# Moment of Inertia of a Sphere

Written by Jerry Ratzlaff on . Posted in Classical Mechanics

This calculation is for the moment of inertia of a sphere.  There are three separate calculations:  a solid sphere, a hollow sphere and a hollow core sphere.  The difference between a hollow sphere and a hollow core sphere is a hollow sphere has a thin shell, or a thickness that is neglible.  Because a sphere is the same dimensions in every dimension, the moment of inertia is the same about every axis.

### Solid Sphere formula

$$\large{ I = \frac {2}{5} \; m \; r^2 }$$

### Hollow Sphere formula

$$\large{ I = \frac {2}{3} \; m \; r^2 }$$

### Hollow Core Sphere formula

$$\large{ I = \frac {2}{5} \; m \; \left( \frac { r_2^5 \;-\; r_1{^5} } { r_2{^3} \;-\; r_1{^3} } \right) }$$

Where:

$$\large{ I }$$ = moment of inertia

$$\large{ m }$$ = mass

$$\large{ r }$$ = radius

$$\large{ r_1 }$$ = radius

$$\large{ r_2 }$$ = radius