The space station takes approximately 20 seconds to complete one rotation and provide artificial gravity for the occupants.
To calculate the time it takes for the cylindrical space station to complete one rotation and provide artificial gravity (g) for the occupants, we can use the concept of centripetal acceleration.
1. First, we need to find the radius of the cylindrical space station, which is half of its diameter:
Radius (r) = 200 m / 2 = 100 m
2. Next, we need to find the centripetal acceleration (a) required to provide artificial gravity (g) at the edge of the station. The formula for centripetal acceleration is:
a = (v^2) / r
Here, v is the linear velocity at the edge of the station.
3. We know that the acceleration due to gravity (g) on Earth is approximately 9.81 m/s². We want the artificial gravity (centripetal acceleration) to be equal to g:
a = g = 9.81 m/s²
4. Rearrange the centripetal acceleration formula to solve for v:
v = √(a * r)
v = √(9.81 m/s² * 100 m)
v ≈ 31.32 m/s
5. Now, we can calculate the time (T) it takes for one rotation using the formula for linear velocity:
v = (2πr) / T
31.32 m/s = (2π * 100 m) / T
6. Solve for T:
T = (2π * 100 m) / 31.32 m/s ≈ 20 seconds
So, the space station takes approximately 20 seconds to complete one rotation and provide artificial gravity for the occupants. Therefore, the correct answer is (B) 20 seconds.