shaft, preventing relative motion and transmitting torque. This problem focuses on calculating the maximum torque a key can transmit based on its shear strength.

Key Torque Calculation Overview

When a key transmits torque, it is subjected to forces that can cause it to fail, typically in shear or crushing. For this problem, we are considering the shear failure of the key, which depends on its dimensions and the material's allowable shear stress. The torque transmitted is directly related to the shear force acting on the key and the radius of the shaft.

Given Key and Shaft Parameters

To determine the maximum torque, we first list the provided dimensions and material properties of the key and shaft:

Formulas for Key Torque

The calculation of maximum torque is based on the key's resistance to shear. The relevant formulas are:

• Shear Area of the Key (\(A_{\text{shear}}\)): The area resisting shear failure in a key is typically its width multiplied by its length. \[ A_{\text{shear}} = w \times L \]
• Shear Force on the Key (\(F_s\)): This force is the product of the allowable shear stress and the shear area. \[ F_s = A_{\text{shear}} \times \tau_{\text{allow}} \]
• Maximum Torque Transmitted (\(T\)): The torque is the shear force acting on the key multiplied by the radius of the shaft where the force is applied. \[ T = F_s \times \frac{D}{2} \]

Step-by-Step Key Torque Calculation

Let's calculate the maximum torque that can be transmitted by the key using the given parameters and formulas. It's important to ensure all units are consistent (e.g., SI units).

1. Convert Units to Standard SI Units:
   • Key width \(w = 14 \text{ mm} = 0.014 \text{ m}\)
   • Key length \(L = 100 \text{ mm} = 0.100 \text{ m}\)
   • Shaft diameter \(D = 50 \text{ mm} = 0.050 \text{ m}\)
   • Allowable shear stress \(\tau_{\text{allow}} = 50 \text{ MPa} = 50 \times 10^6 \text{ N/m}^2\)
2. Calculate the Shear Area of the Key: The shear area is where the key would fail due to shear, which is its width times its length. \[ A_{\text{shear}} = w \times L = 0.014 \text{ m} \times 0.100 \text{ m} = 0.0014 \text{ m}^2 \]
3. Calculate the Maximum Shear Force on the Key: This force is the maximum force the key can withstand in shear before failure. \[ F_s = A_{\text{shear}} \times \tau_{\text{allow}} = 0.0014 \text{ m}^2 \times (50 \times 10^6 \text{ N/m}^2) \] \[ F_s = 70000 \text{ N} \]
4. Calculate the Maximum Torque Transmitted: The torque is the shear force multiplied by the radius of the shaft. \[ T = F_s \times \frac{D}{2} = 70000 \text{ N} \times \frac{0.050 \text{ m}}{2} \] \[ T = 70000 \text{ N} \times 0.025 \text{ m} \] \[ T = 1750 \text{ N} \cdot \text{m} \]

Conclusion on Maximum Torque

Based on the calculations, the maximum torque that can be transmitted by the key, considering its allowable shear stress, is \(1750 \text{ N} \cdot \text{m}\). This value ensures that the key does not fail in shear under the applied load. It's important to always design components to operate within their safe stress limits to prevent failure and ensure the integrity of the mechanical system.