Hello! I'm trying to derive the formula for the moment of inertia of a solid sphere, and I keep running into a strange solution.(adsbygoogle = window.adsbygoogle || []).push({});

I set up the infinitesimally mass of an infinitesimally thin "shell" of the sphere:

dm = 4[itex]\rho[/itex][itex]\pi[/itex]r^{2}dr

And then solved for the moment of inertia:

I = [itex]\int[/itex]r^{2}dm

= [itex]\int[/itex]r^{2}(4[itex]\rho[/itex][itex]\pi[/itex]r^{2}dr)

= 4[itex]\rho[/itex][itex]\pi[/itex][itex]\int[/itex]r^{4}dr

= (4/5)[itex]\rho[/itex][itex]\pi[/itex]r^{5}

And solving for [itex]\rho[/itex] we get the following:

[itex]\rho[/itex] = M/((4/3)[itex]\pi[/itex]r^{3}).

Substituting that into the previously solved equation for I, I get the following:

I = (3/5)Mr^{3}.

What am I doing wrong? I know the formula involves a coefficient of 2/5, not 3/5, but I can't find my problem.

Thank you in advance!

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# Moment of inertia of a solid sphere derivation.

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