Angular Momentum
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Angular Momentum of a Rigid Body
Remote & Local Terms

Angular Momentum of a Rigid Body

 

    Figure 1

A rigid body is both rotating about its center of mass and translating with respect to the origin of the reference frame, O shown in Figure 1. From [1] of Angular Momentum - System of Particles and Figure 1, the angular momentum of a particle composing the rigid body is

   [1]

where m = the mass of the particle, rCM = the position of the body's CM, vCM = the velocity of the body's CM, r = the position of the particle, r' = the relative position of the particle to the body's CM, v = the velocity of the particle, and v' = the relative velocity of the particle to the body's CM. Since all the particles in the body experience the same angular velocity, [1] can be rewritten to

   [2]

since, from Figure 1,

   [3]

where w = the angular velocity of the body.

The angular momentum of the body is the sum of the angular momentums of the particles composing the body:

   [4]

where M = the mass of the rigid body.

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Remote & Local Terms

From [4] of Inertia Tensor and [4]:

   [5]

where pCM = the momentum of the body, ICM = the inertia tensor of the body about its CM, HCM = the angular momentum of the body due to the motion of the CM, and H' = the angular momentum of the body due to its rotation about the CM. As shown in [5], the angular momentum of a rigid body can be reduced to two distinct terms: the angular momentum due to the translation of the body's CM (HCM), and the angular momentum due to the rotation of the body about its CM (H'). The first term in [5] is called the remote term, while the second is the local term of the body's angular momentum:

   [6]

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Young-Hoo Kwon, 1998-