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**Determining Moment of Inertia of a Sphere**

I'm having some troubles determining the moment of inertia of a sphere about it's central axis. My original question was to calculate it for a cylinder, which I've done, but I'd like to know how to find it for a sphere.

Here is the problem solved for a cylinder:

**Problem:**

A uniform solid cylinder has radius

*R*, mass

*M*, and length

*L*. Calculate its moment of inertia about its central axis (the z axis).

**Solution:**

I divided the cylinder into infinitesimally small layers because I knew that [tex]dV = (2\pi*r*dr)*L[/tex]. From here I calculated the integral [tex]I = \int \rho*r^2 dV = \int_{0}^R \rho*r^2*(2\pi*r*L)dr = 2*\pi*\rho*L*R^4 [/tex]

I substituted [tex]\frac {M}{\pi*R^2*L}[/tex] (or [tex]\frac{M}{V}[/tex]) for [tex]\rho[/tex] into the equation to get [tex] I = \frac{1}{2}*\pi*(\frac {M}{\pi*R^2*L})*L*R^4 = \frac{1}{2}*M*R^2[/tex]

I understand this, but when I tried to get it as a sphere I ended up getting the wrong answer. Could anyone please show me how to start the problem with a sphere?

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