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13.3 Angular Momentum and Its Conservation 407

To express the equation for rotational motion in terms of angular momentum, we
proceed as we did in the translational case. We note that if the change of angular veloc-
ity is d , then dL Id . Dividing both sides of this relation by dt, we see

dL d
I
dt dt
If we compare this with Eq. (13.18), we see that the right side can be expressed as the
torque, so
dL equation of rotational motion in
t (13.27)
dt terms of angular momentum

This says that the rate of change of angular momentum equals the torque. Obviously, this
equation is analogous to the equation dp dt F for translational motion.
x x
We now see that the analogy between rotational and translational quantities men-
tioned in Section 12.3 can be extended to angular momentum and momentum.
Table 13.2 lists analogous quantities, including the quantities for work, power, and
kinetic energy.
If there is no torque acting on the rotating body, 0 and therefore dL dt 0,
which means that the angular momentum does not change:

L [constant] (when 0) (13.28) conservation of angular momentum

This is the Law of Conservation of Angular Momentum. Since L I , we can also
write this law as

I [constant] (13.29)

TABLE 13.1 SOME ANGULAR MOMENTA

Orbital motion of Earth 2.7 10 40 J s
Rotation of Earth 5.8 10 33 J s TABLE 13.2
4
Helicopter rotor (320 rev/min) 5 10 J s FURTHER ANALOGIES BETWEEN 1D
2
Automobile wheel (90 km/h) 1 10 J s TRANSLATIONAL AND ROTATIONAL
Electric fan 1 J s QUANTITIES
Frisbee 1 10 1 J s
dW F dx S dW t df
Toy gyroscope 1 10 2 J s
P Fv S P t
Phonograph record (33.3 rev/min) 6 10 3 J s 1 2 1 2
K mv S K I
2
2
Compact disc (plating outer track) 2 10 3 J s
ma F S I t
Bullet fired from rifle 2 10 3 J s
p mv S L I
Orbital motion of electron in atom 1.05 10 34 J s
dp dL
Spin of electron 0.53 10 34 J s dt F S dt t