Table of contents

**Inertia** is the resistance, of any physical object, to any change in its velocity. This includes changes to the object’s speed or direction of motion. In other words: inertia is the reluctance to motion (or to variations of it), it provides us with information on how much the system is willing to implement the “cause” in order to produce a certain “effect”. Inertial forces are fictitious (or apparent) forces involved in the motion of a body in a non-inertial reference system due to its accelerated motion.

## The principle of inertia

The first principle of dynamics (also called the principle of inertia and of inertial systems or Newton’s first law), states that in an inertial system a material point, isolated (i.e. not subjected to external actions), remains in its state of rest or of uniform rectilinear motion, indefinitely until a force or constraint capable of varying its state acts on it.

*No force is required to maintain motion with constant velocity in a straight line, and absolute motion does not cause any observable physical effects*.

The formulation of the first part of the principle of inertia dates back to Aristotle, while the second part was enunciated in a particular case by G. Galilei and in general by Descartes. The law of inertia is closely linked to an absolute reference; for this reason the absolute reference is also called inertial reference, that is the reference for which the law of inertia applies. The inertial system is therefore defined as the system in which the “isolated” material points remain, if they are in a state of rest, in the pre-existing state of rest. In the case of an observer in solidarity with a non-inertial reference system, forces of inertia are those forces which do not represent actual physical actions, but which are introduced to restore validity to the principle of inertia.

## Moment of inertia

The moment of inertia measures the inertia of a body as its angular velocity varies. The moment of inertia has two forms, a scalar form, used when the rotation axis is known exactly, and a more general tensor form, which does not require knowledge of the rotation axis. The scalar moment of inertia is often simply called the moment of inertia.

Analytically, the moment of inertia *I* of a material point of mass *m* with respect to an axis of rotation is given by the product of the mass of the point by the square of the distance *r* between the point and the axis is given by: *I* = *mr*^{2} and turns out to be constant. By extension, the moment of inertia of a system consisting of N material points with respect to an axis is the sum of the moments of inertia of the individual points with respect to the axis: