Final answer:
The mass attached to a spring in a damping medium should be driven at its natural angular frequency of 69 rad/s to achieve maximum speed, assuming damping is not too heavy.
Step-by-step explanation:
To find the angular frequency at which the speed of a mass attached to a spring in a damping medium is a maximum, we refer to the principle of resonance in the context of damped harmonic motion. The maximum speed is achieved when the system is driven at an angular frequency that matches its natural angular frequency, in absence of excessive damping.
In the given scenario, a 290-g mass attached to a spring with a natural angular frequency of 69 rad/s is considered. Despite the presence of damping, the frequency that would produce the maximal response speed remains the same as the natural angular frequency, as long as damping is not too great, which is 69 rad/s. Therefore, one should drive this system at an angular frequency of 69 rad/s to achieve maximum speed.