8. Centripetal Forces & Gravitation

Kepler's Third Law

# Jupiter & Neptune's orbits

Patrick Ford

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Hey, guys, let's take a look at this problem together. So we've got Jupiter and Neptune orbiting the sun, and I'm gonna draw a quick little diagram. What's going on? So the sons in the center, I'm gonna have the orbit of Jupiter out here like that, and I'm gonna have the orbit of Neptune. That's a little bit farther out. Obviously, it's not to scale. So I'm told that if Jupiter is some distance here that Jupiter orbits once. So in other words, the orbital period takes 11.86 years. I'm gonna write that here 11.86 it's at a distance and orbital distance, which I'll call R. J. And so we're told, if Neptune is out here, the Neptune orbits around the sun in some time, which I'll call t n. And it orbits the sun at some distance r n so were asked to find out how long it actually takes to orbit in years. In other words, our target variable is Tien. So what are we working with? We're working with the mass of the sun thing that's in the middle, the orbital distance. It's R and T. So we're gonna use Kepler's third law. Kepler's third law says that there is a relationship between our cube and T squared, and that's just a constant. And specifically, if you have two objects that are orbiting the same thing, you can set up a ratio between the two. So Kepler's Third Law and Ratio form says that if you take the RJ cubed over T j squared, that's just gonna equal a number. And it's the same number as if you were to grab our n cubed divided by T n squared. By the way, remember that these things don't have to necessarily be in S I units. As long as you have this ratio set up, then what happens is as long as these units right here years and a you are consistent, then you can set up thes ratios together. So, in other words, I'm just gonna go ahead and start isolating for T N Square. That's my target variable. So if I have this thing on the bottom, I could either cross multiply or I could just flip the fractions. Let's do that. I'm basically just gonna flip these upside down, so I got t j squared over r J Cube equals t n squared over r n cubed poops R n cubes. Now I've just gotta move this RN over to the other side And then I've isolated TN so I've got to t j squared r n cubed over r j cubed equals t n squared. So, in other words, what is the orbital period of Jupiter? Well, I'm told that that is 11.86 years, and I'm just gonna square that. And then I've got the orbital distance of Neptune, which is 30.0 point 11 a use I've got a cube that and then divided by five point to a use. And also cube that notice how all the units are consistent. And so what I'm gonna get is I am going to get to seven times 10 to the fifth. But remember, I have to take the square roots because I have t n squared. So really, what happens is, uh, the orbital period of Neptune in years, eyes expressed as 1.1 65 25. And that's in years. All right, you can actually look this up, and this is pretty close to what the actual orbit is. 165 years. Let me know if you guys have any questions with this

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