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Problem 49 Easy Difficulty

Pluto's diameter is approximately $2370 \mathrm{km},$ and the diameter of its satellite Charon is 1250 $\mathrm{km}$ . Although the distance varies, they are often about $19,700 \mathrm{km}$ apart, center-to-center. Assuming that both Pluto and Charon have the same composition and hence the same average density, find the location of the center of mass of this system relative to the center of Pluto.

Answer

$x_{\mathrm{cm}}=2520.5 \mathrm{km}$

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Video Transcript

{'transcript': "problem. 8.53 we're trying to find where between plutonium and its moon. Tehran, The center of mass of the two is located. Now we're told the diameter of Fluto the diameter care on. And we're told to assume that they have the same composition, and so their densities air the same. Now, as far as I know, we don't actually know what the mass of masses of Ludo and Karen are. Exactly. Are you even have really good estimate for it? Uh, so this is not just, uh you know, the textbook to being difficult. This is actually sort of a practical thing. It might need to do if you know how big something is, which is fairly easy to figure out for observation. But you don't know how massive it is. Um, if you make some assumptions about, in this case the composition of the two bodies being the same and so their densities would be the same way that you can actually figure out something about this. I believe you can also figure out what their center of mass is, uh, through observing their orbiting around each other. But that's not what this problem is. about. So let's call this the extra action. Just, uh, I have something to call it and we'll put Pluto's center at zero X equals zero so that the export innit of ploo toe is equal to zero. And whenever you can do something like this, it will always make your life easier, because then one of the things is going to end up, not mattering. So we know that the center of mass position in this case we only have one court that X to worry about. It's going to equal the massive Fudo Times, its exposition, which we've called its position zero. So that's nice, like I just said, And then the massive care on times position of care on which we know they're about 19,700 kilometers apart. So this will be 19,700 kilometers because we take this to be zero, and then this will be divided by the sum of the masses. So we need to figure out what what the masses end up being a force. And as we know, the mass of an object is equal to its density times its volume. Hey, and assuming that their little spheres you know, this is ro times for thirds. Aye. R cubed that were given the diameters and might as well this work with those that works out to be balsa. Let's nicer looking. Order 16 Hey, row, I am with you. Of course, all of this isn't going to matter because it's a common factor. All of these things here. So But just for that, you know these things anyway, So we have 16 I road diameter care on you. 16 I, uh huh. Maybe not make it look like pious six. I row times the sum of the cubes of the diameters. And so you see, the 1/6 I rose cancel out as promised, and oh, uh, I forgot that you put the numbers in and we discover that center of mass is 2520 kilometers. This would be from the center of Ludo, because that's how we set up our coordinate system. And now you can see that the center of mass of the clue tocar on system is actually outside of Ludo because this is 2370 this is 2520 So Karen is actually large enough in comparison to Bluebell that the senator masses, you know, somewhere out here this is the fact of the heroin is basically comparable to Plato, and it's not entirely comfortable. But it's a fairly major size thing compared to Pluto's part of why I went the, uh of the International Astronomical Union. Some I forget the name of the organization exactly. But when they came up with an actual precise definition of a planet, this is part of the reason my puto is no longer considered a planet. It's a dwarf planet now, because Pluto's not putting feared its orbit, which means that there's this other gravitationally significant object. And I think there are also some other, uh, some other bodies include those orbit that, uh, a large I don't recall anyway, so that's part of the reason. So this this is sort of hinted the reason why Pluto's not been considered a planet since we have, since we developed an actual precise definition, we could apply, decide on a case by case basis"}