"super earths" are known to have similar densities and compositions to earth but they have a mass 2-10x greater than earth. If one of these super earth's had a mass 5x the mass of earth what would be the super earth's radius ?
a) 1.7x earth's radius
b) 2.2x earth's radius
c) 5x earth's radius
d) 7.3x earth's radius
e) 10x earth's radius
ANY HELP IS HIGHLY APPRECIATED!
since volume grows as the cube of the radial scale factor,
the radius grows as the cube root.
That is, since 8 = 2^3, a planet with 8x earth mass will have ∛8=2x the radius.
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To determine the radius of a super earth with a mass 5x greater than Earth's mass, we can use the concept of density.
Density is defined as an object's mass divided by its volume. Since the composition and density of super earths are similar to Earth's, we can assume their densities are also similar.
If we have an object with a mass 5x greater than Earth, but a similar density, we can conclude that its volume will also be 5x greater than Earth's volume.
The formula for the volume of a sphere is V = (4/3) * π * r^3.
So, if the volume is 5x greater, we can set up the equation:
(4/3) * π * r^3 (for the super earth) = 5 * [(4/3) * π * (Earth's radius)^3]
Now we can simplify the equation:
(4/3) * π * r^3 = 5 * [(4/3) * π * (Earth's radius)^3]
Dividing both sides of the equation by (4/3) * π, we get:
r^3 = 5 * (Earth's radius)^3
Taking the cube root of both sides, we find:
r = (5 * (Earth's radius)^3)^(1/3)
Calculating this expression gives us the radius of the super earth relative to Earth's radius.
Based on the options provided, we can check which option matches our calculated result.
Let's solve for the expression:
r = (5 * (Earth's radius)^3)^(1/3)
Assuming Earth's radius is 1, we will have:
r = (5 * 1^3)^(1/3)
r = (5 * 1)^(1/3)
r = 5^(1/3)
r ≈ 1.71
From our calculations, we find that the radius of the super earth would be approximately 1.71 times Earth's radius.
Out of the given options, the closest one is option:
a) 1.7x Earth's radius,
so that would be the correct answer.
Therefore, the super earth's radius is approximately 1.7x Earth's radius.