Moment of Inertia - Overview


Moment of inertia appears in Newton's second law for the rotation of a rigid body, which states that the torque necessary to accelerate rotation is proportional to the moment of inertia of the body. Thus, the greater the moment of inertia the greater the torque needed for the same acceleration. Many systems use masses with 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.

The moment of inertia of an object is defined by the distribution of mass around an axis. It depends not only on the total mass of the object, but also on the square of the perpendicular distance from the axis to each element of mass. This means the moment of inertia increases rapidly as masses are distributed more distant from the axis. For example, consider two wheels that have the same mass, one that is the size of a bicycle wheel and one that is half that size. The larger wheel has four times the moment of inertia even though it is only twice the diameter.

Moment of inertia around a fixed axis is a scalar, however the rotation of a body in space can occur around the three coordinate axes. In this case, the moments of inertia associated with the three coordinate axes define a matrix of scalars called the inertia matrix, also known as the inertia tensor.

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