A Classical Approach of Newtonian Mechanics

9 ANGULAR MOMENTUM 9.3 Angular momentum of an extended object

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9.3 Angular momentum of an extended object

Consider a rigid object rotating about some fixed axis with angular velocity ω.

Let us model this object as a swarm of N particles. Suppose that the ith particle

has mass mi, position vector ri, and velocity vi. Incidentally, it is assumed that

the object’s axis of rotation passes through the origin of our coordinate system.

The total angular momentum of the object, L, is simply the vector sum of the

angular momenta of the N particles from which it is made up. Hence,

L =

i=1,N

mi ri × vi. (9.10)

Now, for a rigidly rotating object we can write (see Sect. 8.4)

vi = ω × ri. (9.11)

Let

ω = ω k, (9.12)

where k is a unit vector pointing along the object’s axis of rotation (in the sense

given by the right-hand grip rule). It follows that

L = ω

i=1,N

mi ri × (k × ri). (9.13)

Let us calculate the component of L along the object’s rotation axis—i.e., the

component along the k axis. We can write

Lk = L k = ω

i=1,N

mi k · ri × (k × ri). (9.14)

However, since a · b × c = a × b · c, the above expression can be rewritten

Now,

Lk = ω

i=1,N

mi (k ri) (k ri) = ω

i=1,N

mi |k × ri|^2. (9.15)

i

X

=1,N mi^ |k^ ×^ ri|

(^2) = Ik, (9.16)