Angular rotation and torue high school physics question?

A rotating space station uses the centrifugal force to simulate gravity (note that the centrifugal force is not technically a force itself, but arises from the rotational motion of the station).

- Imagine a tennis ball on a string; when you twirl it around fast enough, the ball gets "forced" out. So if you were inside a large spinning tube like the space station, you would experience that same effect. Thus, you would be inhabiting the far side; when you stand, your head would be pointing towards the axis of rotation.

- Earth's gravitational acceleration, g, is equal to 9.8 m/s². To mimic Earth's pull, the station would have to develop a centripetal acceleration equal to 9.8 m/s². Again, recall that this pull is OUTWARD from the axis of rotation. This centripetal acceleration is equal to:

ac = v²/r = w²*r

where w is the rotational speed of the system, v is the tangential speed of the person and r is the distance from the axis of rotation. Setting this expression equal to g provides:

w²*r = g

w = sqrt(g/r)

where sqrt is the square root. Whis works out to:

w = [(9.8 m/s²) / (1.6 km * 1000 m/km)]^(1/2)

w = .0783 rad/s

Thus, the station only needs to rotate at a speed of about .0783 rad/s in order to develop an atrificial gravitational field equal to that of Earth.

- The torque on a body is given by:

tau = I * alpha

where tau is the torque, I is the station's moment of inertia and alpha is the angular acceleration. From your problem statement, we must make some assumptions to answer this question. First, we have to assume that all of the mass is evenly distributed throughout the "hoop" that makes up the station. Second, we must assume a constant angular acceleration of the station.

Since the angular acceleration is constant, we can find its value as:

alpha = (w2 - w1) / t

alpha = ( .0783 rad/s - 0 ) / (2 hrs * 3600 s/hr)

aplha = 0.000010875 rad/s²

Finally, we must calculate the moment of inertia. This can be a bit tricky, and I don't know if you cover it in high school, so check out the wikipedia link below for the specifics. I'll just provide the moment of inertia for a hoop as:

I = (m/2)(r2² + r1²)

I = (187016/2)[(1600 m)² + (1500 m)²]

I = 449773480000 kg-m²

Substituting the known values provides:

tau = (449773480000 kg-m²)(0.000010875 rad/s²)

tau = 4,891,287 N-m

Hope that clears it up a bit for you :)

[Quick Edit - you may want to double-check my numbers. I couldn't find my calculator so I used the one on my computer.]

- they will live standing oposite to the rotation center since the rotation is what is supposed to simulate gravity. That is, head to the rotation center, feet outside

- I am mathematically stupid, so I can not calculate that, but I guess you will have to rotate it in a speed that will drive people out with a force equeal to 9.8m/s^2

- Again mathematically stupid.

At least I knew 1 answer, didn't I?