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I can see why mathematically, but why conceptually?

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  • $\begingroup$ The hemisphere has the same mass as the full sphere, right? $\endgroup$
    – DJohnM
    Dec 18, 2019 at 21:28
  • $\begingroup$ Because the relative distribution of mass is the same about the center. $\endgroup$ Dec 20, 2019 at 13:39

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Because when you think about what fraction of the mass is at what distance from the axis (say, between $r$ and $r+dr$), they’re the same.

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Because moment of inertia depends on the distribution of mass and not on the location of Mass about the axis of rotation.

You can do a thought experiment. Suppose you take a solid sphere and you calculate its moment of inertia about a diametric Axis.

Now in second case, you cut the solid sphere into two halves and merge one half of solid sphere with the other half, increasing its density to twice of its original value, then the moment of inertia of newly formed hemisphere will be same as the moment of inertia of the total solid sphere you had earlier.

This is happening because the distribution of mass about the axis of rotation is not changing though the location of mass changed.

Moment of inertia depends on the distribution of mass with respect to the axis of rotation and not the exact location of the mass.

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