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An interplanetary spaceship passes through the point in space where the gravitational forces from the sun and the earth on the ship exactly cancel. (a) How far from the center of the earth is it? Use the data in Appendix E. (b) Once it reached the point found in part (a), could the spaceship turn off its enginesand just hover there indefinitely? Explain.

(a) 2.59 $\times$ 10$^8$ m (b) no

Physics 101 Mechanics

Chapter 6

Circular Motion and Gravitatio

Physics Basics

Motion Along a Straight Line

Motion in 2d or 3d

Newton's Laws of Motion

Applying Newton's Laws

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University of Michigan - Ann Arbor

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okay. And problem 21. We are considering a satellite. That is No, that's not Smith Stud wits located right between this son. Stop drawing Red here and the earth kind of blue and black will be the satellite. Um, so it is at a point in space, right where the pool from the sun is equal to the poor from the earth so we can represent those forces, as it's called them F s for force of the sun and f d for force from the earth. And those forces can select so well, that's just write them down. First, the force from the sun is G M son. I'm satellite over now. We'll meet, describes distances. So I will call this distance R s because it's the radius of the orbit around the sun. And this is our e cause It's the distance between the satellite and the earth. So for the force exerted by the sun, we will have to use a distance from the sun, Um and then for the force exerted by the Earth, they're similar. But we're gonna use Earth's Constance instead. And so and the problem since the satellite is at this point with those forces Cancel out. Used forces must be equal to each other. Okay, I'm just gonna write it down here, right? Massive. The satellite's doesn't matter as well as this g constant. And now, um, what we want to know is the distance on DDE off the satellite from this, the center of the earth, which is already so what we want to know is already but one thing we don't know here is the distance that the satellite is from, Son. We know the distance, however, off the earth from the sun, that's the orbiting distance between Earth and sun. And we'll call that d on looking at this diagram here, which he realizes that R s plus r E is equal to d. Excellent. So that's right, this one down. Um, it's just rewrite this one that this sweet Artie equals the minus rs. And now what we're gonna do is use those two equations together in order to and find an answer for Artie. All right, so using this equation here, um, I will isolate for rs since that rs equals r E square root of m s over me and then this rs we will use it here. So d minus our eat over and s a square root of a mess over me. And now we have a new art Eat. It appeared. So that's really just a little bit, Um, which means that my first epi's here was, um, actually unnecessary. I'm gonna write down the equation the same equation again without changing anything to it are e equals the minus rs square root of in a silver ari over any Amina, um, misery. This is already so it's rearranged this such that we have our e um one plus I screwed up. M s over. M e equals d. So I took this term Take this term and I brought it from the left side. And then I isolate Ari. And then ar e equals the over one plus the square root of M s over Emmy. Right. So now we have everything we need on the right side so we could get a value for our feet. Distance is, as noted in diagram here, the distance between the sun and the earth, and that would be 1.5 times 10 to 11 meters. And then we have the mass of the Sun, which is £1.99 10 to 30 killed grounds and the Mass of the Earth, which is 5.97 times 10 to 24 programs. All right, but those numbers in your calculator and you will find that the distance of satellite from the Earth is 2.59 times 28 meters. And that is the answer is it's keeping in the value as well room. Now there's a B Do this question and question be we are, um, asked. So, um, you read that again. If the spaceship was turning off, its engine would just stay there basically, So it's going to hold them again. There's no math in here. Just have Thio think about it. So right now we take a picture and we in time froze and we studied this situation or the poor from the two planets was the same. Now, if this patient was to stop here, would this state the same forever? And if we were to resume time unease e argument for no is the fact that different objects of different masses will orbit. Um, we'll have different orbits or different speeds of orbit So this satellite orbiting the sun, we'll do it at a certain speed. If you want, our Earth will do it at a certain speed V two as well. And those will be completely different and that this is just, um, fine by the mass of the object. And so, uh, since they will do this different speeds, there is no way unless you're very lucky with the masses. But I honestly doubt bed with that would ever happen. Um, yeah, this the situation would end up, uh, that's working because But, for example, the earth could be faster. Andi, that would mean that at a certain time past this point, well, the Earth would be here in the cell. I would be here, for example. And then the forces would not cancel out anymore. How would that make sense? That's just one easy argument for it. You could find other arguments. Uh, I'm sure

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