Angular momentum

Consider a single particle that has linear momentum p and is located at position r relative to an origin O, as in Fig. 12.2. Its angular momentum ℓ is defined as

Single particle ℓ = r × p (6.8)

Angular momentum is measured with respect to a point, the origin of the position vector r. It is often convenient to express the magnitude of ℓ in terms of the "moment arm," , which is the perpendicular distance from the origin to the line of motion of the particle. Thus,

Single particle ℓ = rp sin θ = (6.9)

The SI unit of angular momentum is kg·m²/s.

SYSTEM OF PARTICLES

The total angular momentum L of a system of particles relative to a given origin is the sum of the angular momenta of the particles. From Eq. 6.9,

FIGURE 6.10

(System of particles) L= ∑ ℓ_i =∑ r _i × p _i (6.10)

This sum takes a simple form in the special case of a rigid body rotating about a fixed axis. The origin can be chosen at any point on the axis of rotation, which we take to be the z axis. The angular momentum of the i th particle is , where R_i is the radius of its circular path. Thus, the z component of the total angular momentum of a rigid body rotating about a fixed axis is

L_z= ∑ ℓ_iz =∑

Since the moment of inertia is, we have

(Rigid body, fixed axis) L_z = Iω (6.11)

The subscript z is often dropped. Equation 12.7 is not a vector equation; it repre­sents only the component of the angular momentum along the angular velocity. Note that L = Iω has the same form as p = mv: the product of an inertial property and a speed. Thus, angular momentum is the rotational analog of linear momen­tum.

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