Final answer:
Without specific details about the rotating system, such as natural frequency and physical parameters, it isn't possible to calculate the first and second critical speeds in rpm. The provided examples discuss angular velocity and acceleration but do not pertain to critical speed calculations.
Step-by-step explanation:
Understanding Critical Speeds in Rotational Motion
To calculate the first and second critical speeds in rpm (revolutions per minute), a specific formula derived from the characteristics of the rotating system would normally be required, such as the natural frequency of the system and the properties of the rotor. However, the information provided does not contain details about the type of system (like a shaft or a rotor system) for which we are calculating critical speeds, nor does it provide the necessary physical parameters to use such formulas.
The example questions provided mostly deal with the calculations of angular velocity, centripetal acceleration, and angular acceleration in various contexts, but they don't directly address how to calculate critical speeds. If more specific information on the system in question were available, including material properties, dimensions, and bearing types, it would be possible to apply engineering formulas to determine critical speeds.
The critical speeds of a rotating system are important to ensure the safe and stable operation of rotational machinery, as exceeding these speeds can result in destructive resonant vibrations.