It has been suggested that rotating cylinders about 9 mi long and 3.5 mi in diameter?

Question

be placed in space and used as colonies. What angular speed must such a cylinder have so that the centripetal acceleration at its surface equals the free-fall acceleration on Earth?_____rad/s

Hey guys, I have tried this problem many times and its driving me nuts. Can you please show me how to get to the correct answer? Thanks!

umm...I don't understand that completely, I have gotten to the part where I converted to SI units and found velocity. But I get stuck, there after. ? Is there an easier way to do this? Thanks

Answer 1

Hmm... it seems like I was helping some students with this very problem today... WebAssign perhaps?

Well, the centripetal acceleration will be the only acceleration caused by the rotating cylinder, thus the source of our "artificial gravity". We express the centripetal acceleration as:
a=v²/r, where v is the velocity at the radius, and r is the radius.

Now, to match the acceleration of free-fall on earth, the centripetal acceleration will also equal 9.80 m/s² (or 32.2 ft/s²).

You'll want to convert miles to meters (SI units), and then you can solve for what velocity you'll need to obtain the desired centripetal acceleration, since you already know the radius. (You could also keep the empirical units if you use 32.2 ft/s² for the free-fall acceleration on earth.)

After solving for velocity, you can then find angular speed (ω) with the conversion factor 1 revolution = 2π radians. Notice that one revolution will be the circumference of your cylinder. You can thus use dimensional analysis (where you compare units) to properly convert the velocity (v) to angular speed (ω). That is, by dividing your velocity by the conversion factor of 1 [revolution] per [m], you'll obtain [revolutions/second], which you can easily convert to [radians/second].

Alternatively, note v=rω, therefore we could substitute to rewrite our original equation:
a=ω²*r
And solve for ω directly.

While the latter method is probably preferred, I don't believe you've encountered the equation v=rω yet, which is why I presented the other way first.

Either way, be sure to watch your units.

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Also note that, assuming a constant angular speed (ω), the centripetal acceleration will depend on the radius measured from the axis of rotation. If you stood right along the central axis of the cylinder, you would be "weightless" as you would have no centripetal acceleration. As your distance from the axis (r) increases, so too will the centripetal acceleration.

Good luck!