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13.3 Angular Momentum and Its Conservation 409

(a) (b)

Ball of clay.

v v'

FIGURE 13.12 (a) A pottery wheel rotates
Initially, only with angular velocity ; (b) when a ball of clay
pottery wheel Clay and wheel is dropped on the wheel, the angular velocity
is rotating. rotate together.
slows to '.

and the moment of inertia of the clay is that of a uniform sphere (see Table 12.3):

2
2
2
I MR 6.0 kg (0.080 m) 2
clay 5 5
2 2
1.5 10 kg m
Accordingly,
For a particle, all
of mass is a distance
r from axis.
I I
v
œ
I œ I I
clay
2 2
7.5 10 kg m
8.4 radians/s
2
7.5 10 2 kg m 1.5 10 2 kg m 2
r
7.0 radians/s

As already mentioned in Chapter 9, the Law of Conservation of Angular
Momentum also applies to a single particle moving in an orbit under the influence of
FIGURE 13.13 A particle moving with
a central force. Such a force is always directed along the radial line, and it therefore
speed v along a circle of radius r. The moment
exerts no torque. If the particle is moving along a circle of radius r with velocity v (see of inertia of this particle with respect to the
2
2
Fig. 13.13), its moment of inertia is mr and its angular velocity is v r. Hence I center of the circle is I mr .
2
mr v r mvr, and the angular momentum of the particle is
L mvr (circular orbit) (13.34) angular momentum for circular orbit

This formula is valid not only for a circular orbit, but also for the perihelion and
aphelion points of an elliptical orbit, where the instantaneous velocity is perpendi-
cular to the radius. In Chapter 9 we took advantage of the conservation of the angu-
lar momentum L mvr to compare the speeds of a planet at perihelion and at
aphelion.
The angular momentum defined by Eq. (13.34) is called the orbital angular
orbital angular momentum
momentum to distinguish it from spin angular momentum of a body rotating about
and spin angular momentum
its own axis. For instance, the Earth has both an orbital angular momentum (due to its
motion around the Sun) and a spin angular momentum (due to its rotation about its
own axis). Table 13.1 includes examples of both kinds of angular momentum.