Three satellites orbit a planet of radius R, as shown in FIGURE EX13.24. Satellites S₁ and S₃ have mass m. Satellite S₂ has mass 2m. Satellite S₁ orbits in 250 minutes and the force on S₁ is 10,000 N. (b) What are the forces of S₂ and S₃?

Question

Diagram showing three satellites with different masses orbiting a planet at varying distances.

Accepted Answer

Hey, everyone in this problem, we have three asteroids A one, A two and a three orbiting around a star that has a radius R asteroid. A one has a mass of M A two has a mass of three M and a three has a mass of four M. We're asked to calculate the force experienced by A two and a three if the force on a one is 15,000 nodes. And we're told that a one completes an orbit in 290 minutes. We're given a diagram of what we have and we have that asteroid A one is at a distance of two R from the center asteroid A two is at a distance three R and asteroid A three is at a distance four R. OK. And these are circular orbits. We have four answer choices. A through D, each of them just containing a different combination of the force for a two and for a three. So we're looking for a force. OK. We have this circular orbit. So let's recall Newton's law of gravitation which tells us that the force f gonna be equal to G gravitational constant multiplied by big M multiplied by little M divided by R squared. OK. So in general, that's the equation for force we wanna use. So let's look at this for each of our asteroids. So let's start with A two and we're gonna draw a two in blue and we're gonna add it on our diagram in blue. And then we're gonna write the equation in blue just so it's easier to keep track. So we're gonna write the force and we're gonna write it as F subscript A two. So the force on asteroid A two, this is gonna be equal to that gravitational constant capital G multiplied by capital M which is gonna be the mass. Oh The star multiplied by the mass of the asteroid itself. OK. We're told that A two has a mass of three M. So multiplied by three M all divided by the radius squared. OK. We said that the distance from the center out to A two is three R. So the radius is three R. So we have three R all squared. All right. And we can simplify in the numerator. We have G multiplied by MS multiplied by three M in the denominator, applying that square, we get nine R squared. Uh We can simplify the coefficient so we can divide numerator and denominator by three. And this leaves us with G MS um divided by three R squared. And we're gonna call this equation one. And we're gonna come back to that in a minute and we don't really know what R is. We don't really know what M is. We don't know what MS is. So we don't have these values to actually calculate that force. OK. So that's why I said we're gonna come back to it. We've labeled it. Let's keep going with a three and see what else we can figure out. So for asteroid A three, we're gonna draw this guy in green, we're gonna write the equation in green and it's gonna be very similar. So fa three, there's gonna be that gravitational constant multiplied by the mass of the star. MS multiplied by the mass of the asteroid or M all divided by the radius squared. And for asteroid A three, it's at a distance of four R from the center. So that radius is gonna be four R and they get all simplifying and we're gonna skip this middle step this time because we know what's going on here. We have four squared in the denominator, we have four in the numerator. And so one of those factors of four is going to divide out and we get G MS multiplied by M divided by four R squared. And that's gonna be equation two. So we have these two equations for the forces that we try, we're trying to find. But we don't know all of these values MS M and R, let's use what we know about a one to try to figure out these values. OK. So let's move over to 81. And for a one, hm, we know that the force A one, it's gonna be equal to G multiplied by the mass of the star multiplied by the mass of the asteroid, which is just M divided by the radius squared. OK. Which is two R all squared. Simplifying we get G MS multiplied by M all divided by four are squared. And we know that this force on asteroid one is equal to 15,000 newtons. What you'll notice here is that this term, this force of A one G MS M divided by four R squared. This is actually equal to what we had with the force on asteroid three. OK. Equation two. That's exactly what we had in equation two. So what that tells us is that the force F on asteroid three gonna be equal to the force on asteroid one, which we know is 15 moons. OK. So we figured out one of our forces now we just have asteroid two left. OK. All right. So we're gonna go back to this equation for the force on a one because we have this value that it's equal to. OK. So we're gonna go back to that equation and see if we can rearrange in a way that will allow us to solve for fa two. So what we can do is rearrange and write this as G multiplied by MS multiplied by M divided by R squared is equal to four multiplied by 15,000 nodes. And so we've taken that form in the denominator, we've multiplied both sides by four to remove it, move it to the other side. And now we have an expression for G MS M divided by R squared, all of those values that we don't know. Right. So we're gonna call this equation three. Remember we're trying to solve for equation one. Now we know G MS M divided by R squared. So we're gonna substitute equation three into equation one, substitute equation three into equation one. Mhm. All right. So now we have the FA two and let me just rewrite the original equation we had. So we can see it on the screen. G multiplied by MS multiplied by M divided by three R squared is going to be equal to G MS M divided by R squared is four multiplied by 15,000 newtons. And all that's left to do is divide by three. And that's the only term that's not accounted for. We do that and we simplify, we get that fa two is equal to 20,000 nodes. Mhm All right. So we didn't need to know M exactly. And we didn't need to know the mass of the star exactly. We didn't need to know the radius. Exactly. All we needed was this relationship between all three asteroids. OK? And that force for a one. So we found that the force for asteroid two is 20,000 newtons. The force on a three is 15,000 newtons and this corresponds with answer choice B thanks everyone for watching. I hope this video helped see you in the next one.

Three satellites orbit a planet of radius R, as shown in FIGURE EX13.24. Satellites S₁ and S₃ have mass m. Satellite S₂ has mass 2m. Satellite S₁ orbits in 250 minutes and the force on S₁ is 10,000 N. (b) What are the forces of S₂ and S₃?

Verified Solution

Key Concepts

Gravitational Force

Centripetal Force

Orbital Period