Aproposed space station includes living quarters in a circular ring 50.0 m in diameter. at what angular speed should the ring rotate so the occupants feel that they have the same weight as they do on earth?

Given that the space station is free of gravitational force, it is required that it spins an certain speed to acquire centripetal acceleration.

In this case, you want that the centripetal acceleration, Ac, equals g (gravitational acceleration on the earth), becasue this will cause a centripetal force equal to the weight on earth.

The formula for centripetal acceleration is Ac = [angular velocity]^2*R

where R = [1/2]50.0m = 25.0 m

Ac = 9.81 m/s^2

=> [angular velocity]^2 = Ac/R = 9.81m/s^2v/ 25.0m = 0.3924 (rad/s)^2

[angular velocity] = √(0.3924) rad/s = 0.63 rad/s

0.63 rad/s