Mass moment of inertia is the resistance an object has to rotation. It is based off of the objects mass and how far that mass is from the center of rotation. A general equation that can be used to calculate the mass moment of inertia can be seen in equation 1.

**(Eq 1) ** $I=∫r^2dm$

r = radius

I = Mass Moment

Now what if the object its self is not rotating but instead is rotating around a specific point out in space? To calculate the mass moment of inertia for this case, the parallel axis theorem can be used. To use the parallel axis theorem the mass moment of inertia of the object as if it was rotating around its own centroid needs to be calculated, and the distance to the point in space that the rigid body will be rotating about must also be known. If those two variables are known equation 3 can be used to calculate the mass moment of inertia of a rigid body rotating around a point in space. You can also consider the mass as a lumped mass, which will remove make it possible to remove the first part of equation 2.

**(Eq 2) ** $I=I_a+md^2$

I_{a} = Object moment of inertia

d = distance from center of rotation

A mass moment of inertia is sometimes represented as the radius of gyration. What the radius of gyration does is it transforms a mass moment of inertia into a unit length used equation 3.

**(Eq 3) ** $k=\sqrt{\frac{I}{m}}$

## Mass Moment of Inertia for a Sphere

## Mass Moment of Inertia for a Hemisphere

## Mass Moment of Inertia for a Thin Disk

## Mass Moment of Inertia for a Thin Ring

## Mass Moment of Inertia for a Cylinder

## Mass Moment of Inertia for a Cone

## Mass Moment of Inertia for a Thin Plate

## Mass Moment of Inertia for a Slender Rod