Moment of Inertia - Moment of Inertia Tensor

Moment of Inertia Tensor

The moment of inertia for a rigid body moving in space is a tensor formed from the scalars obtained from the moments of inertia and products of inertia about the three coordinate axes. The moment of inertia tensor is constructed from the nine component tensors,

where e_i, i=1,2,3 are the three orthogonal unit vectors defining the inertial frame in which the body moves. Using this basis the inertia tensor is given by

This tensor is of degree two because the component tensors are each constructed from two basis vectors. In this form the inertia tensor is also called the inertia binor.

For a rigid system of particles P_k, k=1,...,N each of mass m_k with position coordinates r_k=(x_k, y_k, z_k), the inertia tensor is given by

where E is the identity tensor

The moment of inertia tensor for a continuous body is given by

where r defines the coordinates of a point in the body and ρ(r) is the mass density at that point. The integral is taken over the volume V of the body. The moment of inertia tensor is symmetric because I_ij= I_ji.

The inertia tensor defines the moment of inertia about an arbitrary axis defined by the unit vector n as the product,

where the dot product is taken with the corresponding elements in the component tensors. A product of inertia term such as I₁₂ is obtained by the computation

and can be interpreted as the moment of inertia around the x-axis when the object rotates around the y-axis.

The components of tensors of degree two can be assembled into a matrix. For the inertia tensor this matrix is given by,

It is common in rigid body mechanics to use notation that explicitly identifies the x, y, and z axes, such as I_xx and I_xy, for the components of the inertia tensor.