8. Centripetal Forces & Gravitation

Period and Frequency in Uniform Circular Motion

# Spinning Space Station

Patrick Ford

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Hey, guys, let's check out this problem here. So a big problem for astronauts in space is gonna be the lack of gravity. So one way we can fix this for future space travel is we can build a spinning space station. So the idea here is that you have this big ring like this that's attached to the main part of the hull of the ship or whatever, and you have this arm that basically attach is it to the central part of the ship, and this entire ring here is going to rotate, and as it rotates, it actually produces some centripetal acceleration for the people on the inside, so they start to accelerate towards the center. So we want to figure out basically how fast we have to spin this thing so that the acceleration inwards equals G, which is 9.8. It's kind of like the same effect as when you're going in a car and you're taking a corner. You'll feel like that kind of push from the walls or the door of your car. It's basically kind of like the same effect they're going to be pushed inwards like this from the walls of the space station, and we want the acceleration to be 9.8. So we have to do here is let's take a look at our problem here. Right? We've got this diameter, which equals 500 which is the diameter of the length of the entire thing. So remember that we use the radius in these problems, and that's just gonna be the diameter divided by two. So we're going to actually use 250 in our problems. Now we're trying to find here is how fast in RPMs we have to actually spend this spaceship here. Okay, so how do we figure RPMs? Well, let's take a look at our problems Are equations we have RPMs is equal to, um sorry. Frequency is equal to RPMs over 60. So, you know, you can find the RPMs if we have the frequency. So this is frequency equals RPMs over 60. So we can do here. Is that if I want to solve for r. P M. S, which is really just like a modified version of frequency, all we have to do is right, is get the frequency and then multiplied by 60. So to find my RPMs. I just have to find the frequency like this and then multiply it by 60. So remember, I don't have the frequency in this problem. So how do I go ahead and find that right? So I'm gonna have to go find the frequency. Now, let's take a look at our problems or equations. We know that frequency is related to one over the period one over T. But I don't have tea in this problem either. So this is not gonna be a good equation for us to use Now, we can also look at our velocity equation because our velocity equation has period and also frequency. Now, unfortunately, what happens is I don't really know what the the tangential velocity of any one of the points of the ring are. So this is not going to be a good equation for us to use either. Instead of what we can do is we know that a C is related to V squared over R. And we know that this centripetal acceleration also has a shortcut equation in which we can use this. This guy over here, we actually do know something about the centripetal acceleration. So that's what we're going to use. So we have here is we have a C equals and then we have four pi squared r times the frequency squared. So if you take a look, I know what the centripetal acceleration is gonna be. It's gonna be 90.8. I don't know are so I can figure out the frequency squared and then I can bring it basically back into this equation to solve for r p. M s. All right, so that's what we're gonna do. So I'm just gonna go ahead and solve for the frequency squared and this is gonna be a C divided by four pi squared times are now All I do is I just take the square roots and I'm just gonna start plugging stuff in so you know, this is gonna be 9.8. That's the centripetal acceleration. And then when you plug in all this stuff on the bottom here, you're just gonna basically plug this in as a as a parentheses. Actually, let me go ahead and write this four pi squared times 250 you're gonna have to, like, surround this whole entire thing, a parentheses otherwise, you get the wrong answer and you're gonna get 0.0 99 hertz. So now we just plug this guy right back into here. So we have 0.0, 99 times 60 and we get an rpm equal to 94. So basically, you have to spin this thing at almost six revolutions per minute, even though the space station's almost half a kilometer long or it is half a kilometer long in order to simulate gravity that's similar to that of Earth. All right, so that's it for this one. Guys, let me know if you have any questions.

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