# 3.4.1 Moment of Inertia for Mass

The moment of inertia turns out to be an essential part for the calculations of rotating bodies. Furthermore, it turns out that the moment of inertia has much wider applicability. Moment of inertia of mass is defined as

Moment of Inertia

\[I_{rrm} = \int_{m} \rho r^{2} dm \tag{13}\]

If the density is constant then equation 13 can be transformed into \[I_{rrm} = \rho \int_{V} r^{2} dV \tag{14}\] The moment of inertia is independent of the coordinate system used for the calculation, but dependent on the location of axis of rotation relative to the body. Some people define the radius of gyration as an equivalent concepts for the center of mass concept and which means if all the mass were to locate in the one point/distance and to obtain the same of moment of inertia. \[r_{k} = \sqrt{\frac{I_{m}}{m}}\tag{15}\] The body has a different moment of inertia for every coordinate/axis and they are \[I_{xx} = \int_{V} r_{x}^{2}dm = \int_{V} \left(y^{2} + z^{2}\right)dm\] \[I_{yy} = \int_{V} r_{y}^{2}dm = \int_{V} \left(x^{2} + z^{2}\right)dm\tag{16}\] \[I_{zz} = \int_{V} r_{z}^{2}dm = \int_{V} \left(x^{2} + y^{2}\right)dm\]

### Contributors

Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.