In the assembly design of shaft, pulley, and key, the weakest member is the key (option B).

- Introduction:
In the assembly design of a shaft, pulley, and key, it is important to consider the strength and reliability of each component. The weakest member refers to the component that is most likely to fail or be damaged under the applied loads and operating conditions.

- Explanation:
1. Pulley: A pulley is a wheel with a grooved rim designed to transmit power and change the direction of a belt or cable. Pulleys are typically made of materials such as cast iron, steel, or aluminum. While pulleys can experience wear and tear over time, they are generally robust and able to withstand significant loads.

2. Shaft: The shaft is a long, cylindrical mechanical component that rotates and transmits power from one part of a machine to another. Shafts are commonly made of materials such as steel, stainless steel, or alloy steel. They are designed to withstand bending, torsional, and axial loads. Shafts are typically larger and stronger than pulleys and are less likely to fail under normal operating conditions.

3. Key: The key is a small, rectangular component that fits into a keyway on the shaft and pulley. It is used to transmit torque between the shaft and the pulley, preventing slippage under load. Keys are usually made of materials such as steel or stainless steel. However, compared to the shaft and pulley, the key is relatively small and susceptible to shear and bending stresses.

- Reasons for the key being the weakest member:
1. Size and geometry: The key is smaller and has a simpler geometry compared to the shaft and pulley. This means it has a lower cross-sectional area and reduced strength.

2. Load distribution: The key is responsible for transmitting torque between the shaft and pulley. It experiences concentrated stresses at the point of contact, leading to potential stress concentration and failure.

3. Weakest link: In an assembly, the strength of the entire system is determined by its weakest component. Since the key has lower strength compared to the shaft and pulley, it becomes the weakest link in the assembly.

- Conclusion:
In the assembly design of a shaft, pulley, and key, the key is the weakest member. This is due to its smaller size, simpler geometry, concentrated stresses, and lower strength compared to the shaft and pulley. Proper design considerations, material selection, and stress analysis are essential to ensure the reliability and longevity of the assembly.