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Taking a uniform solid sphere of radius R and mass M, with the centre of mass at the origin, I divided it into infinitesimal disks of thickness dx, and radius y. I need to find the moment of inertia about the x-axis, so taking an arbitrary disk at some horizontal distance x from the centre of mass, I obtain ;

y^2 + x^2 = R^2, (fairly obviously),

density, rho = dm/dV,

dV = (pi)(y^2)dx => dm = (rho)(pi)(y^2)dx

So using the standard definition for moment of inertia :

I = integral of (y^2)dm

I = integral of (y^2)(rho)(pi)(y^2)dx -with x limits R and -R

= (rho)(pi) integral of ((R^2 - x^2)^2) dx

which simplifies down to I = (16/15)(pi)(rho)R^5,

and using M = (4/3)(pi)R^3, I obtain I = (4/5)MR^2.

Of course my textbook is telling me it should be (2/5)MR^2, and as far as my understanding goes, this is a consequence of each infinitesimal disk having a moment of inertia of (1/2)dm(r^2).

Logically then, using dI = (1/2)dm(r^2), such that :

I = integral of (pi)(rho)((R^2 - x^2)^2)dx with x limits R and 0, the answer comes out correctly as (2/5)MR^2.

Unfortunately, I am not a particularly sophisticated mathematician and I am worried that my own method, using I = integral of (y^2)dm as described, is giving me an answer which is out by a factor of 2.

I fear I may have made a trivial mistake, but if not, I'd greatly appreciate some insight as to the cause of the discrepancy.

Many thanks!

Trev