product of inertia

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A quantity similar to a moment of inertia, but relating to a rigid body and a pair of given perpendicular axes. The products of inertia occur in the inertia matrix which arises in connection with the angular momentum and the kinetic energy of the rigid body. With Cartesian coordinates, let Iyz, Izx and Ixy denote the products of inertia of the rigid body associated with the y- and z-axes, the z- and x-axes, and the x- and y-axes. Suppose that a rigid body consists of a system of n particles P1, P2,…, Pn, where Pi has mass mi and position vector ri, with ri=xii+yij+zik. ThenFor solid bodies, the products of inertia are defined by corresponding integrals.

When the coordinate planes are planes of symmetry of the rigid body, the products of inertia are zero. Furthermore, it can be shown that, for any point fixed in a rigid body, such as the centre of mass, there is a set of three perpendicular axes, with origin at the point, such that the corresponding products of inertia are all zero. These are called the principal axes, and the corresponding moments of inertia are the principal moments of inertia. The use of the principal axes produces a considerable simplification in the expressions for the angular momentum and the kinetic energy of the rigid body.

Subjects: Mathematics.

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