# Moment of Inertia of a Sphere

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.

### Moment of Inertia of a Sphere Formula, Solid Sphere

$$I = \frac {2}{5} \; m \; r^2$$
Symbol English Metric
$$\large{ I }$$ = moment of inertia $$lbm\;/\;ft^2-sec$$ $$kg\;/\;m^2$$
$$\large{ m }$$ = mass $$lbm$$ $$kg$$
$$\large{ r }$$ = radius $$in$$ $$mm$$

### Moment of Inertia of a Sphere Formula, Hollow Sphere

$$I = \frac {2}{3} \; m \; r^2$$
Symbol English Metric
$$\large{ I }$$ = moment of inertia $$lbm\;/\;ft^2-sec$$ $$kg\;/\;m^2$$
$$\large{ m }$$ = mass $$lbm$$ $$kg$$
$$\large{ r }$$ = radius $$in$$ $$mm$$

### Moment of Inertia of a Sphere Formula, Hollow Core Sphere

$$I = \frac {2}{5} \; m \; \left( \; r_2^5 \;-\; r_1{^5} \;/\; r_2{^3} \;-\; r_1{^3}\; \right)$$
Symbol English Metric
$$\large{ I }$$ = moment of inertia $$lbm\;/\;ft^2-sec$$ $$kg\;/\;m^2$$
$$\large{ m }$$ = mass $$lbm$$ $$kg$$
$$\large{ r }$$ = radius $$in$$ $$mm$$
$$\large{ r_1 }$$ = radius $$in$$ $$mm$$
$$\large{ r_2 }$$ = radius $$in$$ $$mm$$

Tags: Moment of Inertia