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Ch 13: Newton's Theory of Gravity
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All textbooksKnight Calc 5th EditionCh 13: Newton's Theory of GravityProblem 13.54b

Chapter 13, Problem 13.54b

In 2014, the European Space Agency placed a satellite in orbit around comet 67P/Churyumov-Gerasimenko and then landed a probe on the surface. The actual orbit was elliptical, but we’ll approximate it as a 50-km-diameter circular orbit with a period of 11 days. (b) What is the mass of the comet?

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Hi, everyone. Let's take a look at this practice problem dealing with Kepler laws. So in this problem, a team of engineers have placed a high altitude balloon in orbit around the planet and we will approximate its path as a circular orbit with a diameter of 35 kilometers and a period of seven days. And the question wants us to estimate the mass of the planet given four possible choices as our answers. A is 3.0 multiplied by 10 to the 12 kg. Ob is 8.7 multiplied by 10 to the 12 kg. C is 6.9 multiplied by 10 to the 13 kg and D is none of these. Now, since I was given a diameter and a period here, I'm actually going to use Kepler's third law. Recall that Kepler's third law states that we have T squared is equal to or pi squared or cubed divided by GM T is the period of the orbit R is the radius of the orbit is gravitational constant and M is the mass of the planet. I'm asked, asked to find the mass of the planet. So I'm gonna solve this equation for M, I'll do that by just multiplying through by M and divided by T square on both sides. So I'll have M is equal to or pi squared R cubed divided by GT squared. Now, I wasn't given the radius of the problem. I was actually given the diameter. So I'll need to calculate my radius to recall that the radius are gonna be equal to half the diameter. So I'll have D divided by two. And so lugging the values for the D have R is equal to 35 kilometers divided by two. However, I'm going to need to have this in si units of meters and that's because meters shows up in our gravitational constant units. So I'm gonna go ahead and convert that now into meters. So I'm gonna multiply this fraction by 1000 m divided by one kilometer and that will convert my kilometers into meters. And once I've calculated that my radius R in meters is going to be equal to 17,500 m. If I look at the period, which is also going to be plugged into our equation for the mass, it has units of days and I'll need that also in SI units. Let's go ahead and convert that into seconds. We'll have t equal to, we'll have seven days and I'm going to need to convert days into hours. I'm gonna multiply this by the conversion factor of 24 hours divided by one day. And now I have hours, I need to convert hours into minutes. I multiply this conversion factor again by 60 minutes divided by one hour. And I'm going to have to convert minutes to second. I'm gonna multiply by the conversion factor of 60 seconds divided by one minute. And this will give you my period in uh uh seconds. So when I plug those into my calculator, I get T is equal to 604,800 seconds. So now I have my radius and period in seconds, I can plug those into my formula or the mass. So I have M equal to or high squared for the R, I'll have the 17,500 meters and that quantity is cubed and this gets divided by RG will have the 6.67 multiplied by 10 to the negative 11 Newton meter squared per kilogram squared. And that gets multiplied by the period squared. So we'll have multiplying the um 600 4800 seconds in that period is squared. So everything on the right hand side is just a number. So I can plug those into my calculator. And I get a mass value just keeping two significant figures of M equals 8.7 multiplied by 10 to the 12 kg. That corresponds to answer B. So while we used um new uh Kepler's third law directly, we did have to remember that we needed everything in SI units. And so we had to first find the radius by dividing the diameter by two and then converting that kilometers into meters. We also had to convert our period from days into seconds. And that's because our gravitational constant has units of meters and seconds in it. And the second is hidden in the newtons, which is a kilogram meter per second squared. So once we had everything in si units, we could plug it into the formula for the mass that we found and calculate the mass. So I hope that this has been useful and I'll see you in the next video.
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