Introduction:
In a belt drive system, the coefficient of friction plays a crucial role in determining the design and performance of the system. The coefficient of friction represents the frictional resistance between the belt and the pulleys. If the two pulleys in a belt drive have different values of the coefficient of friction, the design will be based on the pulley for which the coefficient of friction is minimum.

Explanation:
The coefficient of friction (μ) is a dimensionless quantity that represents the frictional resistance between two surfaces in contact. It is defined as the ratio of the frictional force (F) between the surfaces to the normal force (N) pressing the surfaces together.

μ = F/N

In a belt drive system, the coefficient of friction between the belt and the pulleys affects the power transmission efficiency, slip, and tension in the belt. The coefficient of friction is influenced by various factors such as the materials of the belt and pulleys, surface roughness, tension in the belt, and environmental conditions.

Design Considerations:
When designing a belt drive system with pulleys of different coefficients of friction, it is important to consider the following factors:

1. Slip: Slip occurs when the belt does not grip the pulleys tightly, resulting in a loss of power transmission efficiency. Higher coefficients of friction between the belt and the pulleys help to minimize slip. Therefore, the design should be based on the pulley with the higher coefficient of friction.

2. Tension in the Belt: The tension in the belt is crucial for the proper functioning of the belt drive system. Higher coefficients of friction between the belt and the pulleys help to maintain a higher tension in the belt, reducing the chances of belt slippage. Therefore, the design should be based on the pulley with the higher coefficient of friction.

3. Power Transmission Efficiency: The coefficient of friction affects the power transmission efficiency of the belt drive system. Higher coefficients of friction result in higher power transmission efficiencies. Therefore, the design should be based on the pulley with the higher coefficient of friction.

Conclusion:
In a belt drive system with pulleys of different coefficients of friction, the design should be based on the pulley for which the coefficient of friction is minimum. This ensures that slip is minimized, tension in the belt is maintained, and power transmission efficiency is maximized. By considering these factors, the design can be optimized for efficient and reliable operation of the belt drive system.