# Saturn vs Saturn's Satellites

## Homework Statement

There is a ring outside of Saturn. In order to distinguish if the ring is actually a part of Saturn or is instead part
of the satellites of Saturn, we need to know the relation between the velocity v of each layer in the ring and the
distance R of the layer to the center of Saturn. Which of the following statements is correct?
(A) If v ∝ R, then the layer is part of Saturn.
(B) If v^2 ∝ R, then the layer is part of the satellites of Saturn.
(C) If v ∝ 1/R, then the layer is part of Saturn.
(D) If v^2 ∝ 1/R, then the layer is part of Saturn.
(E) If v ∝ R^2, then the layer is part of the satellites of Saturn.

## The Attempt at a Solution

I used Kepler's 3rd law to get that R^3=period^2 which equals 4pi^2R^2/v^2. Simplifying I get R is proportional to 1/v^2 assuming that the layer will be a satellite of Saturn. However this is not an answer choice, what am I doing wrong?

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Bystander
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get R is proportional to 1/v^2
Can you rewrite this proportion?

V^2 is proportional to R.

Bystander
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"R ∝ 1/V2" is what you've got. You handle the proportion with the same set of "grammatical rules" you use to handle any other algebraic statement, equalities, inequalities, whatever --- whatever you do to one side of the proportion, you do to the other.

Wait typo (had it correct in my written work) , v^2 is proportional to 1/R.

Bystander
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Which is one of the answers.

OK, but wait how do I know that that means that the layer is part of Saturn ( as opposed to a satellite)?

Bystander
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R is proportional to 1/v^2 assuming that the layer will be a satellite
This is good, or the inverse, and holds for satellites. There are no statements that include this relation and satellites. Therefore, the layer is not a satellite. What can you do with the other relations?

I don't understand.

Bystander
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You've correctly shown that the relationship between V and R for a satellite has to be R ∝ 1/V2, or 1/R ∝ V2. Neither of those relationships are included in the list of possible answers that also include satellites. Correct?
So, you're looking at a part of Saturn. Now, which of the statements that call the layer a part of Saturn are consistent with the motions you see in satellite photos of earth of parts of earth?

That is true, however what if the question wasn't multiple choice?

Bystander
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wasn't multiple choice?
Then, the question has to specifically ask you for a specific answer which you can derive from information given in the question, or that the question's author(s) are assuming you to know or be able to access. I'm guessing you know Kepler's laws and other such from the facility you demonstrated using the third.

Well the question I have is know that I know the relation in terms of both variables how do I know which one corresponds to either Saturn itself or a satellite assuming I couldn't figure it out using the answer choices.

Bystander
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From the way the question is written, I'd say you can regard Saturn as a solid body --- V is going to be directly proportional to R, just like rotating a lightbulb and watching the filament rotate with it. Satellite motion you've got from Kepler. If they play multiple choice games with you, you have to watch for what statements are consistent internally with themselves, and with the question/answer you're asked to give.

So I take it you don't like multiple choice questions huh. :)

Bystander