Hey, guys. Let's take a look at this problem together. So we've got Jupiter and Neptune orbiting the sun, and I'm going to just draw a quick little diagram of what's going on. So the sun's in the center, and I'm going to have the orbit of Jupiter out here like that. And I'm going to 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 going to write that here: 11.86. And it's at a distance, an orbital distance, which I'll call RJ. And so we're told, if Neptune is out here, then Neptune orbits around the sun in some time, which I'll call TN, and it orbits the sun at some distance RN. So we're asked to find out how long it actually takes to orbit in years. So in other words, our target variable is TN. So what are we working with? We're working with the mass of the sun, the thing that's in the middle. The orbital distance is R and T. So we're going to use Kepler's third law. Kepler's third law says that there's a relationship between r cubed 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 in ratio form says that if you take the RJ cubed over TJ squared, that's just going to equal a number, and it's the same number as if you were to grab RN cubed divided by TN squared. And by the way, remember that these things don't have to necessarily be in SI units. As long as you have this ratio set up, then what happens is as long as these units right here, years and AU, are consistent, then you can set up these ratios together. So, in other words, I'm just going to go ahead and start isolating for TN squared. That's my target variable. So if I have this thing on the bottom, I can either cross multiply or I can just flip the fractions. Let's do that. I'm basically just going to flip these upside down. So I got TJ squared over RJ cubed equals TN squared over RN cubed. Oops. RN cubed. Now I've just got to move this RN over to the other side, and then I've isolated TN. So I've got TJ squared*RN cubed over RJ cubed equals TN 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 going to square that. And then I've got the orbital distance of Neptune, which is 30.11 AUs. I got to cube that and then divide it by 5.2 AUs and also cube that. Notice how all the units are consistent. And so what I'm going to get is I am going to get \(2.7 \times 10^4\). But remember, I have to take the square root because I have TN squared. So, really, what happens is, the orbital period of Neptune in years, is expressed as 165.25, and that's in years. Alright. 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.
8. Centripetal Forces & Gravitation
Kepler's Third Law
Video duration:
3mRelated Videos
Related Practice
Kepler's Third Law practice set
Problem sets built by lead tutorsExpert video explanations