In which situation would you want to use a smaller diameter driven pulley?

Using a smaller diameter driven pulley is a common mechanical strategy to increase the input speed of a system. When considering pulley systems, the relationship between the diameters of the pulleys and their speeds is governed by the principle of rotational speed proportionality.

In a pulley system, the speed of the driven pulley (the one being turned) is inversely proportional to its diameter compared to the driving pulley (the one providing the energy). If you reduce the diameter of the driven pulley, it will rotate faster than the driving pulley because the same length of belt will be traversing a smaller circumference on the driven pulley in the same amount of time.

This characteristic is particularly useful in applications where a higher output speed is required, making it ideal for increasing input speed. In practical terms, if the speed of the driving pulley is fixed, a smaller driven pulley will allow for a greater speed output, which is advantageous in various machinery setups where rapid movement is crucial.

Meanwhile, using a smaller pulley would not be advantageous for decreasing output speed, maintaining constant speed, or increasing torque, as these situations generally require a larger diameter for the driven pulley to achieve those effects.