How fast should a space station rotate to simulate Earth's gravity?

[orange03]

A space station is in the shape of a hollow ring, 450 m in diameter. At how many revolutions per minute should it rotate in order to simulate Earth's gravity-that is, so that the normal force on an astronaut at the outer edge would be the astronaut's weight on Earth?

All I got so far was finding the circumference=pid=1413 meters. I'm stuck, I don't know what to do next. Help please!

 

[rock.freak667]

So the space station is rotating, thus there is a centripetal force. Do you know what that is?

 

[orange03]

Yeah, it's the force that is the same direction as acceleration, towards the center.

 

[rock.freak667]

Yeah, it's the force that is the same direction as acceleration, towards the center.

Right then. So you want the space station to produce a force which is equal to the weight of the astronaut. Can you make a relation between this force and the weight?

 

[orange03]

would it just be N=mg?

 

[rock.freak667]

would it just be N=mg?

Yes the normal force on the astronaut is =mg. BUT you want the space station to spin to produce a force which equals mg. Do you know any formulas for centripetal force?

 

[orange03]

F=mrw^2?

 

[rock.freak667]

F=mrw^2?

yes, so you want mw²r=mg

can you find w and then convert that to RPM?

 

[orange03]

okay i got w=.20 but I don't know how to convert that to RPM. Wouldn't the units on w by 1/s^2? Is there a formula or conversion factor to get it to RPM?

 

[Astronuc]

Note that in the equation mw²r=mg, the mass is the same.

So dividing by mass, it become w²r=g.

g had unit of m/s², and since r had units of m (meters, length), then w² must give units of 1/s².

w is angular frequency which is expressed in 1/s, and it's actually radians/s. Now since 1 revolution passes through 2pi radians, then 1 rps (revolution per second) = 2pi rad/s.

The frequency in revolutions per unit time = f = w/2pi = 1/T, where T = the period.

Conversely, w = 2pi f

 

[rock.freak667]

okay i got w=.20 but I don't know how to convert that to RPM. Wouldn't the units on w by 1/s^2? Is there a formula or conversion factor to get it to RPM?

Right then w=0.2 rad/s and you want it in revolutions per minute.

In 1 revolution, the station rotates 2pi radians.
So 1 rad = 1 rev/2pi.

60 seconds = 1min

so

$$1 \frac{radian}{seconds} = \frac{60}{2 \pi} revolutions/min$$

 

[orange03]

Okay, I got it now. Thank you!