Find mass of planet, given moon speed and period

by Patrick Ford
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Hey, guys, let's see if we can work it out. This problem together. So we have a moon in orbit around a planet, right? So let's go ahead and draw diagrams. So we have a planet like this and we have a moon that is orbiting around in a circular orbit. Just pretend that that's perfectly circular, right? And this moon has a has a speed around its planet, and we're told that velocity is equal to 7500. And we're also told that it takes 28 hours to go all the way around. So all the way one rotation, you know that T is equal to 28 that is ours. So using that information, how can you find out what the mass of this planet is? So we're finding the mass of the planet. That's the thing in the center. So we're really looking for capital M so capital and big is our target variable. So how do we go about doing that? Well, let's see. We've got a whole bunch of equations evolving forces and gravitation, gravitational acceleration. But we don't have any information about forces and gravitational acceleration, but we do have the motion of satellites. We're gonna use all these equations over here, so let's see, I've got orbital velocity and I also have orbital period. So really, I can use, like, any one of these equations to start off. I'm gonna use the V sad equation. So we've got visa equals, Let's see square root of g times, big M big M Big M over our and let's see, um, I know what the velocity of the satellite is and I'm looking for big end, which is my target variable gravitational constant since the number. So if I If I can figure out what this little our distances, I could find that. But I actually don't have any information about the orbital distance or the height or anything like that. So let's see, Maybe this isn't the approach. Let's see if I could start off with my tea sat equation with the T squared one. So let's start with the tea sat squared equation. Let's see if we have better luck there. So you've got four pi squared r cubed over g g times. Big M again. That Big M is our my target variable here. So I know what the orbital period is But again there's that are that I don't have that orbital distance. So in both of these approaches here, both these equations that I've seen, I ended up with the same unknown variable. I have too many unknowns. So there's gotta be something else I could do to solve for this little our distance. Let's see, what's the one equation I haven't used yet? So I'm gonna go over here and solve for a little r The one I haven't used it is this one. The V sat equals two pi r ver t So I've got visa equals two pi r divided by teeth. Now, in this situation for this equation, I have to knowns and Onley one unknown. So now I can use that to Seoul for my little are So let's see, I've got moving everything over to the left side. I'm gonna get visa times t divided by two pi is gonna give me Little are so that means that that little are here If you just go ahead and plug, everything in is going to be let's see 7500 times the period. Now that T is expressed in hours so first, I need to multiply by 3600 to get into seconds. If you do that, you should get 1.8 times 10 to the fifth. That's in seconds. So I'm just gonna go ahead and plug it in 1.8 times 10 to the fifth. Now, I've just got a divide by two pi. So if I divide by two pi, I'm gonna get the orbital distance that little are That's 1.2 times 10 to the eighth. So now what I could do is now I can take this. Little are that I found I can plug it back in tow either this t squared equation because now I only have this unknown. Or I can plug it back into the V sat equation so that I still only have one known. So really, the choice is up to you. I'm gonna just keep going with the visa approach. So I've got now my my little our distance. So I've got ve sat equals square roots. Whoops. Square root of GM over. Little are So now I'm just gonna go ahead and salt for that big M because I already have everything else, right? So I've got ve sat and I'm gonna square both sides because it's because I want to get rid of the square root. So they've got V sat squared, equals G Mm over our And then I just moved the are over everything over to the left side. So I get V sat squared are divided by G equals Big M. If you plug all that stuff in, what you're gonna get is 7500 squared times the radius 1.22 Sorry. 1.2 times 10 to the eighth. And then we've got to do divided by the gravitational constant. So it times 10 to the 11th. And if you do that, you should get the mass of the planet, which is 1.1 times 10 to the 26. And that's in kilograms. That's it for this one. Let me know if you guys have any questions with that