The Moment of Inertia Tensor

Abstract

We know that Euler’s first law (5.43) relates the total external applied force on a rigid body to the motion of its center of mass, and in the next chapter we shall demonstrate that Euler’s second law (5.44) relates the total external applied torque to the body’s rotational motion through its moment of momentum vector. The latter involves introduction of the moment of inertia tensor studied here; and, of course, the first law involves the location of the center of mass of the body. We begin, therefore, with the concept of the center of mass of a complex structured body and illustrate its application to a materially nonhomogeneous body having a complex shape and cavities. Then the inertia tensor is introduced, and its components for some special homogeneous bodies are determined. Afterwards, some important physical properties of the moment of inertia tensor, properties actually characteristic of all kinds of symmetric tensors, are derived. Consequently, as an additional benefit, study of the inertia tensor provides tools useful, for example, in the study of the mechanics of deformable solid and fluid materials in which stress, strain, and deformation rate tensors play a major role.