Satellite Period

by Patrick Ford
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Hey guys, We saw that the orbital velocity of satellite was how fast the satellite travel in its orbit. There's another variable you need to know to solve problems. And that's the orbital period, which the time that it takes to complete one orbit. So we saw that the satellite has a tangential velocity as it's going around in its orbit, but it takes some amount of time to actually complete one full orbit. That's called T, and we actually relate that velocity to the period just by using circular motion and basic mathematics. How do we get t? How do we get time from velocity? We know from cinematics that V equals distance over time. What's the distance that this thing completes in one orbit? Well, it's traveling in a circle of radius R. So that's just the circumference of that circle two pi r. And if you do that one full circle, then this T just becomes a capital T. And this equation is really useful for us in satellite motion. So I'm actually gonna write it here. Two pi r over tea. So now we can actually solve for this thi this capital T right here most of the time. You'll see it in his T squared form. This is four pi squared R cubed, divided by GM. You could actually get to it pretty quickly by using this equation. So let's go ahead and do it real quickly. V sat equals two pi r over tea. So what happens is if you want tea, you could get t equals two pi r over v sat. So we have this V sat, but we have already an equation that will tell us we have another equation for V sat. We could actually stick that in here, and we're gonna get two pi r divided by square roots of GM over R So you get this nasty formula with square roots and fractions. So what will happen is if you square it, it will become a lot cleaner. Two becomes the four pi becomes pi squared, are becomes r squared. And what happens to the square root is that when you square it, the whole thing just goes away. Um, or the square root goes away. So you get GM over a little are so we've got this situation. We have a fraction of a fraction and what happens is that this denominator of the bottom fraction will actually go up and merge with that top Little are so what'll happen is four pi squared r cubed over gm So you get that equation. This equation actually has a name. It is called Kepler's Third Law. Kepler was a guy who is studying the motions of planets in our solar system and you notice that the relationship between the orbital period of all the planets and the distances from the sun it's pretty cool. So we've got these three equations and these are all of our V sat equations are sorry, none of visa our satellite motion equations. So we've got the speed That's V SATs. We have the period t and the distance, and they're all related by these three equations, and they're all sort of dependent on each other. So what happens is the distance increases. Alright, So happens is our increases we can use. How does V change? So we have these two V sad equations, and if the little are is changing, how does this equation change? It's a little bit tricky to tell just by using this one, because you have all three variables present in these and we don't know how t changes yet. So instead, let's use the other V sat equation. So as you are increases in the denominator Thievy sat has to decrease. All right. And again, let's not use this, uh, this equation 40 because we have a relationship between just t and R. So it's gonna be easier to relate. Those to buy this equation now are is in the numerator here. So if our goes up than T has to increase as well. So that means that as r increases, your velocity decreases. But your period increases. This should make some sense because as you're going farther away, the force of gravity gets weaker on you, which means you don't have to go is fast to stay in a circle. On the other hand, if you were to go really close to the earth, the earth would be pulling on you really, really hard. Would you have to go really fast not to crash into the earth because you're traveling in a larger circle as R increases. The time that it takes also should be increasing, and that's basically it. So let's go ahead and take a look, at example, and use all of these equations together. So we've got the orbital period and speed of the international space station. So first thing is, we're gonna be calculating the orbital period. So let's take a look at our equations, so t equals What? So you've got these equations right here? So I've got V equals two pi r over t. The problem is, I don't know what the velocity of this thing is. Actually, that's part B. So I can't use this equation just yet, So I don't have V. But I can't use this equation because this just relates t and R. So I've got t squared equals I've got four pi r Sorry. Four pi squared are cubes over g times m Now I don't have our but I can relate are using big R plus h right? I know what the radius of the earth is and I also have the height. So I know what actually little are is so I could go ahead and use this equation to solve for t So I got t squared equals. Then I've got four pi squared. So what is this? Our distance? We'll remember this is This is just the radius of the earth. 6.37 times, 10 to the sixth. Oops. Plus this, 400 kilometers above the earth's surface. So that means that H is equal to 400 and then we've got 400 kilometers, so it's actually 400,000 400,000 m. So we've got four times, 10 to the fifth meters, and you're gonna have to cube that, actually. Just be careful. You gotta cube that. And then you've got a divided by 6.67 times 10 to minus 11 and then multiplied by the mass of the earth. 5.97 times, 10 to the 24. If you do that, you're gonna get some crazy number, which is 3.8 times 10 to the seventh. But we're not quite done yet, because remember, you have t squared here, so you have to square both sides square root both sides. Just make sure that's the last step. You're not forgetting that. So if you square root both sides, you're gonna get 5550 seconds, which is actually about 1.5 hours. This is pretty cool. You can actually google this and you'll find the International Space station does orbit about 1.5 hours. Takes 1.5 hours to travel around the entire earth, which is pretty awesome. It's going insanely fast. So that was part a eso now in part B were actually figuring out what the velocity is. So in part B, what is V sat right? Cool. Let's take a look at our equations here. I have this equation and I have this equation. But in this case right here, I actually have what are is. And now I've just figured out what the velocity is, so I can actually use either one of these equations. I'm just gonna use the simpler one. So if I have both equations, then Visa equals two pi. I know what the R is that our distance here is just 6.37 times 10 to the sixth, plus four times 10 to the fifth and then divided by the period, which is 5550 seconds. And if you do that, you should actually get, um, 7660 m per second, which you can also Google and you'll find that it's just about this number, so let me know if you guys have any questions with this.