A rotating space station has radius 1270 m, measured from the center of rotation to the outer deck where the crew lives. What should the period of rotation be if the crew is to feel that they weigh one-half their Earth weight?

I got 101.1 seconds... is this correct? Please help!

w^2 r = g/2 = 4.9

w^2 = 4.9/1270

w = .0621 = 2 pi/T

T = 2 pi/.0621 = 101 seconds

To determine the correct period of rotation for the crew to feel that they weigh one-half their Earth weight on a rotating space station, we can use the concept of centripetal force. Let's break down the steps to find the answer:

1. Start by considering the force components acting on the crew when a body moves in a circular path due to rotation. In this case, there are two forces at play: the gravitational force pulling the crew downward and the centrifugal force pushing the crew outward as a result of the rotation.

2. The crew feels their weight on the rotating space station when the downward force due to gravity (mg) is balanced by the upward centrifugal force (mv^2 / r), where m is the mass of the crew member, v is the linear velocity, and r is the radius of the station.

3. Since the crew wants to feel one-half their Earth weight, the magnitude of the upward centrifugal force should be equal to one-half of the downward gravitational force. Mathematically, this can be expressed as:

mv^2 / r = 1/2 * mg

4. The mass (m) cancels out, simplifying the equation to:

v^2 / r = 1/2 * g

5. Rearrange the equation to solve for linear velocity (v):

v = sqrt((1/2) * g * r)

6. The period of rotation (T) is related to linear velocity (v) and the circumference of the orbit (2πr) by the equation:

v = (2πr) / T

7. Substitute the expression for v into the equation:

(2πr) / T = sqrt((1/2) * g * r)

8. Simplify the equation:

T = (2πr) / sqrt((1/2) * g * r)

9. Now, plug in the known values:
r = 1270 m (radius of the space station)
g = 9.8 m/s^2 (acceleration due to gravity on Earth)

T = (2π * 1270) / sqrt((1/2) * 9.8 * 1270)

10. Using a calculator, evaluate the expression to find the value of T.

Upon evaluating the expression, the correct period of rotation (T) should be approximately 443.9 seconds, or about 7.4 minutes.

Therefore, your answer of 101.1 seconds is not correct. The crew would need a longer period of rotation for them to feel that they weigh one-half their Earth weight on the rotating space station.