Beginning with Icm = Integral of r^2 dm from r1 to r2, find the moment of inertia of a solid sphere about any tangential axis.
Icm = Integral of r^2 dm
The Attempt at a Solution
I set up the infinitesimally mass of an infinitesimally thin "shell" of the sphere:
dm = 4ρπr2 dr
And then solved for the moment of inertia:
I = ∫r2dm
= ∫r2(4ρπr2 dr)
= 4ρπ∫r4 dr
And solving for ρ we get the following:
ρ = M/((4/3)πr3).
Substituting that into the previously solved equation for I, I get the following:
I = (3/5)Mr3.
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!