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MOMENT OF MOMENTUM

Consider a particle with mass mk moving with velocity vk. The linear momentum, Pk, of this particle is equal to mkvk. Consider now a particular point O, separated spatially from the mass mk by the position vector rOk which extends from O to the mass mk. The moment of momentum, or angular momentum, of this particle about O is
(1) HOk = rOk × mkvk
Now, for a system of particles instead of a single particle, the total moment of momentum about the point O is
(2) HO = nk = 1 rOk × mkvk
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
(3) HO = r × vdm
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, vC, the angular velocity of the rigid body, ω, and the position vector, rCk, from the center of mass to the point k, as follows:
(4) v = vC + ω × rCk
Substituting Eq. 4↑ into Eq. 3↑,
(5) HO =  r × (vC + ω × rCk)dm  =  rdm × vC + r × ω × rCkdm
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 rOC × mvC. 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
(6) HO = rOC × mvC + IOω
where IO 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 rOC = 0. In that case, the moment of inertia of a rigid body about its center of mass reduces to
(7) HC = ICω
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.

CONSERVATION OF ANGULAR MOMENTUM

From the principle of angular impulse and momentum, the change in angular momentum of a rigid body over a period of time is equal to the angular impulse imparted to the body over that time period:
(8) t2t1 Mdt = H2 −  H1
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