Rank the following hypothetical planets -- all of which have the same total mass and same radius -- from lowest moment of inertia to highest moment of inertia:
1) A uniform sphere of mixed up rock and iron
2) A hollow sphere
3) A rocky planet with an iron core
4) A rocky planet with an iron crust
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To rank the hypothetical planets based on moment of inertia, we need to understand what moment of inertia is and how it varies for different objects.
Moment of inertia is a measure of how an object's mass is distributed around its axis of rotation. It depends on both the mass and the distance of the mass from the axis of rotation. In simple terms, it describes how difficult it is to rotate an object.
Now, let's analyze each hypothetical planet and determine their relative moment of inertia:
1) A uniform sphere of mixed-up rock and iron:
In this case, the mass is uniformly distributed throughout the sphere. Since all points on the sphere are equidistant from the axis of rotation, the moment of inertia will be the highest. This is because the mass is distributed farther from the axis, making it harder to rotate. Therefore, this planet will have the highest moment of inertia.
2) A hollow sphere:
For a hollow sphere, all the mass is concentrated at the surface, and there is no mass inside. Since the mass is concentrated farther from the axis of rotation, it contributes more to the moment of inertia. Although the total mass is the same as the other planets, the lack of mass near the axis makes it easier to rotate. Hence, the moment of inertia will be lower than that of a uniform sphere.
3) A rocky planet with an iron core:
In this case, the mass is concentrated in two regions, the rocky outer layer and the iron core. The iron core will contribute more to the moment of inertia because it is denser and potentially nearer to the axis of rotation. The rocky outer layer will have less impact on the moment of inertia. Therefore, the moment of inertia will be lower compared to a uniform sphere but higher than a hollow sphere.
4) A rocky planet with an iron crust:
Here, the mass is concentrated primarily in the rocky crust and not distributed equally throughout the planet. The moment of inertia will be lower than the previous three because the mass is concentrated even closer to the axis of rotation. The iron crust may still have some impact on the moment of inertia, but it will be lower in comparison. Therefore, this planet will have the lowest moment of inertia among the options.
Hence, the ranking of the hypothetical planets from lowest moment of inertia to highest moment of inertia would be:
4) A rocky planet with an iron crust
3) A hollow sphere
2) A rocky planet with an iron core
1) A uniform sphere of mixed-up rock and iron