moment of inertia - Scalar moment of inertia

In mechanics, the notion that, for a rotating object, the turning force required to make the object turn faster depends on how the object's mass is distributed about the axis of rotation; symbol I, units kg.m². For example, the force needed to spin a disc more quickly about its centre will be greater if the disc's mass is concentrated towards its rim. For a uniform disc of radius r and mass m spinning horizontally about its centre, I = mr²/2.

Moment of inertia, also called mass moment of inertia and, sometimes, the angular mass, (SI units kg m², English units lbs ft2) quantifies the rotational inertia of a rigid body, i.e. Quantitatively, the smaller wheel has a smaller moment of inertia, whereas the larger wheel has a larger moment of inertia.

The moment of inertia has two forms, a scalar form I (used when the axis of rotation is known) and a more general tensor form that does not require knowing the axis of rotation.

The moment of inertia can also be called the mass moment of inertia (especially by mechanical engineers) to avoid confusion with the second moment of area, which is sometimes called the moment of inertia (especially by structural engineers) and denoted by the same symbol I.

Scalar moment of inertia

Definition

The (scalar) moment of inertia of a point mass rotating about a known axis is defined by

where

m is its mass, and r is its perpendicular distance from the axis of rotation.

The moment of inertia is additive so, for a rigid body consisting of N point masses mi with distances ri to the rotation axis, the total moment of inertia equals the sum of the point-mass moments of inertia

Generalizing to a solid body described by a continuous mass-density function , the moment of inertia for rotating about a known axis can be calculated by integrating the moments of the point masses relative to the rotation axis

where

V is the volume region of the object, r is the distance from the axis of rotation, m is mass, v is volume, ρ is the pointwise density function of the object, and x, y, z are the Cartesian coordinates.

The moment of inertia for non-point objects can also be found or approximated as the product of three terms:

where

k is the inertial constant, M is the mass, and R is the radius of the object from the center of mass.