Which method provides the ability to connect and disconnect while a shaft is running?

The method that provides the ability to connect and disconnect while a shaft is running is the clutch. A clutch is specifically designed to engage and disengage power transmission between the driving and driven shafts, allowing for control over the operation of machinery without needing to stop the system. This ability is crucial in applications where starting and stopping motion while the system is in operation is necessary, such as in vehicles or various industrial machinery.

When a clutch is engaged, it allows the drive shaft to transmit torque to the driven shaft, and when disengaged, it disconnects this transmission, permitting the driven component to come to a stop or operate independently from the power source. This function is essential in many mechanical systems that require the ability to start and stop torque flow while minimizing the disruption to the overall operation.

In contrast, universal joints and couplings serve different purposes—they provide flexibility and connection between shafts but do not offer the ability to disconnect while in motion. Coaxial connectors are used primarily for electrical signals and do not pertain to mechanical shaft connections in the same manner.