A proposed space station includes living quarters in a circular ring 62.0 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
Let's denote the angular speed as ω (in rad/s), which we need to determine. The radius R of the circular ring is half of the diameter, R = 62.0 m / 2 = 31.0 m.
The occupants will feel their weight if the centripetal acceleration equals the acceleration due to gravity (g = 9.81 m/s²). The centripetal acceleration (a_c) is given by the formula:
a_c = R * ω²
We want this to be equal to the gravitational acceleration: g = 9.81 m/s².
R * ω² = g
ω² = g / R
ω = sqrt(g / R)
Now, just plug in the values:
ω = sqrt(9.81 m/s² / 31.0 m) ≈ 0.566 rad/s
So the ring should rotate at an angular speed of 0.566 rad/s for the occupants to feel that they have the same weight as they do on Earth.