8. Centripetal Forces & Gravitation

Geosynchronous Orbits

# Find Mars' period, given synchronous satellite

Patrick Ford

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Alright, guys. So, in this problem we're asked to calculate with a period of Mars rotation is assuming that we have a satellite in synchronous orbit. So let's go ahead and draw the diagram real quick. Got Mars right here. I'm gonna have a synchronous orbit around it. So I'm told that there is a satellite here at some some distance. I'm told that there's a satellite and that satellite is in synchronous orbit and it has a velocity. The velocity is equal to 14. 50. And when I'm supposed to do is figure out what the period of Mars rotation is. So I need to figure out what T planet is, but I'm just gonna write team ours. So that's really our target variable for this problem. What is T Mars? Okay, so we've got a synchronous orbit, right? And so what you need to remember about synchronous orbits is that the period of the satellite that's orbiting it is equal to the period of the planet's rotation. So that's what we need to use for this problem. So t Mars, Where does that? Where does that variable pop up in our equations? It pops up in the synchronous orbits equations. So let's go ahead and start there. So we've got our synchronous is equal to I've got the cube root Then I've got G And then I've got the mass of Mars that I've got the period of Mars squared divided by four pi squared. So I really just solving for what this t Mars is equal to. So let's go ahead and start isolating that variable. I've got a que both sides so I've got our sink Cube is equal to g mass Mars team R squared, divided by four pi squared. And then when I move everything over to the other side, I get four pi squared are sink cubed over g times the mass of Mars and that's equal to t Mars. And that's squared, right? So let's take a look at all of these variables. This is just a constant gravitational constant I have with the mass of Mars is in this table right here. So the only thing I don't know is that orbital distance. I've never told any information about the orbital distance of this thing, so let's see how we can use something else about this problem to solve for that right, if I can figure out what that are synchronous is that I can figure out what team ours is. So how do I figure out what that little are is? Let's see, we've got our equations here and all of them involve our but it will see a lot of them involved T SATs, which is what I'm trying to find or t planet. So I can't use these equations. So let's see, I'm gonna have to use my satellite TV satellite equations and there's two basic ones I can use. I can use the square G ever gm over r. I can use this two pi r over tea. But if I take a look at this right here, one of the variables is the variable I'm trying to look for, which is t and then the other one is are so I can't use that equation because it involves two unknowns. So let's instead start with the V sat equation. So I got V sat equals the square root of G times mass over. Little are I'm trying to find out what this little our is and I actually have with the velocity of this satellite is and then these two are just constants. So let's go ahead and salt for that. Little are so I've got a square, both sides. So I've got ve sat squared equals GM over R. Now, I just gotta basically isolate our by trading places with the visa. So we got our equals. I've got g m over V. Sat squared is equal to then I've got let's see, 6.67 times, 10 to the minus 11. I've got the mass of Mars, right? The massive Mars, which is equal to I've got 6.42 times to the What is that? That is 23 and then I got divided by V Sat square. That's 14 50 that's squared. And if you do that, you should get our is equal to Let's see, I got 2.4 times 10 to the seventh, and that's in meters. So I'm gonna take this our distance. I'm gonna plug it back all the way into this equation and then solve for t Mars. So if I go in to do that and start just plugging in all of these variables, I got four pi squared. Then I've got 2.4 times. 10 to the seventh. I'm gonna cube that. Then you've got a divided by 6.67 times 10 to the minus 11 and then multiplied by six points 42 times 10 to the 23. And if you do that, you're gonna get T Mars squared, which is equal to Let's see, I got 7.83 times 10 to the ninth. But we're not done yet because again, we have that t squared. So if you take the square roots, we're gonna find that the rotation period of Mars is equal to about 88,470 seconds, which is about 24 point six hours. And this is actually the right answer. It's pretty amazing how we use two pieces of information, which is a synchronous orbit and the velocity of that synchronous orbit. We can actually figure out the rotation period of an entire planet. It's pretty cool. Let me know if you guys have any question with this. But that's the final answer. Alright, guys, we'll see the next one

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