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13.4 Torque and Angular Momentum as Vectors 415

Hence (a)
z When you rotate
dL d gyroscope’s axis, angular
L (13.49) momentum changes by d L.
dt dt
According to Eq. (13.49), the magnitude of the torque is
L d L
dL d db d L
t L O
dt dt L y
2
With L 3.0 10 J s and d dt (90 ) (1.0 s) 2 radians/s,
x
p 2
2
t 3.0 10 J s radians/s 4.7 10 N m
2 (b)
z
Since dL/dt, the direction of the torque vector must be the direction of
dL; that is, the torque vector must be perpendicular to L, or initially into the plane Since d L/dt,
is parallel to d L.
of the page (see Fig. 13.22b).To produce such a torque, your left hand must push
up, and your right hand must pull down.This is contrary to intuition, which would
suggest that to twist the axis in the horizontal plane, you should push forward with
O
your right hand and pull back with your left! This surprising behavior also explains r y
F
why a downward gravitational force causes the slow precession of a spinning top,
as considered in the next example. For desired r F,
direction of force must
x be downward!

A toy top spins with angular momentum of magnitude L; the FIGURE 13.22 (a) dL is approximately
EXAMPLE 13
axis of rotation is inclined at an angle with respect to the perpendicular to L, in the x–y plane.
(b) The torque is parallel to dL, also in
vertical (see Fig. 13.23). The spinning top has mass M; its point of contact with
the x–y plane.
the ground remains fixed, and its center of mass is a distance r from the point of
contact.The top precesses; that is, its angular-momentum vector rotates about the
vertical. Find the angular velocity of this precessional motion. If a top has
p
2
r 4.0 cm and moment of inertia I MR 4, where R 3.0 cm, find the period
of the precessional motion when the top is spinning at 250 radians/s.

SOLUTION: From Fig. 13.24a, we see that the weight, Mg, acting at the center L
of mass, produces a torque of magnitude

t rMg sin u (13.50)
CM
As in Example 12, the change in angular momentum dL will be parallel to the
torque, since dL dt. In a time dt, the top will precess though an angle d given
by (see Fig. 13.24b)

dL
d
Lsinu
Using dL dt rMg sin dt, we thus have

rMg sin u dt rMg
d dt FIGURE 13.23 A tilted top spinning
L sin u L
with angular velocity .
The precessional angular velocity is the rate of change of this angle:
d rMg
(13.51)
p
dt L
Thus the angular velocity of precession is independent of the tilt angle .