Inertia Tensor

Inertia Tensor

In classical mechanics, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass (SI units kg·m2, US units lb_m ft2), is a property of a distribution of mass in space that measures its resistance to rotational acceleration about an axis. This scalar moment of inertia becomes an element in the inertia matrix when a distribution of mass is measured around three axes in space. This inertia matrix appears in the calculation of the angular momentum, kinetic energy and resultant torque in the dynamics of a rigid body.

Newton's first law, which describes the inertia of a body in linear motion, can be extended to the inertia of a body rotating about an axis using the moment of inertia. That is, an object that is rotating at constant angular velocity will remain rotating unless acted upon by an external torque. In this way, the moment of inertia plays the same role in rotational dynamics as mass does in linear dynamics, describing the relationship between angular momentum and angular velocity, torque and angular acceleration. The symbols I and sometimes J are usually used to refer to the moment of inertia or polar moment of inertia.

The moment of the inertia force on a single particle around an axis multiplies the mass of the particle by the square of its distance to the axis, and forms a parameter called the moment of inertia. The moments of inertia of individual particles in a body sum to define the moment of inertia of the body rotating about an axis. For rigid bodies moving in a plane, such as a compound pendulum, the moment of inertia is a scalar, but for movement in three dimensions, such as a spinning top, the moment of inertia becomes a matrix, also called a tensor.

Many systems use a mass with a large moment of inertia to maintain a rotational velocity and resist small variations in applied torque. For example, the long pole held by a tight-rope walker maintains a zero angular velocity resisting the small torque applied by the walker to maintain balance. Another example is the rotating mass of a flywheel which maintains a constant angular velocity resisting the torque variations in a machine.