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If the radii of two planets are π
sub one equals 1,552 kilometers and π
sub two equals 6,208 kilometers and the ratio between their gravitational accelerations π sub one to π sub two is equal to one to two, find the ratio between their masses π sub one to π sub two.
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Well, to help us solve this problem, what we have is a formula for our acceleration due to gravity.
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And that is that π is equal to capital πΊπ over π squared, where π is the acceleration due to gravity, capital πΊ is the universal gravitational constant, π is the mass, and π is the radius.
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And this is actually derived by combining Newtonβs law of universal gravitation with Newtonβs second law.
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Okay, great, so we have this.
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But howβs it gonna help us?
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Well, what we can do is actually work out the acceleration due to gravity of both of our planets.
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So therefore, we can say that the gravitational acceleration of the first planet is gonna be equal to capital πΊ multiplied by π sub one over π sub one squared, which is gonna give us π sub one is equal to capital πΊ multiplied by π sub one over 1,552,000 squared.
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And the reason we have this is because if we look at the radius of the first planet, itβs 1,552 kilometers.
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However, whenever weβre looking to work acceleration due to gravity, the units we want our radius to be in are in fact meters.
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So what weβve done is we multiplied this by 1,000 to give us 1,552,000 meters.
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Now, if we repeat the same for the second planet, weβll have π sub two is equal to capital πΊ multiplied by π sub two over π sub two all squared.
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So then what weβre gonna get is π sub two is equal to capital πΊ multiplied by π sub two over 6,208,000 squared.
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And once again, what weβve done is converted our kilometers to meters, so weβve done 6,208 kilometers multiplied by 1,000, which gives us 6,208,000 meters.
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Well, next, what weβre going to do is take a look at another bit of information weβve been given in the question.
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And that is that the ratio between the gravitational accelerations is one to two.
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So therefore, what this means is the acceleration due to gravity on the second planet, so π sub two, is twice that of the acceleration due to gravity on the first planet, π sub one.
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So therefore, what we can do is form an equation.
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And we can do that by multiplying our expression for π sub one by two.
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Because what we get is that two multiplied by capital πΊπ sub one over 1,552,000 squared is gonna be equal to, then our expression for π sub two, which is capital πΊπ sub two over 6,208,000 squared.
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Well, what we can do is divide through by capital πΊ, our universal gravitational constant, βcause this is unchanged on each side of our equation.
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And then we can change the subject to make π sub two the subject.
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So we have π sub two in terms of π sub one.
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And to do that, what weβve done is multiplied through by 6,208,000 squared.
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So what we have is two multiplied by 6,208,000 squared over 1,552,000 squared π sub one is equal to π sub two.
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So calculating this, we get 32π sub one is equal to π sub two.
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So π sub two is 32 times bigger than π sub one.
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So therefore, what we can say is that the ratio between the masses of our two planets, so π sub one to π sub two, is going to be one to 32.