Consider a particle with mass m_k moving with velocity v_k. The linear momentum, P_k, of this particle is equal to m_kv_k. Consider now a particular point O, separated spatially from the mass m_k by the position vector r_Ok which extends from O to the mass m_k. The moment of momentum, or angular momentum, of this particle about O is

Now, for a system of particles instead of a single particle, the total moment of momentum about the point O is

where n is the total number of particles. The concept of moment of momentum may now be readily extended to a rigid body by considering that the rigid body is a continuous system of particles with infinitesimal mass dm. For a rigid body, the moment of momentum about the point O is

where both r and v may vary over the rigid body and r represents the vector from point O to the infinitesimal mass dm. For a rigid body, the velocity of any point k may be related to the velocity of the center of mass, v_C, the angular velocity of the rigid body, ω, and the position vector, r_Ck, from the center of mass to the point k, as follows:

Substituting Eq. 4↑ into Eq. 3↑,

The first term in Eq. 5↑ is the moment of momentum of the rigid body about point O due to the velocity of its center of mass, or r_OC × mv_C. The second term may be written in terms of the moment of inertia tensor, which is a matrix consisting of the principal moments of inertia along the diagonal and products of inertia off the diagonal. After these simplifications, Eq. 5↑ becomes

where I_O is the moment of inertia tensor of the rigid body about point O. A typical simplification that is often made is to consider the moment of momentum about the center of mass, which implies that O = C and r_OC = 0. In that case, the moment of inertia of a rigid body about its center of mass reduces to

Note the similarity of the form of this equation to the linear momentum of a rigid body, P = mv. In rotational motion, the moment of inertia is analogous to mass in linear motion, and the angular velocity is analogous to linear velocity.